the results obtained for the quantum compass model on a square lattice. ... Keywords: frustration, magnetization plateaus, supersolid phases, compass model, ...
Exotic Phases of Quantum Frustrated Magnets: Magnetization Plateaus, Nematic Order and Supersolid Phases
THÈSE NO 4204 (2008) PRÉSENTÉE le 27 novembre 2008 À LA FACULTE SCIENCES DE BASE CHAIRE DE THÉORIE DE LA MATIÈRE CONDENSÉE PROGRAMME DOCTORAL EN PHYSIQUE
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES
PAR
Julien Dorier ingénieur physicien diplômé EPF de nationalité suisse et originaire de Arzier (VD)
acceptée sur proposition du jury: Prof. R. Schaller, président du jury Prof. F. Mila, directeur de thèse Dr C. Berthier, rapporteur Prof. D. Ivanov, rapporteur Dr G. Misguich, rapporteur
Suisse 2008
Abstract In this work, we study two examples of frustrated magnetic systems whose degrees of freedom are spins (or pseudo-spins) on two-dimensional lattices. The first part presents the results obtained for the quantum compass model on a square lattice. This model is a minimal model for the orbital degrees of freedom of transition metal compounds. We use exact diagonalizations, quantum Monte Carlo simulations as well as a perturbative approach in order to study the ground state degeneracy in the thermodynamic limit and the type of order in the ground state. The second part of this work is devoted to the study of the magnetic properties of SrCu2 (BO3 )2 , which is one of the few quasi two-dimensional quantum magnets whose magnetization curve as a function of the magnetic field shows plateaus. The magnetic degrees of freedom are described by a spin 1/2 Heisenberg model on the Shastry-Sutherland lattice in the presence of an external magnetic field. In order to study the magnetic properties at zero-temperature, we use perturbative continuous unitary transformations to derive an effective Hamiltonian in which the triplets S z = 1 are particles moving on a background of singlets. This Hamiltonian is characterized by a kinetic energy dominated by correlated hopping, which let a particle hop only if there is another particle nearby. In order to better understand the effect of the correlated hopping, we start by studying a minimal model. By using exact diagonalizations, quantum Monte Carlo and a semi-classical approximation we show that the correlated hopping can stabilize a phase characterized by a condensation of pairs of particles and strongly favor supersolid phases. Next, we determine the zero-temperature phase diagram of the effective model by using a classical approximation. A comparison to previous theoretical works and experimental measurements shows that the Shastry-Sutherland model cannot reproduce the series of magnetization plateaus measured experimentally. This suggest that the residual interactions, which were neglected, could be crucial for SrCu2 (BO3 )2 . Keywords: frustration, magnetization plateaus, supersolid phases, compass model, ShastrySutherland model, SrCu2 (BO3 )2 .
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Résumé Dans ce travail, nous étudions deux exemples de systèmes magnétiques frustrés dont les degrés de liberté sont des spins (ou pseudo-spins) situés sur des réseaux bidimensionnels. La première partie présente les résultats obtenus pour le compass model quantique sur réseau carré qui est un modèle minimal pour la description des degrés de liberté orbitaux dans les composés de métaux de transition. En utilisant des diagonalisations exactes, des simulations Monte Carlo quantiques ainsi qu’une approche perturbative nous déterminons la dégénérescence de l’état fondamental dans la limite thermodynamique ainsi que le type d’ordre dans l’état fondamental. La seconde partie de ce travail est consacrée à l’étude des propriétés magnétiques du SrCu2 (BO3 )2 , un des rare aimants quantiques quasi bidimensionnels dont la courbe d’aimantation en fonction du champ magnétique présente des plateaux. Les degrés de liberté magnétiques sont modélisés par un ensemble de spins 1/2 sur un réseau de ShastrySutherland avec des couplages de type Heisenberg et en présence d’un champ magnétique. Afin de déterminer les propriétés magnétiques à température nulle, nous utilisons des transformations unitaires continues pour dériver un Hamiltonien effectif de manière perturbative qui décrit des triplets S z = 1 comme des particules se déplaçant sur un vide formé de singulets. Cet Hamiltonien est caractérisé par une énergie cinétique des triplets dominée par un terme de saut corrélé. Ce terme permet à une particule de sauter seulement si une autre particule est présente sur un site voisin. Afin de mieux comprendre l’effet du saut corrélé, nous commençons par étudier un modèle minimal à l’aide de diagonalisations exactes, de simulations Monte Carlo quantiques ainsi que d’une approximation semi-classique. Nos résultats montrent que que le saut corrélé peut stabiliser une phase caractérisée par une condensation de paires de particules et favorise fortement les phase supersolides. Le diagramme de phase à température nulle du modèle effectif est ensuite déterminé en utilisant une approximation classique. En comparant nos résultats à de précédents travaux théoriques ainsi qu’aux résultat expérimentaux, il apparait que le modèle de Shastry-Sutherland ne permet pas de reproduire la suite de plateaux mesurée expérimentalement. Ceci suggère que les interactions résiduelles négligées jusqu’ici pourraient être importante pour le SrCu2 (BO3 )2 . Mots-clés: frustration, plateaux d’aimantation, phases supersolides, compass model, iii
modèle de Shastry-Sutherland, SrCu2 (BO3 )2 .
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Acknowledgments First of all I would like to thank Frédéric Mila, who gave me the opportunity to do a doctoral thesis in his group and who guided me during these four years. Special thanks goes to Kai Phillip Schmidt whose contribution was essential for the second part of this thesis. I appreciate his support, our scientific discussions, his help for the correction of this thesis. For the first part of this thesis, I thank Federico Becca who spent a lot of time to explain me the Green’s function Monte Carlo and exact diagonalization methods. I thank Andreas Läuchli for his contribution to the second part of this thesis, his advices and our collaboration on various projects. I also thank Arnaud Ralko for the discussions about the Green’s function Monte Carlo, Andreas Abendschein and Sylvain Capponi for the discussions about the Shastry-Sutherland model and Rachel Bendjama who explained me how to perform the Bogoliubov transformation with several sites per unit cell. During these four years, it was a pleasure to work and also to spend some time outside of work with all the peoples in CTMC and IRRMA groups: Arnaud Ralko, Kai Phillip Schmidt, Ioannis Rousochatzakis, Jean-David Picon, Salvatore Manmana, Tamas Toth, Maged Elhajal, Valery Kotov, Alain Gellé, François Vernay, Thomas Gloor, Cedric Weber, Nicolas Laflorencie, Brijesh Kumar, Rachel Bendjama, Andreas Läuchli, Anreas Lüscher, and Julien Sudan. Of course this work could not have been done without computers. I would like to thank Japhet Bagilishya who helped me whenever I had a problem with computers, Vittoria Rezzonico who managed our cluster and Pascal Jermini who helped me with the computing grid Greedy at EPFL (greedy.epfl.ch) that I used for the calculations on the Shastry-Sutherland model. Finally I want to thank my family and my friends, and in particular my mother and my fiancee Mélanie for their love and support during all these years.
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Contents
1 General Introduction
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Frustrated orbital models
2 Compass model 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Classical compass model . . . . . . . . . . . . . . . . . . . . 2.3 Semi-classical compass model . . . . . . . . . . . . . . . . . 2.4 Quantum spin 1/2 compass model . . . . . . . . . . . . . . . 2.4.1 Low-energy spectrum . . . . . . . . . . . . . . . . . 2.4.2 Ground state degeneracy in the thermodynamic limit 2.4.3 The transition at Jx = Jz . . . . . . . . . . . . . . . 2.5 General spin quantum compass model . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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SrCu2(BO3 )2 , a frustrated magnet
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3 Introduction
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4 Correlated hopping, a minimal model 4.1 Low density limit and paired superfluid phase . . . . . 4.1.1 n − t′ phase diagram . . . . . . . . . . . . . . . 4.1.2 V − t′ phase diagram . . . . . . . . . . . . . . . 4.1.3 Reduced density matrices and pair condensation 4.1.4 Quantum Monte Carlo . . . . . . . . . . . . . . 4.1.5 Summary . . . . . . . . . . . . . . . . . . . . . 4.2 Supersolid phases . . . . . . . . . . . . . . . . . . . . . 4.2.1 Semi-classical approximation . . . . . . . . . . . 4.2.2 Quantum Monte Carlo . . . . . . . . . . . . . .
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4.2.3 Summary . . . . . . . . . . . . 4.3 Generalizations of the model . . . . . . 4.3.1 Negative t′ . . . . . . . . . . . . 4.3.2 Next-nearest neighbor hopping 4.4 Conclusion . . . . . . . . . . . . . . . .
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5 The Shastry-Sutherland model 5.1 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . 5.1.1 Triplet operators . . . . . . . . . . . . . . . . . 5.1.2 Perturbative continuous unitary transformations 5.1.3 Description of the effective model . . . . . . . . 5.2 Zero-temperature phase diagram . . . . . . . . . . . . . 5.2.1 Method . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 A minimal model for the low-density plateaus . 5.2.4 Comparison with other theoretical works . . . . 5.2.5 Comparison with SrCu2 (BO3 )2 experiments . . 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
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6 General conclusion
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Appendix
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A Green’s function Monte Carlo A.1 Single walker GFMC . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Power method . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Stochastic process . . . . . . . . . . . . . . . . . . . . . . . A.1.3 Ground state energy . . . . . . . . . . . . . . . . . . . . . A.1.4 Importance sampling . . . . . . . . . . . . . . . . . . . . . A.2 Many walkers GFMC . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Reconfiguration . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Ground state energy and stochastic average . . . . . . . . A.2.3 Expectation value of diagonal operators (forward walking)
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B Exact diagonalization 139 B.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 B.2 Lanczos approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 C Perturbation theory 145 C.1 Second order perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 145 C.2 Gap in the ground state manifold . . . . . . . . . . . . . . . . . . . . . . . 148 viii
D Classical approximation 149 D.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 D.2 Finite size clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 E Semi-classical approximation E.1 Model . . . . . . . . . . . . . . . E.2 Spins rotation . . . . . . . . . . . E.3 Holstein-Primakoff transformation E.4 Fourier transformation . . . . . . E.5 Bogoliubov transformation . . . .
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F Perturbative continuous unitary transformations
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CHAPTER
1
General Introduction This work is devoted to the study of two-dimensional frustrated magnetic systems and focuses on systems whose degrees of freedom are described by spin operators acting on the sites of a two-dimensional lattice. A frustrated system is a system which cannot simultaneously minimize the energy for each of its entities. This definition can be illustrated by the famous example of three spins on a triangle with antiferromagnetic Heisenberg couplings. Although the energy can be minimized separately on each bond by having the two spins in opposite directions, it is not possible to do it simultaneously for the three bonds of the triangle. Unfrustrated systems are characterized by a ground state with long-range magnetic order. This type of order can be well described within a classical approximation where the quantum spins are replaced by classical vectors. In order to determine the properties of the quantum system, one can simply use a semi-classical approximation in which the quantum spins are described by small quantum fluctuation around the classical spins (spin-wave approximation). This approximation can give really good results for unfrustrated systems like the antiferromagnetic spin 1/2 Heisenberg model on the square lattice [1]. Although some frustrated systems like the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice also have long-range magnetic ordered ground states [2], it is in general not the case. At the classical level, a strong frustration manifest itself by a large or infinite ground state degeneracy and one can hardly extract any relevant information regarding the quantum system. Indeed, one can hardly know whether the quantum ground state will be close to one of the classical ground states, or will correspond to a superposition of all these states, or will be totally different. In order to answer this question, one can try to estimate the effect of the quantum fluctuation with a semi-classical approximation. However, this is not trivial, as there is no longer a natural classical starting point. One should therefore perform the semi-classical approximation around all the classical ground states and compare the energy of the resulting states. This procedure usually results in a lifting of the degeneracy by the mechanism of order by dis-
2
General Introduction
order [3]. However, this is not the end of the story since the large classical degeneracy in frustrated systems usually comes along with diverging quantum fluctuations around the classical states selected by the order by disorder mechanism. These divergences signal that the classical ground state is not a good approximation for the quantum ground state. Because of the large quantum fluctuations which can destroy the classical magnetic order, frustrated systems are an interesting playground for physicists, since one can hope to find new exotic purely quantum phases. The spin 1/2 antiferromagnetic Heisenberg model on the kagome lattice is a good example to illustrate these considerations. This model is highly frustrated because of the lattice topology which consists in corner sharing triangles. The classical ground state manifold is highly degenerate and contains all the states which have zero total spin on each triangle. Quantum fluctuations lift the classical degeneracy by an order by disorder mechanism but within the linear spin wave approximation the quantum fluctuations diverge [4], suggesting that the quantum system cannot be described by a semi-classical approximation. This is confirmed by several studies which suggest a ground state with resonating valence bond order, a state which does not break any symmetry of the system and in particular has no long-range magnetic order [5]. Another example is given by the spin 1/2 J1 − J2 model on the square lattice. The frustration in this system is due to the competing antiferromagnetic Heisenberg interactions between nearest neighbor spins, which favor a Néel order, and between the next nearest neighbor spins which favor a Néel order on each sub-lattice. This competition results in an infinite degeneracy of the classical ground state when J1 = 2J2 where the system is the most frustrated. This degeneracy is lifted by the quantum fluctuations which select the collinear states [6]. However, within a linear spin wave approximation the quantum fluctuations diverge, which strongly suggest a ground state without long range magnetic order for the quantum system. This is confirmed by various approaches which suggest that a valence bond crystal phase is stabilized around the frustrated point J1 = 2J2 [7–10]. A valence bond crystal ground state only breaks the lattice symmetries but not the SU(2) symmetry, and can be seen as a state where pairs of nearest neighbor spins form singlets. In this work, we concentrate on two specific frustrated models. In the first part, we study the compass model, which is a minimal model appearing in the context of Mott insulators with orbital degeneracy. This model describes the orbital degrees of freedom expressed in terms of pseudo-spins 1/2 on a square lattice. Although this lattice is bipartite, the compass model is strongly frustrated because each bond tends to align the pseudo-spins in a given direction, reflecting the underlying geometry of the orbitals. The main motivation is to better understand the zero-temperature properties of Mott insulators with orbital degeneracy, and in particular to determine the type of orbital ordering, if any. As we shall see, the frustration in this model leads to a nematic ground state in which the spins are parallel to a given axis but are still free to be in any of the two directions. The second part concentrate on the frustrated quantum antiferromagnet SrCu2 (BO3 )2 which is usually described by the spin 1/2 antiferromagnetic Heisenberg model on the Shastry-Sutherland lattice. This compound has attracted a lot of interest in the past years, because it is one of the few two-dimensional materials which is a realisation of a va-
3 lence bond crystal. This is related to an interesting properties of the Shastry-Sutherland model: its ground state is exactly given by a product of singlet in the absence of magnetic field. The interest in this material is also due to its properties in the presence of a magnetic field. Indeed, SrCu2 (BO3 )2 was the first example of two-dimensional quantum spin system whose magnetization curve as a function of the applied magnetic field displays a series of plateaus. These plateaus can be understood as phases where triplet excitations crystallize into a super-lattice. Moreover, recent experiments suggest that the translational symmetry is still broken above the low-magnetization plateau, and this phase could be understood as a supersolid of triplets, where a finite fraction of the triplets condense while the other triplets stay localized in a super-lattice. If this could be confirmed, it would correspond to the first observation of a supersolid phase in a frustrated magnet. Frustrated magnetic systems are difficult to study, mainly because of the lack of a general theoretical method. In this work, we will combine several methods, including exact diagonalizations, quantum Monte Carlo simulations, classical and semi-classical approximations as well as strong coupling expansions. In addition, we will derive a high order perturbative effective Hamiltonian for the Shastry-Sutherland model, whose ground state properties will be evaluated within a classical approximation. None of these methods is sufficient in itself to study every possible frustrated magnetic system. When directly used on a frustrated magnetic system, classical and semi-classical approaches usually fail because the frustration tends to destroy the classical long-range order. Exact diagonalizations usually suffer from the small cluster available, which make the size scaling difficult, if not impossible. Quantum Monte Carlo methods usually suffer from the minus sign problem, which is more the rule than the exception in frustrated magnetic systems. Although perturbative effective models can simplify the problem by extracting the relevant degrees of freedom, this approach is also not perfect. Indeed, frustrated systems are usually in a region of the parameter space where the small parameter used for the perturbative expansion is not so small. As a consequence, high order expansions are required, which is not a trivial task. However, by combining all these methods appropriately, we will be able to extract a consistent physical picture of the various models.
Part I Ordering in frustated orbital models of transition metal compounds
CHAPTER
2
Compass model 2.1
Introduction
Orbital degeneracy is a common feature in transition metal (TM) compounds [11]. In these materials, a d sub-shell of the TM ions is only partially filled. When a TM ion with partially filled 3d sub-shell is placed in an octahedral or tetrahedral environment of negative ions, the coulomb interaction between the negative ions and the 3d electrons lift partially the five-fold degeneracy (figure 2.1). The five-fold degenerate 3d orbitals split into a eg level with two-fold degeneracy and a t2g level with three-fold degeneracy. In an octahedral environment the t2g level will have the lowest energy whereas in a tetrahedral environment it will be the eg level (figure 2.2). If both eg and t2g levels are completely or half filled then the electronic state will be unique (Hund’s rule). For any other filling, the orbital degeneracy will be two-fold or three-fold depending whether it is the eg or the t2g level which is not completely or half filled. The description of a crystal with TM ions can be done by using a Hubbard Hamiltonian and taking properly into account the orbital degeneracy. It is possible and useful to express the orbital degrees of freedom in terms of pseudo-spins 1/2 (or 1) τ for a twofold (or a three-fold) orbital degeneracy. For Mott insulators, where the kinetic energy is much smaller than the coulomb interaction, a perturbative approach can be used to build an effective Hamiltonian, the so-called Kugel-Khomskii Hamiltonian [11], for the description of the low-energy physics. Although the orbital part of the Hamiltonian can be expressed in a generalised Heisenberg form, it is generally highly anisotropic and frustrated, reflecting the geometry of the crystal and the orbitals. Apart from the orbital part and spin Heisenberg-like exchange, the Hamiltonian contains also an interaction between spins and orbital pseudo-spins. Thanks to the frustration and the interaction between orbital and spin degrees of freedom, the ordering of the spin and orbital degrees of freedom can be highly non-trivial and interesting. In real materials, the usual scenario is a first transition (Jahn-Teller
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Compass model
Figure 2.1: The 3d levels of the transition metal ion splits in eg levels (orbitals 3z 2 − r 2 and x2 − y 2 ) and t2g levels (orbitals xy, yz and zx). In an octahedral environment of negative ions, the coulomb interaction is stronger for the eg electrons whose orbitals point toward the negative ions.
Figure 2.2: Splitting of the 3d sub-shell. transition) to a phase with orbital long-range order, followed at lower temperature by a transition to a phase with both orbital and magnetic long-range order. Another scenario, which seems to be realised in V2 O3 , is a single transition towards a low temperature phase with both orbital and magnetic order [12]. Finally, one could also expect that some systems stay unordered at any finite-temperature, as it has apparently been observed in LiNiO2 [13]. What makes these models interesting also makes them difficult to study. A first step
2.1 Introduction
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towards the understanding of their properties consist in studying only the orbital part of the Hamiltonian and try to understand the effect of the anisotropy. A minimal model with these features has been proposed by Kugel and Khomskii [11] X X α H = −J τrα τr+e . (2.1) α r
α=x,y,z
By analogy to the dipolar coupling between compass needles, this model is called the compass model. The problem can be further simplified by looking at the two-dimensional model X z x (2.2) H = −J + τrz τr+e τrx τr+e z x r
where the τr are spins 1/2 operators on the square lattice. This model is also relevant in other physical areas. In the context of Mott insulators without orbital degeneracy, frustration can generate extra degrees of freedom which play a similar role to the orbital degrees of freedom. For instance, for the problem of spins 1/2 on the trimerized kagome lattice [14], the low-energy physics can be described by two spins 1/2 on each triangle. In a magnetic field, this model exhibit at 1/3 magnetization plateau [15–18] and the low-energy physics of the plateau can be described by a kind of compass model for one of the two spins 1/2 [19]. In the context of quantum computers, Douçot and collaborators [20] proposed a realization of the anisotropic two-dimensional compass model X z x (2.3) + Jz τrz τr+e H=− Jx τrx τr+e z x r
with an array of Josephson junctions. This system should serve as a qubit, which is the quantum equivalent to a bit in classical computers. A necessary condition for this system to be useful as a qubit is a two-fold degenerate ground state, well protected from the excited states by a large gap. Although, when Douçot and collaborators wrote this paper, it was known that for finite-size systems the compass model had a two-fold degeneracy, what happened in the thermodynamic limit was not clear and it was an important question in this context. Despite its apparent simplicity, the two-dimensional quantum compass model (2.3) is really challenging and not well understood. The complexity of the problem can already be seen in the high degeneracy of the classical model with Jx = Jz , as described by Nussinov and collaborators [21] (see section 2.2). Thermal fluctuations lift partially the degeneracy, leading to a one-dimensional nematic order. Using Monte Carlo simulations, Mishra and collaborators [22], have shown that the system undergo a finite-temperature phase transition of the Ising universality to a disordered phase. For the quantum spin 1/2 compass model (2.3), most of the results before the present work have been obtained by Douçot and collaborators [20]. Using elegants symmetry argument they showed that each eigenstate has to be at least two-fold degenerate. In the limit Jx ≪ Jz or Jx ≫ Jz , they showed that all the states which evolve adiabatically from the ground state of the decoupled Ising chain collapse onto each other in the thermodynamic limit. But this perturbative approach is not valid close to the symmetric case
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Compass model
Jx = Jz which is important in the context of quantum bits. More recently, Nussinov and Fradkin [23] have shown that these models are dual to models of p + ip superconducting arrays, and have discussed general properties of order parameters and phase transitions. In this chapter, we present our results (published in [24]) about the zero temperature properties of the two-dimensional quantum compass model. Section 2.2 gives an overview of the ground state properties of the classical model. Section 2.3 presents a spin-wave calculation which is the first step towards the quantum model. The spin 1/2 compass model is discussed in section 2.4 which starts by a presentation of the symmetries and their consequences. It is followed in 2.4.1 by a presentation of the low-energy spectrum obtained by exact diagonalizations and in 2.4.2 by an analysis of the ground state degeneracy in the thermodynamic limit with Green’s function Monte Carlo simulations. The analysis of the spin 1/2 model is completed in 2.4.3 with a Green’s function Monte Carlo study of the spin-spin correlations in the ground state and a characterization of the phase transition at Jx = Jz . Finally, section 2.5 presents some results on the compass model with general spins, with an emphasis on the spin 1 model.
2.2
Classical compass model
This section reviews what is known about the ground state properties of the classical compass model. It is strongly based on the results of Nussinov and collaborators [21]. The classical compass model is defined by the Hamiltonian X x z z (2.4) H=− Jx Srx Sr+e + J S S z r r+ez x r
where Sr = S cos φr ex + S sin φr ez are classical vectors of norm S = 1/2 lying on a L × L square lattice with periodic boundary conditions. Depending on Jx and Jz , this Hamiltonian can have two types of discrete symmetries in addition to the lattice translational symmetries. The first type, which is present for any Jx and Jz , is given by the transformations Qi which flip the z component of all the spins of the column rx = i and the transformations Pj which flip the x component of all the spins of the row rz = j (figure 2.3). In anticipation of the next sections, it is useful to realise that these operations Qi (respectively Pj ) can be seen as a rotation of all the spins of the column (respectively row) about the ex axis (respectively ez ) by an angle π. The second type, which is a symmetry only when Jx = Jz , is given by the simultaneous rotation Ry (π/2) of all the spins and of the lattice about an ey axis by an angle π/2. Note that this Hamiltonian does not have any continuous symmetry. In order to find a ground state, it is useful to rewrite the Hamiltonian in the following form X 2 1X X α (2.5) H = −Jz L2 S 2 + (Jz − Jx ) S 2 cos2 φr + Jα Srα − Sr+e α 2 r α=x,z r where φr is the angle between the x axis and the direction of the spin at site r. When Jx = Jz , the second term of the Hamiltonian (2.5) vanish, and it is clear that any uniform
2.2 Classical compass model
11
Figure 2.3: The symmetry Qi flips the z component of all the spins of the column rx = i and the symmetry Pj flips the x component of all the spins of the row rz = j.
state with φr = φ is a ground state, regardless of its orientation φ. Starting from a uniform ground state with fixed 0 < φ < π (φ 6= 0, π/2), it is possible to build 22L different ground states by applying the symmetries Qi and Pj . When φ = 0, π/2 it gives only 2L different ground states. Figure 2.4 present an example of a ground state when φ 6= 0, π/2 (a) and when φ = 0 (b). The ground state manifold has a discrete degeneracy of order 22L (or 2L depending on φ) associated to the symmetries of the Hamiltonian, but it also has a continuous degeneracy related to an emergent rotational symmetry of the ground state. Because the continuous degeneracy is not related to any symmetry of the Hamiltonian, it can be lifted by any thermal or quantum fluctuations. And this is indeed what happens: The thermal fluctuations select the 2 · 2L states which are parallel to the direction ex and ez . When Jx > Jz , the last term of the Hamiltonian (2.5) is minimized by any uniform state, while the second term is minimum when φr = 0, which correspond to the uniform state with spins parallel to the x axis. This state is invariant under the symmetries Qi , and applying the symmetries Pj leads to 2L different ground states having all the spins parallel to the x axis, as depicted in figure 2.4 (b). Similarly, when Jx < Jz , one ground state is given by the uniform state with spins parallel to the z axis and the symmetries Qi give 2L different ground states with spins parallel to the z axis. When Jx 6= Jz there is no continuous degeneracy, and the discrete degeneracy is related to symmetries of the Hamiltonian. Thermal fluctuations therefore have no effect on the degeneracy.
12
Compass model
Figure 2.4: Examples of state obtained by applying the symmetries Qi and Pj onto the uniform state with spin in the direction φ 6= 0, π/2 (a) and φ = 0 (b).
2.3
Semi-classical compass model
In the last section, we saw that the classical ground state was highly degenerate and that thermal fluctuations can partially lift the degeneracy by selecting only the ground states with spins parallel to the x and z direction. The next step towards the quantum model consist in studying the quantum fluctuations around the classical ground states, and see if the effect is similar to the thermal fluctuations. This section presents a summary of the results regarding the semi-classical treatment of the compass model. All the details can be found in my diploma thesis [25]. The first approximation consists in using linear spin-wave theory (see appendix E) around a ferromagnetic classical ground state with spins in the direction φ (Sr = S cos φex + S sin φez ). It is sufficient to do the expansion around ferromagnetic states, thanks to the symmetries Qi and Pj which can transform any classical ground state into a ferromagnetic state without changing the form of the Hamiltonian. In the symmetric case Jx = Jz = J, the Hamiltonian can be brought into the form of a quadratic bosonic Hamiltonian X ωk(φ)b†kbk (2.6) H = E0 (φ) + k
where pE0 (φ) is the ground state energy which can be computed numerically and ωk(φ) = 2JS 1 − cos2 φ cos kx − sin2 cos kz . As it can be seen in figure 2.5, E0 (φ) reaches its minimum when φ = 0, π/2, π, 3π/2, which means that the classical ground states with all the spins in the x or z direction are selected by quantum fluctuations. Applying the symmetries Qi and Pj to these states gives 2 · 2L equivalent favored states corresponding to 2L states parallel to ex and 2L states parallel to ez . Within this approximation one can evaluate the correction to the magnetization, which is a measure of the quantum fluctuations around the classical ground state, and check that it is small. As in many
2.4 Quantum spin 1/2 compass model
13
Figure 2.5: Ground state energy of the ferromagnetic states defined by Sr = S cos φex + S sin φez as a function of the angle φ. Ecl = −JNS 2 is the classical ground state energy and N is the number of sites. frustrated magnets like the J1 − J2 model on the square lattice for J2 /J1 = 1/2, this correction to the magnetization diverges when φ = 0, π/2, π, 3π/2, which means that the spin-wave approximation is not fully consistent. In the present case, the divergence comes from a line of zero energy along the kx = 0 direction when φ = 0, π and along the kz = 0 direction when φ = π/2, 3π/2. The next approximation consists in keeping also the four-boson terms in the spin-wave expansion and to do a self-consistent mean-field decoupling. Within this approximation scheme, the divergence of the quantum fluctuations is suppressed and long-range order is preserved even for spins 1/2. The main change for the spectrum is that it becomes gapped, which does not violate any general theorem since the Hamiltonian does not possess any continuous symmetry. In the anisotropic case Jx 6= Jz , the classical ground state manifold has no continuous degeneracy and it contains only the 2L states with spins parallel to the x axis when Jx > Jz or parallel to the z when Jx < Jz . Since these 2L classical ground states are related by symmetries of the Hamiltonian, quantum fluctuations cannot lift this degeneracy.
2.4
Quantum spin 1/2 compass model
This section presents some results on the zero-temperature properties of the quantum spin 1/2 compass model defined by the Hamiltonian H=−
X r
z x + Jz σrz σr+e Jx σrx σr+e z x
(2.7)
14
Compass model
where σrx and σrz are Pauli matrices acting on the spin at site r of a L×L square lattice with periodic boundary conditions. We choose the parametrization of the exchange integral Jx = J cos θ and Jz = J sin θ with θ ∈ [0, π/2], and denote the total number of spins by N = L2 . In addition to the lattice translation symmetries Tx and Tz , Hamiltonian (2.7) has up to two types of discrete symmetries depending on Jx and Jz . The first of symmetries, Q set x which is present for any Jx and Jz , is given by the operators Q = σ , i j i,j which are the Q z x product of the σr on the column rx = i and Pj = i σi,j , which are the product of the σrz on the row rz = j. One can easily see that Qi is a symmetry. Indeed, the operator x of the Hamiltonian (2.7). For the Qi obviously commutes with the first term σrx σr+e x z z second term σr σr+ez , note that two different Pauli matrices anti-commutes if they are on the same site and commutes if they are on different sites. Because the sites r and r + ez are on the same column, if Qi acts on another column then it obviously commutes with z x σrz σr+e , while if Qi act on the same column, it contains two operator σrx and σr+e which z z z z z z both anti-commute with σr σr+ez and Qi commutes with σr σr+ez . The same reasoning can be used to see that the operator Pj is a symmetry. These operators have the same effect on the Pauli matrices than their classical counterpart on the classical spins: the operator Qi corresponds to a rotation of all the spins of the column rx = i by an angle π about the ex axis: x x Q−1 i σi,j Qi = σi,j
(2.8)
y y Q−1 i σi,j Qi = −σi,j
(2.9)
z Q−1 i σi,j Qi
=
z −σi,j
(2.10)
while the operator Pj corresponds to a rotation of all the spins of the row rz = j by an angle π about the ez axis: x x Pj−1 σi,j Pj = −σi,j y Pj−1 σi,j Pj −1 z Pj σi,j Pj
=
=
y −σi,j z σi,j .
(2.11) (2.12) (2.13)
When Jx = Jz , another discrete symmetry is given by the operator Ry (π/2) which corresponds to the rotation of all the spins and the lattice by an angle π/2 about the ey axis. All the Qi operators obviously commute with each other, as well as the Pj , but [Qi , Pj ] 6= 0 ∀i, j. Indeed, all the σrx operators in Qi and the σrz in Pj act on different sites and commute, except the operators acting on the site r = (i, j) which anticommute. Therefore Qi and Pj anti-commute. Clearly, applying twice the same rotation by an angle π should corresponds to the identity, which means Q2i = 1 for all i as well as Pj2 = 1 for all j. This can be also easily checked by using the Pauli matrices property (σrx )2 = (σry )2 = (σrz )2 = 1. A consequence is that each operator Qi has two eigenvalues qi = ±1, the same being true for the operator Pj which has eigenvalues pj = ±1. Regarding the translation Tx by one lattice spacing in the direction x, it commutes with Pj ∀j, but not with Qi which is transformed in Qi+1 under the action of Tx (Tx−1 Qi Tx = Qi+1 ).
2.4 Quantum spin 1/2 compass model
15
Similarly, Tz commutes with Qi and transforms Pj in Pj+1. Finally, the rotation Ry (π/2) does not commute with any other symmetry. A first consequence of these properties is that all eigenstates of the Hamiltonian (2.7) must be at least twofold degenerate. This can be shown by using the following argument given by Douçot and collaborators [20]. We consider two operators Pj and Qi . These operators satisfy [Pj , Qi ]|ψi = 6 0 for all |ψi = 6 0. Indeed, because these operators anticommute ({Pj , Qi }|ψi = 0 for all |ψi in the Hilbert space), we have [Pj , Qi ]|ψi = 2Pj Qi |ψi. On the other hand they satisfy Q2i = 1 and Pj2 = 1 and therefore (Qi Pj )(Pj Qi )|ψi = |ψi which implies Pj Qi |ψi = 6 0 for all |ψi = 6 0. Next, since H and Qi commute, one can work in the basis of the common eigenstates of H and Qi . Starting from any of these eigenstates |ψi, it is possible to build another state Pj |ψi. This state has the same energy because H and Pj commute, and it is not proportional to |ψi. Indeed, if it was proportional to |ψi, it would implies that [Pj , Qi ]|ψi = 0 which we have just shown to be wrong for this system. A second consequence is that the Hamiltonian (2.7) can be diagonalized in a basis whose elements are eigenvectors of the operators {Q1 , Q2 , · · · , QL }. Although, in principle, this set of operators can also contain the translation Tz , it will not be used in practice. The states of this basis can be grouped in 2L different symmetry sectors characterized by the set of eigenvalues (q1 , q2 , · · · , qL ), with qj = ±1. The diagonalization can also be done with eigenvectors of the operators {P1 , P2 , · · · , PL }, leading to 2L different symmetry sectors characterized by the set of eigenvalues (p1 , p2 , · · · , pL ), with pj = ±1. Because the Qi and Pj do not commute, it is not possible to diagonalise the Hamiltonian (2.7) simultaneously in a symmetry sector defined by the set (q1 , q2 , · · · , qL ) and the set (p1 , p2 , · · · , pL ). In order to get a feeling about these properties, it is better to start with the trivial case Jx = 0. In this limit, the system is a setQof decoupled Ising chains, and the eigenstates z of H are given by the 2N states |mi Q = i,j |mi,j i, where |mi,j i is the eigenstate of σi,j with eigenvalue mi,j = ±1 and is the tensor product. The ground state manifold L contains the 2 states having ferromagnetic columns mi,1 = mi,2 = · · · = mi,L = ±1 ∀i = 1, · · · , L. Clearly, all these states are eigenstates of the operators Pj with eigenvalues pj corresponding to the parity of the number of spins down in the row with rz = j. In the ground state, each column has all its spins in the same state, and thus all the rows have to be identical. It implies that the ground state manifold contains 2L−1 states in the sector p1 = p2 = · · · = pL = 1 and 2L−1 states in the sector p1 = p2 = · · · = pL = −1. Although the states |mi are not eigenstates of the Qi operators, it is possible to classify the ground state manifold according Q to the (q1 , q2 , · · ·Q, qL ). In order to do this, one has to introduce the notations | ↑ii = j | ↑ii,j and | ↓ii = j | ↓ii,j for the ground states of the column i with all the spins up and all the spins down respectively. For each column i, the √ two ground states can be replaced by the two linear combinations |+ii = (| ↑ii + | ↓ii )/ 2 √ and |−ii = (| ↑ii − | ↓ii )/ 2 which are eigenstates of Qi with eigenvalue qi = ±1. This can be done for any column, which can be independently in any of the two state |+ii or |−ii , leading to 2L ground states in each of the 2L sectors (q1 , q2 , · · · , qL ), qi = ±1. The case Jz = 0 is connected to the case Jx = 0 by the rotation Ry (π/2). Since Ry (π/2)−1 Qi Ry (π/2) = Pi and Ry (π/2)−1 Pj Ry (π/2) = (−1)L QL−j , the symmetry sectors
16
Compass model
are interchanged. Using the classification according to the operators Pj , the ground state is made up of 2L states in each of the 2L sectors (p1 , p2 , · · · , pL), pi = ±1, while according to the operators Qi , there are 2L−1 states in the sector q1 = q2 = · · · = qL = 1 and 2L−1 states in the sector q1 = q2 = · · · = qL = −1. These results on the symmetry sectors can be conveniently used to determine the structure of the low-energy spectrum. Indeed, in the limit Jx ≪ Jz , each of the 2L low-energy state is in a different symmetry sector (q1 , q2 , · · · , qL ) and corresponds to the non-degenerate ground state of this symmetry sector. In the limit Jx ≫ Jz , the role of the Qi and Pj are exchanged and each of the 2L low-energy state is the ground state of a different symmetry sector (p1 , p2 , · · · , pL ). It is particularly interesting because it allows to get information about the excited states with methods like Green’s function Monte Carlo which can only give properties of the ground state. Another interesting conclusion is the non-degeneracy of the ground state in each symmetry sector, which allows the Green’s function Monte Carlo to give meaningful results about the properties of this state.
2.4.1
Low-energy spectrum
Using exact diagonalization (see appendix B) with the symmetries {Tx , P1 , P2 , · · · , PL }, it is possible to get the complete energy spectrum for lattice sizes up to 4 × 4 and, with a Lanczos approximation, the low-energy spectrum for the 5 × 5 lattice. The resulting low-energy spectrums for the 4 × 4 and 5 × 5 lattices versus the anisotropy parameter θ are given in figure 2.6. In the limit θ = 0 (Jz = 0), the ground state has the predicted 2L -fold degeneracy with each of the state in one of the 2L symmetry sectors (p1 , p2 , · · · , pL ) with pi = ±1. When θ increase, the Jz term of the Hamiltonian lifts partially the degeneracy, leaving only a 2-fold degenerate ground state (line 1 in figure 2.6) with one state in the sector p1 = p2 = · · · = pL = 1 and the other in the sector p2 = · · · = pL = −1. Of all the states coming from the ground state in the limit θ = 0, the states with the highest energy (line 2 in figure 2.6) are in the sectors (p1 , p2 , · · · , pL ) with pi = (−1)i and (p1 , p2 , · · · , pL ) with pi = −(−1)i . In the limit θ = π/2 (Jx = 0), there are only the two symmetry sectors p1 = p2 = · · · = pL = 1 and p1 = p2 = · · · = pL = −1, and each sector contains half of the 2L states. By comparing to the semi-classical results one expects, in the thermodynamic limit, a L 2 -fold degeneracy when Jx 6= Jz and a 2 · 2L -fold degeneracy when Jx = Jz . The former can be achieved if the gap ∆(θ) between the lowest and highest energy states adiabaticaly connected to the ground state manifold of the Jz = 0 limit (lines 1 and 2 in figure 2.6) goes to zero in the thermodynamic limit for each θ 6 π/4 (Jx > Jz ). The behaviour on these small lattices indeed seems to be consistent with it. In that case, it naively leads to a (2 · 2L − 2)-fold degeneracy in the symmetric case Jx = Jz . The reason is that the lowest pair of states is common to the two families of states coming from θ = 0 (Jz = 0) and θ = π/2 (Jx = 0). In order to have a 2 · 2L -fold degeneracy, one needs that two of the higher-energy states approach the ground state when scaling to the thermodynamic limit. A closer look at the spectrum shows that such a pair of states indeed exists, and
2.4 Quantum spin 1/2 compass model
17
Figure 2.6: Energy of the low-lying states versus the anisotropy parameter θ obtained by exact diagonalization on 4x4 (left) and 5x5 (right) lattices (Jx = J cos θ and Jz = J sin θ). Due to the Lanczos approximation, some states are missing in the upper part of the 5x5 lattice plot. they are given by the first two-fold degenerate excited state in the symmetry sectors p1 = p2 = · · · = pL = 1 and p1 = p2 = · · · = pL = −1 (line 3 in figure 2.6). The energy gap between these states and the ground state, denoted by ∆2 (θ) is presented in figure 2.7 as a function of 1/N for the symmetric case θ = π/4 (Jx = Jz ). These results are indeed consistent with a vanishing of this gap in the thermodynamic limit. Because these two excited states come from a highly excited state in the limit θ = 0 and are not ground states of any of the symmetry sectors, it is difficult if not impossible to use other methods than exact diagonalization to evaluate ∆2 (θ) on larger lattice sizes.
2.4.2
Ground state degeneracy in the thermodynamic limit
Perturbation theory A perturbative approach is useful to get information about the lifting of the ground state degeneracy in the limit Jx ≫ Jz , thus giving access to the gap ∆(θ). To do this, the Hamiltonian is written as H = H0 + V (2.14) P P x z with H0 = −Jx r σrx σr+e and V = −Jz r σrz σr+e the small perturbation. Let {|ν, ki : x z k = 0, 1, · · · } be a basis of the ν-th eigenspace of H0 . A simple choice is the basis of the σrx eigenstates, which are made√of tensor product on all the sites √ r of one of the two states | →ir = (| ↑ir + | ↓ir )/ 2 and | ←ir = (| ↑ir − | ↓ir )/ 2. The ground state manifold {|0, ki : k = 0, 1, · · · } is given by the 2L state having all the rows ferromagnetic,
18
Compass model
Figure 2.7: Scaling of the gap ∆2 (θ) between the ground state and the first two-fold degenerate excited state in the symmetry sectors p1 = p2 = · · · = pL = 1 and p1 = p2 = · · · = pL = −1. Exact diagonalization results for the case θ = π/4 (Jx = Jz ). The scaling is consistent with a vanishing of this gap in the thermodynamic limit. with either | →ir or | ←ir on all the sites of the row. The ground state energy is then Eν = −Jx N. The excited states are obtained from the ground state by flipping spins. The energy cost is 2Jx for each interface between region of spins | →i and spins | ←i along a row. Since there is always an even number of interfaces, the energy spectrum is given by Eν = −Jx N + 4Jx ν with ν = 0, 1, · · · , N/2 when L is even and ν = 0, 1, · · · , (N − L)/2 when L is odd. The upper bound for ν comes from the fact that the maximum number of interfaces is realised by the state with each rows having alternating spins and is given by L2 when L is even and L(L − 1) when L is odd. Under the effect of the perturbation V , the degeneracy of the H0 ground state should be lifted and the ground state manifold should be described by an effective Hamiltonian whose specific form at high order is complicated and of no interest here (see appendix C). Because the only information needed is the gap ∆(θ) between the states with lowest and highest eigenvalues of this effective Hamiltonian, only the terms which actually lift this degeneracy are of interest. In the {|0, ki} basis, all the diagonal matrix elements of the effective Hamiltonian are equal (see later), so they do not contribute to the gap and can be neglected. The only terms of the effective Hamiltonian which can lift the degeneracy are those which have at least one non-zero matrix elements between two different states of the set {|0, ki}. Since each term of V flips a pair of spins on neighboring rows (see figure 2.8), for a L×L lattice one needs to apply the perturbation at least L times before flipping all the spins of the two neighboring rows and reach another ground state. That is, only the terms of the effective Hamiltonian containing a product of at least L perturbation V have to be considered. Thanks to this, the problem is slightly easier and amounts to
2.4 Quantum spin 1/2 compass model
19
z z Figure 2.8: The term σi,j σi,j+1 of the perturbation V flips two spins on neighboring rows j and j + 1.
finding the gap between the lowest and highest eigenvalues of the effective Hamiltonian (L)
Heff =
X
ν1 >0
···
X
νL−1
P0 V Pν1 V · · · V PνL−1 V P0 . (E0 − Eν1 ) · · · (E0 − EνL−1 ) >0
(2.15)
Here Pν is the projector on the ν-th eigenspace of H0 and Eν = −Jx N + 4Jx ν is the energy of this subspace. Let us start by evaluating the matrix elements
=
h0, k|V Pν1 V · · · V PνL−1 V |0, k ′ i X X (−Jz )L ···
iL ,jL i1 ,j1 z z ×h0, k|σi1 ,j1 σi1 ,j1 +1 Pν1 σiz2 ,j2 σiz2 ,j2+1
· · · PνL−1 σizL ,jL σizL ,jL+1 |0, k ′ i.
(2.16)
Starting from a ground state |0, k ′ i, which has ferromagnetic rows, one has to apply L z z different operators σi,j σi,j+1 which flip a pair of spins on two neighboring rows as depicted in figure 2.8. The resulting state |0, ki has to be in the ground state manifold. Since z z the number of operators σi,j σi,j+1 matches the number of sites in a row, in order to get z z a resulting state different from the initial state (k 6= k ′ ), the operators σi,j σi,j+1 have to flip all the spin in two neighboring rows. It corresponds to j1 = j2 = · · · = jL = j and i1 = p(1), i2 = p(2), · · · , iL = p(L) with p ∈ SL and SL the set of the L! permutations. z z z z Because in 6= im ∀n, m = 1, · · · , L, the state σp(m),j σp(m),j+1 · · · σp(L),j σp(L),j+1 |0, k ′ i, with 1 < m < L, must be in one of the excited eigenspace, say the νm (p)-th eigenspace. Then
20
Compass model
the matrix element can be written
=
h0, k|V Pν1 V · · · V PνL−1 V |0, k ′i XX (−Jz )L p∈SL
j
z z z z z z σp(1),j+1 σp(2),j σp(2),j+1 · · · σp(L),j σp(L),j+1 |0, k ′ i ×h0, k|σp(1),j
Y
δνm ,νm (p) . (2.17)
m
Since the operators σ z act on different sites, they commute and can be rearranged
=
h0, k|V Pν1 V · · · V PνL−1 V |0, k ′ i XX (−Jx )L p∈SL
j
z z z z z z ×h0, k|σ1,j σ1,j+1 σ2,j σ2,j+1 · · · σL,j σL,j+1 |0, k ′i
Y
δνm ,νm (p) .
(2.18)
m
By using this result, and realizing that E0 −Eν = −4Jx ν, the off-diagonal matrix elements of the effective Hamiltonian become X 1 JzL L−1 (4Jx ) ν1 (p)ν2 (p) · · · νL−1 (p) p∈SL X z z z z z z ×h0, k| σ1,j σ1,j+1 σ2,j σ2,j+1 · · · σL,j σL,j+1 |0, k ′ i.
(L)
h0, k|Heff |0, k ′i = −
(2.19)
j
Regarding the diagonal matrix elements k = k ′ , the only way to go back to the same state |0, ki after having done L flips of spin pairs on a row is to have an even number of flips on each site, with the restriction that it should not give an intermediate state in the ground state manifold. But there is no need to get the exact results. It is sufficient to realise that the result does not depend on the actual state |0, ki. So the diagonal matrix elements are all equal and do not contribute to the gap. In the following, they will simply be set to zero. As announced before, this argument is still valid if there are m + 1 < L operators V in the matrix element h0, k|V Pν1 V · · · V Pνm V |0, ki. In that case, all the diagonal matrix element are again equal, and do not contribute to the gap. Note that even if expression (2.19) was derived only for off-diagonal matrix elements k 6= k ′ , it can also be used for the diagonal matrix elements k = k ′ , as it simply gives zero in that case. The operator part of this expression can be seen as the Hamiltonian of a one-dimensional z z z Ising model for the pseudo spins 1/2 Sjz = σ1,j σ2,j · · · σL,j /2 X j
z z z z z z σ1,j σ1,j+1 σ2,j σ2,j+1 · · · σL,j σL,j+1 =4
X
z Sjz Sj+1 .
(2.20)
j
Similarly to what has been done in the begining of section 2.4, the ferromagnetic states
2.4 Quantum spin 1/2 compass model
21
Figure 2.9: Log linear plot of LP (L) versus the linear lattice size L. The squares are the numerical results and the line is an exponential fit. of the row j are denoted by | →ij = | ←ij =
L Y
i=1 L Y i=1
| →ii,j
(2.21)
| ←ii,j .
(2.22)
√ On each row, these two states can be replaced by the two states |±ij = (| →ij ±| ←ij )/ 2, which are eigenstates of Sjz with eigenvalues ±1/2. Now a basis of the ground state manifold is given by the set of states obtained by taking the 2L combinations of |+ij and |−ij on each row. In this basis, whose elements are eigenvectors of the operators Sjz with eigenvalues ±1/2, the Ising Hamiltonian is diagonal and its eigenvalues are easy to obtain. In particular, the lowest and highest eigenvalues are given by −L and L if L is even and −L + 2 and L if L is odd, which gives a gap 2L if L is even and 2(L − 1) if L is odd. With the latter results and the expression (2.19), the gap ∆ between lowest and highest eigenvalues of the effective Hamiltonian is given by L Jz 8LP (L) 4J if L is even ∆ x L (2.23) = Jx 8(L − 1)P (L) Jz if L is odd 4Jx with
P (L) =
X
p∈SL
1 . ν1 (p)ν2 (p) · · · νL−1 (p)
(2.24)
22
Compass model
Figure 2.10: Ground state energy E0 obtained with GFMC versus the number of sites N for various value of the anisotropy parameter θ. Now the only problem is to evaluate the sum P (L) which is done numerically. A fit to the numerical results (see figure 2.9) gives LP (L) ≃ exp(0.754L − 0.694), which leads to L Jz 3.997 0.531 if L is even ∆ Jx = (2.25) Jx 3.997 1 − 1 0.531 Jz L if L is odd. L
Jx
It has the same (Jz /Jx )L behavior predicted by Douçot and collaborators [20] for the two rows system. So, when Jz /Jx ≪ 1, this approximation predicts ∆ → 0 in the thermodynamic limit. Moreover, because 0.531Jz /Jx < 1 even for Jx = Jz , it seems likely that this scaling will remain true up to the symmetric limit Jx = Jz . However, the symmetric limit is far from the region of validity of this approximation and other approaches are needed. Green’s function Monte Carlo Green’s function Monte Carlo (GFMC) is a stochastic approach which allows to get information on the ground state properties (see appendix A), and in particular to get the ground state energy in a given symmetry sector (p1 , p2 , · · · , pL). Recalling the results of section 2.4.1, it gives access to the energies of all the 2L states coming from the ground state in the limit θ = 0 (see figure 2.6) and in particular to the energy gap ∆(θ) between the states with lowest and highest energy. The advantage of this method over the exact diagonalization method is that it allows to study larger systems. For this model, we treated up to 16×16 site lattices. It is however not clear if these sizes are sufficient to describe properties in the thermodynamic limit. In
2.4 Quantum spin 1/2 compass model
23
Figure 2.11: Gap ∆ versus the linear lattice size L for various values of the anisotropy parameter θ. For L = 2, 3, 4, 5 the results were obtained by exact diagonalization while for L > 5 the results were obtained with GFMC.
order to get a feeling about the scaling, the ground state energy per site is calculated first as a function of the total number of sites. The results, for various values of θ, are shown in figure 2.10. For small systems, the ground state energy is strongly size dependent up to a certain size (8 × 8 for Jx = Jz ) and becomes almost size independent for larger sizes. It means that strong finite-size effects are expected, especially close to the symmetric limit Jx = Jz , justifying the use of quantum Monte Carlo to get information on large clusters. For θ > π/4 ( Jx < Jz ), the convergence becomes really slow and GFMC cannot give reliable results. It is probably because of the guiding function which is no more a good approximation for the ground state. By evaluating the ground state energies in the symmetry sectors (p1 , p2 , · · · , pL ) with pi = 1 and (p1 , p2 , · · · , pL ) with pi = (−1)i , GFMC allows a direct calculation of the gap ∆(θ). Figure 2.11 shows a log-linear plot of ∆(θ) versus the linear lattice size L for various values of θ. These results seem to be consistent with the power law ∆/Jx ∝ αL with α ≃ 0.531Jz /Jx predicted by perturbation theory. In order to check these predictions, the GFMC results were fitted by a power law, keeping only the sizes beyond which the scaling is approximately linear. The resulting values of α are plotted in figure 2.12 as a function of Jz /Jx , as well as a linear fit to these values. The linear fit gives α ≃ 0.55Jz /Jx , which is in good agreement with the perturbation theory prediction α ≃ 0.531Jz /Jx . Moreover, the limit Jz /Jx = 1 is also in agreement with it, which seems to justify the perturbative approach in this limit. For small Jz /Jx , the prefactor and the even odd effect predicted by perturbation theory agree with the results of figure 2.11.
24
Compass model
Figure 2.12: α versus Jz /Jx . α is the gap scaling constant ∆ ∼ αL. The symbols are the results obtained by Green’s function Monte Carlo and the straight line is a fit to these values. Summary Perturbation theory, as well as GFMC predict a power law scaling of the gap ∆/Jx ∝ αL with α < 1 when Jx > Jz (0 6 θ 6 π/4), implying that the gap vanishes in the thermodynamic limit. As a consequence, all the 2L states coming from the ground state manifold in the Jz = 0 (θ = 0) limit (see figure 2.6) will collapse exponentially fast onto each other, leading to a 2L fold degenerate ground state manifold for all Jx > Jz . Using the rotation Ry (π/2), one can map these results to the case Jx 6 Jz (π/4 6 θ 6 π/2), which leads to a 2L fold degenerate ground state for all Jx < Jz . As discussed in section 2.4.1, in the symmetric limit Jx = Jz the collapse of all the states coming from the ground state manifolds in the limits Jz = 0 (θ = 0) and Jx = 0 (θ = π/2) does not lead to a 2 · 2L fold degeneracy of the ground state but only to a 2 · 2L − 2 fold degeneracy because of one pair of state which is shared by both limits (see figure 2.6). As argued in section 2.4.1, the ground state manifold in the symmetric limit is probably 2·2L fold degenerate, thanks to a pair of higher energy state collapsing onto the ground state int the thermodynamic limit. Because this state is not a ground state of any sector of symmetry nor a first excited state state, GFMC is unfortunately not able to give any results about its energy and cannot confirm this possibility.
2.4.3
The transition at Jx = Jz
A crucial question in the context of orbital models, as well as in Mott insulators, is to know if the system exhibit long-range order and if it is the case, what type of order is realized. In
2.4 Quantum spin 1/2 compass model
25
Figure 2.13: Long-range correlations in the ground state along the z direction C z (∞) (see text) as a function of the anisotropy parameter θ for various lattices sizes. the limit Jx ≫ Jz (θ ≃ 0), the system consists of weakly coupled chains in the x direction. In the classical picture, one expects a one-dimensional nematic order with independent ferromagnetic rows having all the spins pointing in the +x or −x direction. Similarly, in the limit Jx ≪ Jz (θ ≃ π/2), a one-dimensional nematic order in the z direction is expected. When increasing θ from 0 to π/2, the system has to go from a one-dimensional order in x direction to a one-dimensional order in z direction and one expects a transition around the symmetric limit Jx = Jz (θ = π/4). Different scenarios are possible regarding the nature of this transition. A first possibility is a smooth crossover between the two limits without any critical behavior. Another possibility is that the system undergoes a true transition at Jx = Jz which could be either second order or first order. A useful quantity to clarify this important issue is the static correlation function z C (r) = hσrz0 σrz0 +r i, which is the ground state expectation value of the correlation between the z components of the spins at distance r. The long distance behaviour of C z (r) will give information about a possible long-range ordered of the ground state. These correlations can be easily evaluated with exact diagonalization for lattice sizes up to 5 × 5 and, this operator being diagonal in the Ising basis, also with GFMC. Because the GFMC calculation require a lot of computational power, it was not possible, to study systems larger than 7 × 7. It turns out that C z (r) is non-zero only in the z direction and, therefore, the only relevant quantity is C z (r) = C z (rez ). In order to investigate the long distance behavior, C z (r) is fitted to an exponential form C z (r) = a exp(−br) + C z (∞). In principle, one should evaluate C z (∞) for large enough system size and extract its value in the thermodynamic limit. If it is non-zero, it means that the system exhibit long-range order. 7 × 7
26
Compass model
Figure 2.14: Long-range correlations in the ground state along the z direction C z (∞) (see text) as a function of the linear lattice size L for various values of the anisotropy parameter. lattices are probably too small to extract exact thermodynamic limit properties, but it nevertheless gives an estimate. Figure 2.13 presents C z (∞) as a function of the anisotropy parameter θ for various lattice sizes, while figure 2.14 presents C z (∞) as a function of the linear lattice size L for various values of the anisotropy parameter θ, which is more appropriate for a scaling analysis. As in section 2.4.2, when θ > π/4 GFMC cannot give reliable results and only exact diagonalization results are plotted. When 0 < θ < π/4 (Jx > Jz ), C z (∞) strongly decreases as the lattice size increases, which is consistent with a vanishing value in the thermodynamic limit. When θ = π/4 (Jx = Jz ), the thermodynamic limit can be extracted by supposing that the scaling is linear. The resulting value is C z (∞) ≃ 0.34. When π/4 < θ < π/2 (Jx < Jz ), although there are only exact diagonalization results the results is clearly consistent with a finite value of C z (∞) in the thermodynamic limit, indicating the presence of a spin order in the z direction. Thanks to the Ry (π/2) rotation, the correlations between the x component of the spins along the x direction C x (r) = hσrx0 σrx0 +rex i are related to C z (r) by a mirror symmetry around θ = π/4. Therefore, in the thermodynamic limit, C x (∞) is expected to have a finite value for all 0 < θ 6 π/4 (Jx > Jz ) and to vanish for all π/4 < θ < π/2 (Jx < Jz ). Figure 2.15 presents a sketch of the expected behavior of C x (∞) and C z (∞) in the thermodynamic limit. Although these results cannot give a definite answer regarding the phase transition, they give a strong indication in favor of a first order phase transition between a Jx > Jz phase with a spin order in the x direction and a Jx < Jz phase with a spin order in the y direction. At the transition point Jx = Jz , both types of order seem to coexist.
2.4 Quantum spin 1/2 compass model
27
Figure 2.15: Sketch of the expected behavior of C x (∞) (dashed line) and C z (∞) (plain line) in the thermodynamic limit. After the publication of these results, the first-order nature of this transition has been confirmed by Chen and collaborators [26]. This was done by first mapping the spins 1/2 onto fermions with a Jordan-Wigner transformation: Y + Si,j = (2c†i′ ,j ′ ci′ ,j ′ − 1) c†i,j (2.26) (i′ ,j ′ )∈K(i,j)
z Si,j = c†i,j ci,j −
1 2
(2.27)
where c†i,j and ci,j respectively create and annihilate a fermion at site (i, j) and K(i, j) = {(i′ , j ′ )|j ′ < j} ∪ {(i′ , j ′ )|j ′ = j and i′ < i}. The Jordan-Wigner transformation is particularly interesting for this model because the non-local gauge interactions between the fermions which usually appear are absent. This is a consequence of the special structure of Hamiltonian (2.7) which does not depend on the y-components of the spins and couples the x-components of the spins only along the x direction. Note that they considered Hamiltonian (2.7) with spins 1/2 operators Si,j instead of the Pauli matrices σi,j . The fermionic Hamiltonian is therefore simply given by X Jx † † H=− Jz ni,j ni,j+1 − Jz ni,j + ((ci,j − ci,j )(ci+1,j + ci+1,j )) . (2.28) 4 i,j In order to determine the zero-temperature phase diagram of this Hamiltonian for Jx < Jz , they replaced the density-density interactions by the mean-field decoupling ni,j ni,j+1 = hni,j ini,j+1 + ni,j hni,j+1 i − hni,j ihni,j+1i and solved the problem self-consistently. This
28
Compass model
approximation was motivated by the fact that in the limit Jx = 0, the system consists of decoupled chains along the z direction and the ground state manifold is given by the 2L states having completely filled or completely empty columns. When 0 < Jx < Jz and by considering our results, they expected the ground state to be still characterized by almost filled or almost empty uniform columns and therefore the mean-field approximation to be valid as long as Jx < Jz . They solved the self-consistent equations by supposing a uniform z density hni,j i and obtained a uniform non-zero hSi,j i when 0.7446 · Jx < Jz as well as a x uniform hSi,j i = 0 when Jx < Jz . Because this approximation is only valid valid for x z i = 0 and hSi,j i > 0 for all Jx < Jz . The other case Jx < Jz , this results means that hSi,j Jx > Jz can be obtained by exchanging the x and z indices everywhere in the previous x z development, which gives hSi,j i > 0 and hSi,j i = 0 for all Jx > Jz . When combined, these results clearly show that the transition is at Jx = Jz is first order. In addition to this calculation and in order to check the validity of the mean-field approximation, they also treated the terms ignored in the mean-field Hamiltonian with second order perturbation theory and concluded that the mean-field approximation is robust again such corrections.
2.5
General spin quantum compass model
The symmetry arguments used for spins 1/2 can be easily extended to arbitrary spins. Consider a system of N spins S on a L × L lattice described by the compass model Hamiltonian X x z z . (2.29) H=− Jx Srx Sr+e + J S S z r r+e x z r
The generalizations of the Qi and Pj symmetries are given by Ql =
Y
x
ie−iπSl,j
(2.30)
j
Pj =
Y
z
ie−iπSl,j
(2.31)
l
for half-integer spins and by Ql =
Y
x
e−iπSl,j
(2.32)
j
Pj =
Y
z
e−iπSl,j
(2.33)
l
for integer spins. The action of Qi on the operators Si,j is similar to a rotation of all the spins of the column rx = i by an angle π about the ex axis: x x Q−1 i Si,j Qi = Si,j
(2.34)
y Q−1 i Si,j Qi z Q−1 i Si,j Qi
(2.35)
= =
y −Si,j z −Si,j
(2.36)
2.5 General spin quantum compass model
29
while Pj is similar to a rotation of all the spins of the row rz = j by an angle π about the ez axis: x x Pj−1 Si,j Pj = −Si,j y Pj−1 Si,j Pj z Pj−1 Si,j Pj
=
=
(2.37)
y −Si,j z Si,j .
(2.38) (2.39)
Although for integer spins the operators Qi and Pj are the usual rotation operators, it is not the case for half-integer spins because of the factor i in front of each exponential. Using these properties, it is straightforward to check that these operators commute with the Hamiltonian. As for spins 1/2, when S is half-integer the operators Qi and Pj anti-commute {Qi , Pj } = 6 0. However, when S is integer Qi and Pj commute [Qi , Pj ] = 0. Indeed, the only terms in the product which do not trivially commute are those on the site (i, j), so one just have to x z x z show that {e−iπSi,j , e−iπSi,j } = 0 for half-integer spins and [e−iπSi,j , e−iπSi,j ] = 0 for integer z spins. Using the fact that e−iπSi,j = Rz (π) is a rotation by an angle π about the ez axis x and applying it to e−iπSi,j gives x
Rz−1 (π)e−iπSi,j Rz =
X (−iπ)n n!
n
=
X (−iπ)n n x iπSi,j
= e x
x
n!
x n Rz−1 (π)(Si,j ) Rz
x n Rz−1 (π)(−Si,j ) Rz
.
(2.40) x
x
But for half-integer spins eiπSi,j = −e−iπSi,j , and for integer spins eiπSi,j = e−iπSi,j , which z finalizes the proof. By applying the Pj operator on the basis of the Si,j eigenvectors and x the Qi operator on the basis of the Si,j eigenvectors, it turns out that, as for the spin 1/2 system, these operators have two eigenvalues pi = ±1 and qi = ±1 for both integer and half-integer spins. One important consequence of these properties is that for half-integer spins, each eigenstate should have at least a two fold degeneracy while for integer spins the eigenstates are generically non-degenerate. Indeed, the argument used by Douçot and collaborators [20] to show that each eigenstate is at least two fold degenerate rely on the fact that [Qi , Pj ] 6= 0, and it can be directly applied for half-integer spins but not for integer spins. Another important consequence for integer spins is that all Qi commute with all Pj , therefore all these symmetries can be used simultaneously, leading to 22L symmetry sectors defined by the set of eigenvalues (p1 , p2 , · · · , pL , q1 , q2 , · · · , qL ) with qi = ±1 and pi = ±1. In order to go further, one has to restrict to a specific spin value. An example of half-integer spins having already been studied in detail in the previous sections, the next step consist in studying a spin one system. In the limit Jx = 0 the system is made of decoupled Ising chains. A basis of the Hilbert space is given by the 3N states |mi Q Q= z is i,j |mi,j i, where |mi,j i is the eigenstate of Si,j with eigenvalue mi,j = 0, ±1 and the tensor product. In this basis, it is easy to find the ground state manifold, which
30
Compass model
is given by the 2L states with ferromagnetic columns having all the spins in the fully polarized states (mi,1Q = mi,2 = · · · = mQ i,L = ±1 ∀i). For each column i, the two ground states |1ii = j |1ii,j and | − 1ii = j | − 1ii,j can be replaced by the two linear √ √ combinations |+ii = (|1ii + | − 1ii)/ 2 and |−ii = (|1ii − | − 1ii)/ 2. This can be done for any column, which can be independently in any of the the two state |+ii or |−ii , leading to 2L ground states. The main reason for working with this new set of ground states is that they are eigenstates of all the operators Qi and Pj . Indeed, for z the operator Pj , applying e−iπSi,j on any of the two ground states of the column i gives z e−iπSi,j |±ii = −|±ii . Since the lattice has L columns, all the 2L states are eigenstates of Pj with eigenvalue pj = (−1)L . In order to check that the states |±ii are eigenstates x of the operators Qi , the first step consists in expanding the exponential in e−iπSi,j and x x to realize that all odd powers of Si,j are equal to Si,j and all non-zero even power are x −iπSi,j x 2 x 2 equal to (Si,j ) . Rearranging the terms gives e = 1 − 2(Si,j ) . Finally, by using 1 x 2 x 2 x 2 (Si,j ) |1ii,j = 2 (|1ii,j + | − 1ii,j ), (Si,j ) |0ii,j = |0ii,j and (Si,j ) | − 1ii,j = 12 (|1ii,j + | − 1ii,j ), one can show that Qi |±ii = ±(−1)L |±ii . Therefore all the 2L states are eigenstates of Qi with eigenvalue qi = (−1)L if column i is in a state |+ii and qi = −(−1)L if column i is in a state |−ii . In conclusion, the ground state manifold contains 2L states with each state in one of the 2L sectors (p1 , p2 , · · · , pL , q1 , q2 , · · · , qL ) with pi = (−1)L and qi = ±1. As for the spins 1/2 system, the limit Jz = 0 is connected to the limit Jx = 0 by the rotation Ry (π/2). Since Ry (π/2)−1 Qi Ry (π/2) = Pi and Ry (π/2)−1 Pj Ry (π/2) = (−1)L QL−j , the symmetry sectors are interchanged. Therefore the ground state manifold in the limit Jz = 0 consists in 2L states with each state in one of the 2L sectors (p1 , p2 , · · · , pL , q1 , q2 , · · · , qL ) with pi = ±1 and qi = (−1)L . In order to get an idea of what happens when Jx , Jz 6= 0, exact diagonalization of clusters of sizes L = 2 and L = 3 has been performed. The resulting spectrum versus the anisotropy parameter θ (defined by Jx = J cos θ and Jz = J sin θ) is presented in figure 2.16. As expected for integer spins, the ground state is non-degenerate for all Jx , Jz 6= 0 and it is in the only symmetry sector p1 = · · · = pL = q1 = · · · = qL = (−1)L shared by both limits Jx = 0 and Jz = 0. The 2L fold degeneracy of the Jz = 0 ground state is partially lifted by the Jx term of the Hamiltonian, creating a gap ∆(θ) between the ground state (line 1 in figure 2.16) and the state with highest energy (line 2 in figure 2.16) which is in the symmetry sector p1 = · · · = pL = −(−1)L and q1 = · · · = qL = (−1)L . As in the spin 1/2 model, the gap between the ground state and the first excited state which has the same degeneracy and is in the same symmetry sector (line 3 in figure 2.16) is denoted by ∆2 (θ). The two gaps ∆(θ) (with θ 6 π/4) and ∆2 (π/4) are smaller for L = 3 than for L = 2, and one can conjecture that these gaps go to zero in the thermodynamic limit. In this case, the ground state would have the same degeneracy as for the spin 1/2 system. A definite conclusion would require a study of larger clusters though.
2.6
Conclusion
In this chapter, the quantum spin 1/2 compass model has been analysed with a variety of approaches, leading to a coherent picture of the zero-temperature properties. Symmetry
2.6 Conclusion
31
Figure 2.16: Energy of the low-lying states versus the anisotropy parameter θ obtained by exact diagonalization with spins 1 on 2x2 (left) and 3x3 (right) lattices (Jx = J cos θ and Jz = J sin θ). arguments as well as exact diagonalizations have been used to understand the low-energy spectrum for finite-size systems. It has been shown that it consists of 2L states coming from the ground state manifold in the limits Jx = 0 and Jz = 0. The two fold degeneracy of the ground state predicted by Douçot and collaborators [20] has been confirmed. Thanks to Green’s function Monte Carlo simulations, the thermodynamic limit properties of the model could also be tackled. These simulations have shown that the 2L low-energy states collapse onto each other upon increasing the system size, leading to a 2L-fold degeneracy of the ground state when Jx 6= Jz and a 2 · 2L -fold degeneracy when Jx = Jz , in agreement with the semi-classical predictions. Although the collapse of the states has already been shown in the limits Jx ≪ Jz and Jx ≫ Jz , our results have ruled out the possibility of a two-fold degenerate ground state around Jx = Jz . Exact diagonalizations as well as Green’s function Monte Carlo simulations have shown that when Jx 6= Jz the system realizes a one-dimensional order, with long-range correlations only in one spatial direction x (when Jx > Jz ) or z (when Jx < Jz ). When Jx = Jz , although the results suggest the absence of simple long-range order, the system is expected to develop long-range correlations in both x and z directions. It is not possible to give definite conclusions about the nature of the phase transition because of the small system sizes studied, but the results indicate a first order phase transition at Jx = Jz . This has been confirmed by Chen and collaborators [26]. The symmetry arguments by Douçot and collaborators [20] have been extended to general spins. For half-integer spins it predicts at least a two-fold degeneracy of the eigenstates, while for integer spins it predicts generically non-degenerate eigenstates. Exact diagonalization of the spin 1 compass model confirmed the non-degeneracy of the
32
Compass model
ground state for finite-size system. In the thermodynamic limit, the results are consistent with a collapse of the 2L low-energy states, leading to a possible 2L -fold degeneracy when Jx 6= Jz and a 2 · 2L -fold degeneracy when Jx = Jz as in the spin 1/2 system. Physically, this has two consequences. Regarding Mott insulators and orbital ordering in transition metal compounds, where the symmetric version Jx = Jz seems more appropriate, the present results suggest the absence of simple long-range order in the thermodynamic limit at zero temperature. Nevertheless, the system is expected to develop long-range correlations in one spatial direction, and, as in the case of thermal fluctuations for classical spins, the possibility to choose between the x and z directions should still lead to a finite-temperature Ising transition. Regarding Josephson junction arrays and quantum computers, one of the important issues is to ensure that the two-fold degenerate ground state is well protected by a gap to all exited states. Our results clearly ruled out this possibility in the thermodynamic limit. It is however still possible to have systems with a finite gap, but it requires working with not too large systems, as noticed by Douçot and collaborators [20].
Part II SrCu2(BO3)2, a frustrated magnet
CHAPTER
3
Introduction The second part of this work is devoted to the theoretical study of the magnetic properties of the frustrated quantum antiferromagnet SrCu2 (BO3 )2 . This quasi two-dimensional Mott insulator has a tetragonal crystal structure with a layered structure of successive CuBO3 and Sr planes [27]. The magnetic degrees of freedom are given by the spins 1/2 of the Cu2+ ions. In each CuBO3 layer (see figure 3.1 (a)), the arrangement of Cu2+ ions form a set of orthogonal dimers (thick black lines) made of two nearest neighbor Cu sites connected through O sites. The two Cu sites of each dimer are then connected to the neighboring dimers through BO3 molecules (dashed lines). The resulting lattice is topologically equivalent to the Shastry-Sutherland lattice [28]. In the direction perpendicular to the layers, the arrangement of the Cu sites is such that the dimers are on top of each other, but in perpendicular directions (see figure 3.1 (b)). As stated by S. Miyahara and K. Ueda in their excellent review about SrCu2 (BO3 )2 [27], this material shows various unique features. Firstly, the existence of a spin gap ∆ ∼ 35 K has been established at zero field with various experiments [30–37]. This result suggests a non-magnetic ground state, possibly made of spin-singlets. Secondly, the magnetic excitations are unusual. Inelastic neutron scattering spectrum shows an almost flat lowest branch of excitations with a large continuum above [31, 38]. The flat band signals strongly localized triplet excitations, while the continuum indicates that bound states of two triplets can move much more easily than isolated triplets. Finally, the magnetization curve (see figure 3.2) obtained by measuring experimentally the magnetization of the compound as a function of the applied magnetic field, presents an interesting sequence of anomalies which are interpreted as plateaus at 1/8, 1/4 and 1/3 of the maximal magnetization [29,30,39]. Although all the experiments agree on the presence of a 1/3 plateau, the situation is more controversial at lower magnetization. The possibility of other plateaus has been pointed out by Sebastian and collaborators [40], who have interpreted their highfield measurements as evidence for plateaus at 1/q with 2 6 q 6 9 and at 2/9. Nuclear magnetic resonance (NMR) measurements have shown that the translational symmetry
36
Introduction
Figure 3.1: (a) SrCu2 (BO3 )2 structure projected on a CuBO3 plane. The arrangement of the Cu sites is topologically equivalent to the Shastry Sutherland lattice. (b) Two neighboring Cu layers have their dimers on top of each other but in perpendicular directions.
Figure 3.2: The magnetization curve for SrCu2 (BO3 )2 at 1.3K with a magnetic field perpendicular to the Cu planes. (Reproduced from [29]) is broken at the 1/8 plateau [32], which is compatible with a localization of triplets in a super-lattice. More recently, other NMR and torque measurements [41, 42] have shown that the phase directly above the 1/8 plateau still breaks the translational symmetry, even though the magnetization increases. This result would be compatible with a supersolid phase, where a finite fraction of the triplets condense while the other triplets stay localized in a super-lattice. It would be fascinating if the presence of the supersolid phase could be confirmed, as it would correspond to the first experimental observation of a supersolid phase in a frustrated magnet. In order to obtain theoretical predictions which could help the understanding of the SrCu2 (BO3 )2 magnetic properties, a minimal model is given by the two-dimensional spin
37
Figure 3.3: The phase diagram of the Shastry-Sutherland model in the absence of magnetic field. 1/2 Heisenberg model on the Shastry-Sutherland lattice with antiferromagnetic couplings and an external magnetic field: X X X Srz (3.1) Sr · Sr′ − B H = J′ Sr · Sr′ + J
≪r,r′ ≫
r
where the ≪ r, r ′ ≫ bonds build an array of orthogonal dimers while the < r, r ′ > bonds correspond to inter-dimer couplings, as depicted in figure 3.1 (a). In the absence of magnetic field this model has been intensively studied with various theoretical approaches [43–46]. As depicted in figure 3.3, when J ′ /J is smaller than a critical value (J ′ /J)c1 ∼ 0.7, the ground state is given exactly by the tensor product of singlets on each dimer. There is a gap to the spin excitations, which are obtained by replacing singlets by triplets. In agreement with the inelastic neutron scattering experiments for SrCu2 (BO3 )2 , the lowest branch of excitations is almost flat and the spectral densities show a two-triplet bound state continuum [47, 48]. When J ′ /J is larger than a second critical value (J ′ /J)c2 ∼ 1, the ground state has antiferromagnetic long-range order and the excitations are gapless. Between these two critical values, an intermediate phase is likely to be realized. The nature of this phase is not clearly established, possible candidates are an helical ordered phase [44] or a plaquette phase [43, 45]. By fitting various thermodynamic properties such as the magnetic susceptibility and the specific heat, the coupling constants for the SrCu2 (BO3 )2 have been estimated in the range 0.6 < J ′ /J < 0.68 with J of the order of 100K [27], which corresponds to the singlet phase. In order to understand what happens when the magnetic field is turned on, it is easier to start with the limit of isolated dimers (J ′ = 0). For a single dimer, without √ magnetic field, the ground state is given by the singlet state |si = (| ↑↓i − | ↓↑i) / 2 while the excited states √ are given by the three triplets |t1 i = | ↑↑i, |t−1 i = | ↓↓i and |t0 i = (| ↑↓i + | ↓↑i) / 2 which are degenerate. When the magnetic field is turned on, the triplet degeneracy is lifted and the energy of triplet |t1 i goes down towards the energy of the singlet which does not change. When B = J, both states are degenerate (see figure 3.4). For a system of N isolated dimers, as long as B < J, the ground state is given by the state with each dimer in the singlet state. At B = J the ground state manifold is highly degenerate and contains all the states with dimers in either the singlet or the triplet |t1 i states. When B > J the state with all dimers in the triplet |t1 i state becomes
38
Introduction
Figure 3.4: For one dimer, the magnetic field lift the degeneracy of the triplets but does not change the energy of the singlet and the triplet |t0 i. When B = J, the singlet and the triplet |t1 i are degenerate. the ground state. Therefore, in order to describe the low energy physics of this model, one can restrict the Hilbert space to the set of states having the dimers only in the singlet and triplet |t1 i states. Note that the lowest energy states containing |t−1 i or |t0 i triplets are separated from the ground state by a gap of energy J. When the inter-dimer coupling J ′ is non-zero, one expects the various energies to change, but only on a scale J ′ . Therefore, as long as J ′ ≪ J, the states containing |t−1 i or |t0 i triplets will still be separated from the ground state by a finite gap of energy ∼ J − J ′ > 0 and the low energy physics will be correctly described by the set of states containing only singlets and triplets |t1 i. The picture will be similar for B ≪ J and B ≫ J, but the lifting of the degeneracy at B = J should lead to an interesting physics on a scale J ′ around this value of the magnetic field. Although in the absence of magnetic field, the ground state is exactly known for the parameters corresponding to the SrCu2 (BO3 )2 , in the presence of a magnetic field the problem becomes really challenging because one has to deal with a finite density of strongly interacting triplets. The complexity of the problem attracted a lot of interest in the past few years and several attempts to study directly the Shastry-Sutherland model have been done by using exact diagonalizations [49, 50], Schwinger bosons [44] or ChernSimons mean-field approximations [40, 51, 52]. Although all these approaches agree on the presence of magnetization plateaus at 1/3 and 1/2 of the maximal magnetization, the structure below 1/3 is rather controversial. The finite size clusters available to exact diagonalizations prevent reliable predictions for high-commensurability plateaus, and the accuracy of the Chern-Simons mean-field approach initiated by Misguich and collaborators [51] and recently used by Sebastian and collaborators [40] is hard to assess. Another approach which has been intensively used consists in deriving and studying an effective model for the low energy physics. In these effective models, the triplets are treated as particles (hard-core bosons) while the singlets correspond to the absence of particle. Following the previous considerations on the low energy physics in the limit of weakly coupled dimers, a natural way of building an effective model consists in using a perturbative approach in the small parameter J ′ /J ≪ 1. A lot of effort has been made in this direction, with effective Hamiltonians derived up to third order in J ′ /J [27, 49, 53]. However, this approach could not produce completely satisfactory results. The first problem comes from the fact that SrCu2 (BO3 )2 is expected to be close to the transition
39 towards the intermediate phase. In this region, the higher order terms become important and cannot be neglected anymore. These terms include the long-range interactions, which are required in order to stabilize the low-magnetization and high-commensurability plateaus. Therefore high-order expansions are required in order to use this approach. Another problem consists in solving the effective model. Indeed, the effective Hamiltonian obtained with high-order perturbation theory contains a huge number of terms and in order to describe high-commensurability plateaus one needs to work on large clusters. Quantum Monte Carlo suffers from the minus sign problem as well as the complexity of the effective Hamiltonian, exact diagonalizations still suffer from the small clusters available and analytical approaches are difficult because of the high-commensurability and the complexity of the effective Hamiltonian. Another method to obtain a non-perturbative effective model has been used very recently by Abendschein and Capponi [54]. They used the non-perturbative Contractor-Renormalization method (CORE) to derive an effective Hamiltonian, which was then treated by exact diagonalizations. Although the approach is interesting, the resulting Hamiltonian is not easier to solve and has the same problems as the perturbative Hamiltonian. Despite the problems mentioned previously, we decided to use the perturbative approach, mainly because this approach has also several advantages. Firstly, the approximation is clearly excellent in the limit J ′ /J ≪ 1 and should be valid in a finite J ′ /J range. More importantly, the quality of the approximation can be estimated by comparing the results obtained for various orders. Secondly, the description of the system in terms of triplet particles moving on a background of singlets seems appropriate in view of the experimental results. In particular, the low magnetization plateaus could be understood as a localization of the triplets in a super-lattice due to the long-range interactions appearing at high order, while the smooth transitions between the plateaus would correspond to a phase with a Bose-Einstein condensation of the triplets (superfluid). Additionally, a finite fraction of the triplets could still localize and form a super-lattice (supersolid). Finally, although the ground state of the Shastry-Sutherland model cannot be obtained with a classical treatment of the spins, we believe that a classical treatment of the effective model (see section 4.2.1) can give a qualitatively correct description of the phase diagram. Indeed, most of the complexity due to the quantum nature of the problem is encoded in the successive transformations which first maps the spins onto the singlets and triplets and then maps the initial Hamiltonian onto the effective Hamiltonian. In addition, as we shall see in section 4.2, the classical treatment of a simpler model of interacting hardcore bosons gives a qualitatively correct description of the phases which dominate the phase diagram and these predictions are particularly accurate for the insulating phases (plateaus). Thanks to the simplicity of this approximation, it becomes possible to analyse and compare all non-equivalent clusters having up to ∼ 60 dimers and therefore to predict the existence and the structure of the high-commensurability plateaus. In this respect, this approximation is better than the other approaches which cannot stabilize high-commensurability plateaus. In view of the aforementioned advantages, we decided to use perturbative continuous unitary transformations (PCUT) in order to derive a high order effective Hamiltonian (see section 5.1.2 and appendix F), whose phase diagram is then determined within a classical
40
Introduction
approximation. This effective Hamiltonian contains a large number of terms (∼ 104 ) and presents several interesting features. Firstly, the usual two-body interactions have by far the largest coefficients with a nearest neighbor interaction appearing already at order 1 in J ′ /J. So one expects the magnetization curve to consist mainly of plateaus, separated by jumps or by very narrow intermediate phases. Secondly, the three-body interactions (density-density-density terms) appear already at order 2 and should therefore have an impact at least at not too low densities. Interestingly, the dominant three-body interaction is attractive. Finally, as already pointed out by several authors [27, 49, 53], the usual hopping term is extremely small and appears only at order 6 in J ′ /J . Yet, the kinetic energy is not so small, since another type of kinetic processes appears already at order 2. These processes are the correlated hoppings, which allow a particle to hop only if there is a another particle nearby. This situation where correlated hopping dominates the kinetic energy is very unusual and is a consequence of the strong frustation of the model. In order to have a better understanding of these processes, we decided to start by studying a simple model of interacting hard-core bosons with correlated hopping, simple hopping, and two-body repulsive interactions (see chapter 4). By combining exact diagonalizations, a semi-classical approximation and quantum Monte Carlo simulations, we obtained a phase diagram with several interesting features. Firstly, as discussed in section 4.1, correlated hopping stabilize a low-density paired superfluid phase in which pairs of particles condense but without single-particle condensation. Secondly, as discussed in section 4.2, correlated hopping stabilizes large regions of supersolid phase, and this effect is so strong that a supersolid phase can even be stabilized when the two-body interaction is too small to stabilize an adjacent solid phase. This analysis was also an occasion to compare the results obtained by a classical approximation with the quantum Monte Carlo results. It turns out that the phase diagrams obtained with the classical approximation are in qualitative agreement with quantum Monte Carlo results, although the size of the supersolid phases are clearly overestimated. The next part of this work concentrates on the Shastry-Sutherland model (chapter 5). This chapter starts with the derivation and a detailed description of the effective model (section 5.1) which is followed by a presentation and discussion of the classical phase diagram (section 5.2). Before starting with the analysis of the various models, it might be useful to remind some important notions which will appear all along the next chapters. A first important definition concern the hard-core bosons. The two operators b†r and br are creation and annihilation operators of a hard-core boson at site r if they satisfy the following commutation relations ∀r 6= r ′ : {br , b†r } {br , br } [br , b†r′ ] [br , br′ ]
= = = =
1 0 0 0.
(3.2) (3.3) (3.4) (3.5)
Therefore, hard-core bosons are similar to bosons because two operators on different sites commute, but with an additional hard-core constraint which prevents two particles from
41 being on the same site. Throughout the introduction, we discussed about superfluid, supersolid and solid phases. In order to define these phases, one needs to introduce two important symmetry groups. The first one is the U(1) gauge symmetry. It transforms the hard-core boson operators according to br → e−iα br . The whole group is spanned by varying the continuous variable P † α. Note that the corresponding conserved quantity is the total number of particle r br br . The second one is the usual translational symmetry. Clearly, all the Hamiltonians which will be discussed in the next chapters are translationally invariant, although the Shastry-Sutherland Hamiltonian and therefore the effective Hamiltonian are only invariant under a translation of two (dimer) lattice spacing in both x and y direction. The various phases are defined as follows: a state is superfluid if it breaks the U(1) gauge symmetry, it is solid if the density is not uniform (and therefore the state breaks the translational symmetry) and it is supersolid if the density is not uniform and if it breaks the U(1) gauge symmetry. In order to detect solid order, we will use the static structure factor 1 X ′ (3.6) hnr nr′ ieik(r −r) S(k) = 2 N r,r′ which vanishes for all k 6= 0 except if there is solid order. To detect a breaking of the U(1) symmetry, we will use hbr i, which vanishes for all r except if the state breaks the U(1) symmetry. The link between a breaking of the U(1) symmetry and a Bose-Einstein condensation of the particle (superfluid) can be seen in the following way. First, we suppose that the state breaks the U(1) symmetry (∃r such that hbr i = 6 0). In order to simplify the reasoning, we will only consider the uniform case hbr i = η ∀r. Next, we consider the Fourier transformation of the order parameter at k = 0 √ 1 X hbk=0 i = √ hbr i = η N . N r
(3.7)
By using the Cauchy-Schwarz inequality hb†kbki > hb†kihbki, we obtain hb†k=0 bk=0 i > |hbk=0 i|2 = |η|2N.
(3.8)
Therefore, the breaking of the U(1) symmetry (hbr i = η) implies that there is a finite fraction of the total number of particles which condenses in the mode k = 0.
CHAPTER
4
Correlated hopping, a minimal model One striking feature of the effective model for the Shastry-Sutherland model is a kinetic energy dominated by correlated hopping. This is a consequence of the strong frustration in the Shastry-Sutherland model and because it is quite unusual, it has not thoroughly been studied yet. Indeed, although models of interacting bosons arise naturally in different fields of condensed matter and of atomic physics, the dominant processes are usually the local repulsion and nearest neighbor hopping. In these models, the interplay of kinetics and interaction is known to give rich phase diagrams with phases like superfluid (SF), Mott insulators (MI), supersolid (SS) or phase separation (PS). In order to understand the effect of correlated hopping on these models, a first approach consists in studying a simple model of interacting hard-core bosons with correlated hopping: X X X X X nr + V nr nr+δ − t b†r br+δ H = −µ r
′
−t
r
XX X r
r
δ=x,y
nr (b†r+δ br+δ′
+
δ=±x,±y
b†r+δ′ br+δ ).
(4.1)
δ=±x δ′ =±y
nr = b†r br is the bosonic density at site r in the two dimensional square lattice, µ is the chemical potential, V is the nearest neighbor repulsion, t > 0 is the nearest neighbor hopping amplitude and t′ > 0 is the correlated hopping amplitude (see figure 4.1). In the following, the energy scale will be fixed by t + t′ = 1 and all the results will only be discussed as a function of t′ . Clearly, this Hamiltonian is invariant under the translations of one lattice spacing in both x and y directions as well as the U(1) gauge transformation. In the limit t′ = 0, the Hamiltonian (4.1) can be mapped onto a XXZ model in a magnetic field which has already been studied and whose phase diagram is well known [55–58]. Due to the particle-hole symmetry, the phase diagram is symmetric about the density n = 1/2. At large negative or positive chemical potential, the system is empty
44
Correlated hopping, a minimal model
Figure 4.1: The different processes in Hamiltonian (4.1) are the nearest neighbor interaction (a), the nearest neighbor hopping (b) and the correlated hopping (c). The particles are depicted by black circles, while white circles corresponds to empty sites. With nearest neighbor interaction (a), the system pay an energy V if two neighboring sites are occupied. With nearest neighbor hopping (b), the system gains an energy t when a particle hops from one site to an empty neighboring site. Finally, with correlated hopping (c), the system gains an energy t′ when a particle hop from one site to a next-nearest neighbor empty site, provided that there is a particle in a neighboring site. or full. When the repulsion is weak (V < 2t), apart from empty and full phases, there is only a superfluid phase. When V = 2t, the model maps to the Heisenberg model, and an insulating phase with checkerboard order appears at density n = 1/2. When the repulsion is larger (V > 2t), phase separation appears between the solid and superfluid phases. This model does not realize any supersolid phase [56]. The effect of the correlated hopping on the model (4.1) in the limit V = 0 has been investigated with a mean-field analysis [59]. This analysis suggests that correlated hopping can favor a paired superfluid (PSF) phase characterized by a condensation of pairs 6 0) without single-particle condensation (hbr i = 0), similar to phases already (hbr br′ i = discussed in atomic physics in the context of ultra-cold atoms [60–64], and which could correspond to the two-triplet bound state proposed by Momoi and Totsuka [53]. This chapter presents our results (published in [65–67]) on the model defined by Hamiltonian (4.1). The first part (section 4.1) focuses on the low density physics and the paired superfluid phase. Using exact diagonalizations, we show that a paired superfluid phase is stabilized at large correlated hopping in a small window of finite nearest neighbor repulsion. This project has been done in collaboration with K. P. Schmidt who has performed quantum Monte Carlo simulations on the same problem. These results are briefly presented in 4.1.4. They complement the physical picture deduced from the exact diagonalization study. The second part of this chapter (section 4.2) is devoted to the effect of correlated hopping on the stabilization of supersolid phases. Within a semi-classical approximation (in 4.2.1), we show that correlated hopping indeed stabilizes a large supersolid phase. The effect is so strong that this supersolid can even be stabilized without the corresponding solid. Again, in order to confirm the conclusions of the semi-classical approximation, quantum Monte Carlo results are presented in 4.2.2. The last part of this
4.1 Low density limit and paired superfluid phase
45
Figure 4.2: The various clusters used with exact diagonalization. chapter (section 4.3) presents some classical results on generalizations of the model (4.1). The first generalization is rather obvious and consists in model (4.1) but with negative t′ . The frustration induced by the competing hopping and correlated hopping produces an interesting phase diagram with various superfluid and supersolid phases. The tendency to form supersolid phases is even stronger for this model. The second generalization replaces the correlated hopping by a next-nearest neighbor hopping. We show that this model also stabilizes large supersolid phases, but never without an adjacent solid phase.
4.1
Low density limit and paired superfluid phase
P Thanks to the U(1) gauge symmetry, the total number of particles r nr is a conserved quantity. Therefore, by working in the eigenbasis of the nr operator, the Hamiltonian can be separately diagonalized in each subspace with a fixed total number of particles. The ! , where N is size of the subspace with total number of particles Np is given by Np !(NN−N p )! the number of sites in the lattice. It is minimal for the empty (Np = 0) and full system (Np = N) and is maximal at half filling Np = N/2. Taking this into account, as well as the translation symmetry, exact diagonalizations (see appendix B) could be performed for all densities with up to 36 sites clusters and the low density limit could be done with clusters having up to 58 sites. The different clusters are shown in figure 4.2. Because exact diagonalization works with a fixed number of particles, it is natural to work in the canonical ensemble. One has therefore to diagonalize Hamiltonian (4.1) without the chemical potential term, whose only effect is to shift the whole energy spectrum. ˜ In the following, this Hamiltonian will be denoted by H: ˜ = V H
X X r
′
−t
δ=x,y
nr nr+δ − t
XX X r
X X r
b†r br+δ
δ=±x,±y
nr (b†r+δ br+δ′
+ b†r+δ′ br+δ ).
(4.2)
δ=±x δ′ =±y
˜ for a given density n and to extract Exact diagonalization is then used to diagonalize H ground state expectation values of the relevant operators. All relevant quantities are then given as a function of the density n = Np /N. In order to get these quantities as a function of the chemical potential, which is directly related to the magnetic field in
46
Correlated hopping, a minimal model
˜ (black curve) and envelope E ˜env (light Figure 4.3: (a) Ground state energy E blue) as a function of the total number of particles Np . (b) The chemical potential µ is given by the slope of the curve E˜env (Np ). Note that these plots do not present exact diagonalization results, they are only sketches to illustrate how to extract the chemical potential. the initial model, one can do a Legendre transformation. The chemical potential is then obtained as a function of Np by µ=−
dE˜env (Np ) dNp
(4.3)
˜env (Np ) is the envelope of the curve E(N ˜ p ) and E(N ˜ p ) is the ground state energy where E ˜ in the subspace with a fixed total number of particles Np (see figure 4.3). of H
4.1.1
n − t′ phase diagram
A first characterization of the different phases can be done by studying the behaviour of the total number of particles Np when the chemical potential µ increases. An example of such a curve is given in figure 4.4. Supposing that the system is in a superfluid phase, the number of particles is expected to increase by one upon increasing the chemical potential. If the system is in a paired superfluid phase, the pairing gap prevents the insertion of one particle and the number of particles increases by two. In the case of phase separation, the system tends to form clusters with a finite number of particles, and the density n = Np /N jumps by a finite amount. For large system sizes N it will correspond to a large jump in the total number of particles Np . However, for small systems, it can correspond to a jump of two or even one in Np . In the following the criterion for phase separation will be that Np increases by more than two. The resulting n − t′ phase diagram is shown in figure 4.5 for different values of the nearest neighbor repulsion. White regions represent superfluid phases, grey regions phase
4.1 Low density limit and paired superfluid phase
47
Figure 4.4: Total number of particle versus chemical potential. Exact diagonalization results on a 25 sites cluster with V = 2, t′ = 0.9. The different phases can be characterized by the way in which the total number of particles changes upon changing the chemical potential. Steps of one particle corresponds to superfluid (SF), steps of two to paired superfluid (PSF) and larger steps to phase separation (PS). separation and black regions paired superfluid. The small shaded regions in figure 4.5 (a) and (b) corresponds to region where Np increase by two upon increasing µ and it should be interpreted as a paired superfluid phase. However this small region seems to disappear as the system size increases (see section 4.1.2). When V = 0 the phase diagram presented in figure 4.5 (a) shows only a superfluid phase and a phase separation region. The superfluid phase is realized for not too large values of t′ or for large densities. This phase becomes unstable to phase separation at lower densities for large correlated hopping. Phase separation is a consequence of the correlated hopping which acts as an attractive interaction. Indeed the system can only gain energy through correlated hopping when the particles are close to each other. Therefore, at low densities, the system phase separates between either an empty system or a system at a finite density where correlated hopping can lower the energy. The effect of a finite nearest-neighbor repulsion is shown in figure 4.5 (b) for V = 2 and figure 4.5 (c) for V = 2.8. This repulsive interaction should compete with the attractive interaction coming from the correlated hopping and destabilize the phase separation region which can be seen as a many-boson bound state. The hope is that it is not too strong and stabilizes only the two-particle bound states which should generate the paired superfluid phase. That is indeed what happens as it can be seen in the low density and large t′ region in figure 4.5 (b) and (c). The phase separation region becomes smaller as V increases and is replaced by a superfluid phase coming from higher densities or smaller t′ . But
48
Correlated hopping, a minimal model
Figure 4.5: Low density n − t′ phase diagram obtained by exact diagonalization. (a) V = 0 on the 25 sites cluster. (b) V = 2 on the 25 sites cluster. (c) V = 2.8 on the 32 sites cluster. White regions corresponds to superfluid phase (SF), grey corresponds to phase separation (PS) and black corresponds to paired superfluid phase (PSF). The small shaded regions corresponds to regions of paired superfluid appearing only for finite size system. The insets presents the total number of particles Np as a function of µ for t′ = 0.9 in (b) and t′ = 0.95 in (c).
there is also the appearance of a paired superfluid which is first surrounded by phase separation (V = 2) and then almost replaces phase separation at V = 2.8. For larger values of V (not shown here), the paired superfluid region becomes smaller up to V ∼ 4 where it completely disappears. At V = 2.8, another phase separation region can be seen below density 1/2. It is related to the first order transition between the superfluid and
4.1 Low density limit and paired superfluid phase
49
checkerboard solid at n = 1/2 which will be discussed later. The main conclusion of this simple finite size analysis is that a possible paired superfluid phase is stabilized in a finite window of nearest neighbor repulsion for large correlated hopping and at low densities. However, this criterion in itself is not sufficient to prove that a paired superfluid is indeed realized. Firstly, for the small systems discussed here, a jump of two in Np could as well correspond to a small region of phase separation instead of a paired superfluid. Secondly, this criterion only gives an indication that the system tends to form pairs of particles, but gives no information regarding a possible condensation of these pairs.
4.1.2
V − t′ phase diagram
In order to check, in the low-density limit, that the region tagged as paired superfluid does not correspond to phase separation, one can use another criterion based on the binding energies. The two and four-particle binding energies are defined as ∆2 = E(2) − 2E(1) and ∆4 = E(4) − 2E(2) where E(Np ) denotes the ground states energies of the the Hamiltonian (4.2) in the subspace with Np particles. In a phase where the system tends to form pairs of particles, the energy of the two-particle system is expected to be lower than the energy of two independent particles and therefore ∆2 is expected to be negative. On the contrary, a positive ∆2 will indicate the absence of two-particle bound states. Similarly, a negative or positive ∆4 will indicate the presence or the absence of a fourparticle bound state. Regarding the different phases, in analogy to the binding of holepairs in the two dimensional t − J model [68], a bound state of four particles (∆4 < 0) will indicate a tendency to phase separation, a bound state of two particles without a four-particle bound state (∆2 < 0, ∆4 > 0) will correspond to a pairing phase while no bound states (∆2 > 0, ∆4 > 0) will be attributed to a superfluid phase. Because only a small number of particles are needed for this calculation, the binding energies could be obtained by exact diagonalization on clusters having up to 58 sites. The binding energies have been evaluated on a finite set of points on the grid defined by V between 0 and 4 with steps of 0.2 and t′ between 0 and 1 with steps of 0.05. The zeroes of the binding energies have been found by doing a linear interpolation between the points of the grid (see figure 4.6). The resulting V − t′ phase diagram on the 58 site cluster is given in figure 4.7. White, grey and black regions corresponds to superfluid phases (SF), phase separation (PS) and paired superfluid phases (PSF) respectively. The small shaded regions correspond to a region of paired superfluid appearing only for finite size systems. Figure 4.8 presents the linear size ∆V of this region measured along a fixed t′ line versus the number of sites N in logarithmic scale. A fit of the exact diagonalization results (symbols) to a power law (straight lines) gives almost a perfect match, indicating that ∆V indeed vanishes according to a power law in the thermodynamic limit. In agreement with the low-density limit of the n − t′ phase diagram in figure 4.5, the V − t′ phase diagram in figure 4.7 predicts that the zero V limit is dominated by a superfluid phase at low t′ and phase separation at large t′ . Even the small paired superfluid region (shaded region) is also present with both approaches. Upon increasing
50
Correlated hopping, a minimal model
Figure 4.6: Four-particle binding energy ∆4 obtained by exact diagonalization as a function of V and t′ . The intersection of the ∆4 surface (dark grey) with the zero energy plane (light grey) is determined with a linear interpolation between the exact diagonalization results.
Figure 4.7: V − t′ phase diagram obtained by exact diagonalization on a 58 site cluster. White, grey and black regions corresponds to superfluid phases (SF), phase separation (PS) and paired superfluid phases (PSF) respectively. The small shaded regions corresponds to a region of paired superfluid appearing only for finite size systems. ∆V is the height of this region. the nearest neighbor repulsion V , a paired superfluid phase appears at large t′ while the low t′ superfluid phase becomes larger and slowly replaces the phase separation region.
4.1 Low density limit and paired superfluid phase
51
Figure 4.8: Height ∆V of the small paired superfluid region (see figure 4.7) versus the system size N in logarithmic scale for different values of t′ . The symbols are the results obtained by exact diagonalization and the straight lines are fitted to these values. At V = 4 and above, both paired superfluid and phase separation are not stabilized and only the superfluid phase remains. In the limit t′ = 1 (t = 0), a pair of nearest neighbor bosons can hop with amplitude t′ and gain at most a kinetic energy of 4t′ equal to one-half of the bandwidth. Because the two particles are nearest neighbor, it also costs an energy V . This simple reasoning gives an exact upper bound V = 4t′ for the pairing phase, in agreement with the exact diagonalization results.
4.1.3
Reduced density matrices and pair condensation
Up to now, the different approaches have only shown that a pairing phase can be realized, but nothing has been done regarding a possible condensation of the pairs. In principle, one should study the asymptotic behavior of the ground state two-point correlation functions hb†r1 br2 i for the single-particle superfluidity and the four-point correlation functions hb†r1 b†r2 br3 br4 i for the pair superfluidity. The presence of single-particle superfluidity can be checked by an easy criterion: a non-zero two-point correlation function in the limit |r1 − r2 | → ∞ signals single-particle superfluidity. The case of pair superfluidity is more problematic. Intuitively, pair superfluidity should be signaled by a non-zero four-point correlation function in the limit of two distant pairs of particles, as depicted in figure 4.9 (a) with r1 , r2 as well as r3 , r4 nearest neighbor. On the other hand a non-zero four-point correlation functions in the limit where all the particles are far apart, as depicted in figure 4.9 (b), should only signal single-particle superfluidity. In practice, this approach is problematic because one has to analyse the asymptotic
52
Correlated hopping, a minimal model
Figure 4.9: (a) A non-zero four-point correlation function hb†r1 b†r2 br3 br4 i in the limit of two distant pairs signal pair superfluidity. (b) A non-zero four-point correlation function when all the particles are far apart signal single-particle superfluidity. behavior of the numerous four-point correlation functions with r1 , r2 and r3 , r4 nearest neighbor. Hopefully, Yang proposed a simpler approach to solve this problem for a bosonic system, based on the reduced one and two-particle density matrices [69] (1)
ρi,j
= hb†r1 br2 i
(4.4)
† † ρ(2) m,n = hbr1 br2 br3 br4 i.
(4.5)
Supposing that the sites of the lattice are numbered from 1 to N, the indices i, j ∈ {1, 2, · · · , N} are the numbers corresponding to sites r1 and r2 respectively. Similarly, one can number all the pairs of sites (r, r ′ ) from 1 to N 2 . The indices n, m ∈ {1, 2, · · · , N 2 } are the numbers corresponding to the pairs (r1 , r2 ) and (r3 , r4 ) respectively. Therefore ρ(1) is a N × N matrix while ρ(2) is a N 2 × N 2 matrix. Yang proposed to characterize the (1) (2) different phases according to the behavior of the largest eigenvalues λmax of ρ(1) and λmax (1) (2) of ρ(2) as a function of the system size N for constant density. If λmax , λmax ∝ 1 there is (1) (2) no superfluidity. The case λmax ∝ 1 and λmax ∝ N corresponds to paired superfluidity, (1) (2) without single-particle superfluidity. And finally, the case λmax ∝ N and λmax ∝ N 2 corresponds to the usual single-particle superfluidity. (1)
Phase Eigenvalue λmax No superfluidity O(1) Paired superfluid (PSF) O(1) Single-particle superfluid (SF) O(N) (1)
(2)
Eigenvalue λmax O(1) O(N) O(N 2 )
The different behaviors of λmax can be understood as follows. In a phase with no single-particle superfluidity, the two-point correlation function decreases exponentially (1) when the distance between the particles increases i.e. all entries in ρi,j with |i − j| ≫ 1 (1) are very small. In the limit of no correlations, only the diagonal elements ρi,i are non-zero and one expects a maximum eigenvalue of order 1. In contrast, for a standard superfluid, (1) one expects the two-point correlation functions to be long range and all the ρi,j are of
4.1 Low density limit and paired superfluid phase
53
the same order. For the case where all correlations are exactly the same, one can easily convince oneself that the maximum eigenvalue is of order N. (2) The behaviour of λmax is more complex. In the case of no single-particle and no pair superfluidity, all four-point correlation functions are suppressed exponentially and one again obtains a maximum eigenvalue of order 1. In the case of pair superfluidity without single-particle superfluidity, four-point correlation functions between two pairs are long (2) ranged while the other correlations are suppressed exponentially. Therefore all the ρi,j involving large distance are basically zero except the ones where the two distances |r1 −r2 | and |r3 −r4 | (or the other combination of indices) are within a pairing length. This gives a (2) maximum eigenvalue λmax of order N. Finally, in the case of single-particle superfluidity, (2) all four-point correlation functions are long ranged i.e. all the ρi,j are of the same order and the maximum eigenvalue is of order N 2 . Note that these results are derived for a bosonic system, but in the low density limit hard-core bosons behave like bosons. Therefore one expects these results to be valid also for hard-core bosons in the low density limit. (i) (i) In order to study the behavior of λmax at fixed density, one first evaluates λmax for every possible system sizes N and even total number of particles Np . The curves (i) λmax (N, n = const) as a function of N for constant density n are then extracted with a (i) linear interpolation, as depicted in figure 4.10 (a). Each resulting curve λmax (N, n = const) is then fitted to a power law ai N αi (see figure 4.10 (b)) whose exponents αi can be used to characterize the different phases. Note that figure 4.10 is only a sketch to explain the procedure and it does not correspond to exact diagonalization results. (1) (2) Using exact diagonalization, λmax and λmax could be evaluated for systems with N = 16, 25, 36 and 49 sites and a total number of particles Np = 0, 2, 4, 6, 8 except for the N = 49 system where Np = 0, 2, 4, 6. An example of exact diagonalization results at V = 2 and t′ = 0 is given in figure 4.10. The black dots in figure 4.11 (a) correspond to (2) (i) λmax obtained by exact diagonalization and the black crosses are the λmax (N, n = const) (i) obtained by linear interpolation. Figure 4.11 (b) presents the resulting λmax (N, n = const) as a function of N (black crosses) as well as the results from the fit to the power law ai N αi (light gray curves). This procedure is then repeated for several t′ and V . The exponents α1 and α2 as a function of t′ for a density n = 0.1 and different values of V are shown in figure 4.12. Black dots correspond to α1 and grey dots corresponds to α2 . The regions of phase separation obtained in section 4.1.1 are shown in light grey. Figure 4.12 (a) presents the exponents for V = 0. At low t′ the system realizes a superfluid phase (α1 = 1 and α2 = 2). At larger t′ , in the phase separation region, the exponents should not give any useful information. When V = 2, as presented in figure 4.12 (b), the low t′ limit is compatible with a superfluid phase. At intermediate t′ there is a phase separation region, where the exponents change drastically, followed at large t′ by a region which is compatible with a paired superfluid phase (α1 = 0 and α2 = 1). Finally figure 4.12 (c) presents the exponents when V = 2.8. The results are compatible with a superfluid phase for t′ < 0.9 and a paired superfluid in the limit t′ ≃ 1. The exponents obtained by exact diagonalizations are not strictly equal to 0, 1 or 2 but deviate from the predicted values. This is however not a surprise. One should
54
Correlated hopping, a minimal model
Figure 4.10: Sketch to explain how the different phases are characterized. (2) (a) λmax is evaluated for all possible system sizes N and even total number (2) of particles Np (grey surface). The curves λmax (N, n = const) as a function of N for constant density n (blue curves) are then extracted with a linear (2) interpolation. (b) Each curve λmax (N, n = const) is fitted to a power law a2 N α2 whose exponent α2 can be used to characterize the different phases. (1) The same procedure can be used to extract the exponent α1 from λmax .
Figure 4.11: Determination of α2 for different densities. Exact diagonaliza(2) tion results for V = 2 and t′ = 0. (a) Black dots correspond to λmax obtained (2) by exact diagonalization. Black crosses are the values of λmax (N, n = const) (2) obtained by linear interpolation. (b) λmax (N, n = const) as a function of N (black crosses) and fit to a power law a2 N α2 (grey curves).
4.1 Low density limit and paired superfluid phase
55
Figure 4.12: Exponents α1 (black dots) and α2 (grey dots) obtained by exact diagonalization as a function of t′ for a density n = 0.1. (a) V = 0, (b) V = 2 and (c) V = 2.8. The regions of phase separation (PS) obtained in 4.1.1 are shown in light grey. indeed expect finite-size effects, as this theory is based on asymptotic behaviors in the (1) (2) thermodynamic limit. Another reason is the small number of values for λmax and λmax which can be obtained by exact diagonalizations as well as the linear interpolation, which (i) is clearly too simple (see figure 4.11). As a consequence the extracted λmax (N, n = const), and therefore the exponents αi , are not exact. Nevertheless, these results are in qualitative agreement with the phase diagrams shown in figures 4.5 and 4.7. They confirm the presence of the superfluid phase from the low t′ limit up to intermediate or large t′ depending on V . They also give an evidence for the realization of a paired superfluid phase at large t′ and in a finite window of V .
56
4.1.4
Correlated hopping, a minimal model
Quantum Monte Carlo
In order to complement the physical picture deduced from the exact diagonalization results, K. P. Schmidt used a quantum Monte Carlo technique to establish the existence of the paired superfluid phase. As it is not my work, I will only briefly present the main results. All the details can be found in reference [65]. The Hamiltonian (4.1) does not suffer from a sign problem and is therefore accessible to quantum Monte Carlo techniques. However the treatment of three site terms (correlated hopping) is not standard. The algorithm used here is a SSE algorithm [70–72], which has been shown to be able to treat multi-site terms [73]. This algorithm works in the grand canonical ensemble, that is the total number of particles is only fixed in average and controlled by the chemical potential µ. Therefore, in addition to the total density, the algorithm also gives access to the density histogram, which counts how many times a system with a certain total number of particles is realized during the simulation. The density histogram should allow to distinguish between a pairing phase and a standard superfluid phase, by showing a pronounced even or odd effect which should appear only in the pairing phase. This algorithm can also be used to evaluate the equal-time oneparticle Green’s function G(r − r ′ ) = hb†r br′ i. This quantity is expected to decrease exponentially in the paired superfluid phase, due to the pairing gap, and to be long ranged in the conventional superfluid phase. For a given system size, a convenient measure of the presence of one-particle superfluidity is given by the ratio of the one-particle Green’s function at the maximal distance allowed by the cluster to the density G(r = rmax )/n. Note that this algorithm only works at finite temperatures, therefore all the quantities defined previously correspond to thermal averages at a given temperature T . In the following, only T = 0.05 results are discussed. Exact diagonalization results predict a paired superfluid phase at low densities and large correlated hopping in a finite window of nearest neighbor repulsion. In the following, we will concentrate on the two parameter sets t′ = 0.9, V = 2 and t′ = 0.95, V = 2.8. In the first case, a paired superfluid phase is expected at low densities, separated from the standard superfluid by an excluded density region of phase separation (see figure 4.5). In the second case, exact diagonalization results predicts a paired superfluid phase at low densities, with possibly a direct transition to standard superfluid. The density n as a function of the chemical potential µ is shown in figure 4.13 for the two parameter sets t′ = 0.9, V = 2 (a) and t′ = 0.95, V = 2.8 (c). The black curves with error bars corresponds to quantum Monte Carlo results for a N = 100 site cluster and t = 0.05, while the grey curve shows the T = 0 exact diagonalization results on a N = 36 site cluster. Exact diagonalizations predict a paired superfluid phase for n < 0.11 (µ < 0.77) with the first parameter set and for n < 0.17 (µ < 0.27) with the second parameter set. Figure 4.13 also presents the ratio G(r = rmax )/n as a function of the chemical potential for the two parameter sets t′ = 0.9, V = 2 (b) and t′ = 0.95, V = 2.8 (d). In both parameter sets this quantity vanishes for large negative chemical potential, signaling the absence of single-particle superfluidity. This is obviously true for the empty system, but also for larger values of the chemical potential corresponding to finite values of the density, signaling a possible paired superfluid phase. For larger chemical potential it
4.1 Low density limit and paired superfluid phase
57
Figure 4.13: (a) Density n as a function of the chemical potential µ for t′ = 0.9, V = 2, T = 0.05 obtained by quantum Monte Carlo with N = 100. Error bars are one standard deviation. The grey line corresponds to exact diagonalization results with N = 36. (b) Ratio G(r = rmax )/n with √ rmax = 50 as a function of the chemical potential for the same parameter set. The region between the two dotted lines (guide to the eyes) signals the finite-temperature coexistence region (COEX). (c) and (d) presents the same quantities but for the second parameter set t′ = 0.95, V = 2.8.
becomes finite, as expected for a standard superfluid phase. The two phases are separated by a finite region of chemical potential corresponding to a finite-temperature coexistence region (COEX). Density histograms are shown in figure 4.14 for the two parameter sets t′ = 0.9, V = 2 (a) and t′ = 0.95, V = 2.8 (b). In a standard superfluid phase, the histogram should be a Gaussian with no difference between even and odd total number of particles. By contrast, the histogram of a paired superfluid phase should be a Gaussian with a vanishing probability for an odd total number of particles. Finally, phase separation should be reflected by a superposition of a Gaussian with large even or odd effect corresponding to a paired superfluid and a Gaussian corresponding to standard superfluid. Because the temperature is finite, the superposition corresponding to the zero temperature phase separation is expected to be realized in a finite range of chemical potential. In figure 4.14, for small chemical potential, both parameter sets clearly display a Gaussian with a strong even or odd effect expected in a paired superfluid phase. For the first parameter set t′ = 0.9, V = 2, it is followed at larger chemical potential by a superposition of two Gaussians corresponding to a phase separation region. For the second parameter set
58
Correlated hopping, a minimal model
Figure 4.14: (a) Density histogram obtained by quantum Monte Carlo with N = 100, T = 0.05, t′ = 0.9, V = 2 and different values of the chemical potential. (b) Density histogram with N = 100, T = 0.05, t′ = 0.95, V = 2.8. Error bars are one standard deviation. t′ = 0.95, V = 2.8, it is not clear whether there is a superposition or not. At even larger chemical potential, the histogram is a Gaussian without even or odd effect, consistent with a standard superfluid. The quantum Monte Carlo results show that the system realizes a low density phase characterized by the absence of single-particle condensation, signaled by a vanishing oneparticle Green’s function, and a tendency to create pairs of particles signaled by a strong even or odd effect in the density histogram. This phase could be an insulating phase, which would also be characterized by the absence of single-particle condensation and an even number of particles. However, this possibility is ruled out by the absence of plateaus in the density as a function of the chemical potential. Although the quantum Monte Carlo results are not sufficient to prove it, it is natural to conjecture that this phase corresponds to a paired superfluid.
4.1.5
Summary
Using numerical approaches, we wanted to check that the system realizes a low density paired superfluid phase characterized by a condensation of pairs without single-particle condensation, as suggested by a previous mean-field analysis [59]. Exact diagonalization as well as quantum Monte Carlo results show that a pairing phase is stabilized at low density and large correlated hopping for a finite window of
4.1 Low density limit and paired superfluid phase
59
nearest neighbor hopping. The pairing nature of this phase has been clearly established. Indeed both exact diagonalization and quantum Monte Carlo results show that the total number of particles can only take even values. Furthermore, the analysis of the two and four-particle binding energies shows that the system tends to form pairs of particles, but not larger clusters. Regarding the single-particle or pair condensation in this phase, the size scaling of the reduced one and two-particle density matrices strongly suggests a condensation of pairs without single-particle condensation. This size scaling should in principle be sufficient to prove the realization of the paired superfluid. In practice, it is not totally conclusive because of finite size effects. However, the evaluation of the one-particle Green’s function with quantum Monte Carlo confirms that there is no single-particle condensation. Furthermore, the density curve as a function of the chemical potential obtained with quantum Monte Carlo has clearly no plateau, which rules out the possibility of an insulating phase. Although none of these results is sufficient by itself to prove it, by taking everything together one can be confident that this pairing phase is indeed a paired superfluid phase. Another important issue is the nature of the possible quantum phase transition out of the paired superfluid phase. In the single-particle superfluid the U(1) symmetry is completely broken while in the paired superfluid a residual discrete Z2 symmetry remains. Therefore, during the transition between paired superfluid and single-particle superfluid, the Z2 symmetry has to be broken and one expects either a first order transition or a second order transition in the Ising universality class [59–63]. Because of a convergence problem in the paired superfluid phase, the available data are not sufficient to answer this question in a rigorous fashion. Nevertheless, important trends can be deduced. At zero temperature, a first order transition corresponds to a jump in the total density as a function of the chemical potential. Finite temperature should smear this jump, but a superposition of the two phases still signals the tendency to phase separate, which would lead to a jump at zero temperature. For V = 2, as shown in figure 4.5, exact diagonalizations predict a first order transition (phase separation) when t′ ≃ 0.9. When t′ > 0.95 exact diagonalizations do not predict phase separation, but one cannot exclude it. Indeed, if the density jump is smaller than 2/N (with N = 32), it will be interpreted as a paired superfluid or a superfluid phase. The same reasoning holds for V = 2.8 where exact diagonalizations do not display phase separation and are therefore not conclusive. Now with finite temperature quantum Monte Carlo results, for both parameter sets V = 2, t′ = 0.9 and V = 2.8, t′ = 0.95, the density as a function of the chemical potential does not show any clear jump (figure 4.13). Nevertheless the transition is steeper for the first parameter set and very far from a jump for the second parameter set. The density histograms (figure 4.14) are more useful. For the first parameter set V = 2, t′ = 0.9, there is a clear range of the chemical potential where both phases coexist, suggesting phase separation and therefore a first order transition. For the second parameter set, it is not clear whether there is a coexistence region or not. Taking into account both exact diagonalizations and quantum Monte Carlo results, a first order transition seems very likely at V = 2, t′ = 0.9. For other parameters, these results are not sufficient to decide whether it is a first or second order transition.
60
Correlated hopping, a minimal model
Figure 4.15: (a) With correlated hopping, the system gain an energy t′ when a particle hop from one site to a next-nearest neighbor empty site, provided that there is a particle in a nearest neighboring site. (b) When a particle is added to the checkerboard solid, there is always a nearest neighboring site occupied and correlated hopping act as a standard next-nearest neighbor hopping with amplitude t′ . (c) The resulting phase is a supersolid with one sub-lattice completely filled (black circles) and the other sub-lattice partially filled where the particles condense (blurred circles).
4.2
Supersolid phases
In the absence of correlated hopping, Hamiltonian (4.1) has been thoroughly studied and it is now clearly established that it does not realize any supersolid phase. The only phase which breaks the translational symmetry is an insulating phase with checkerboard order appearing at density n = 1/2 when V > 2t. In this model, when V > 2t, the checkerboard solid is separated from the higher density superfluid by a first order transition. The physical picture is that if particles are added to the solid, the system will phase separate between a checkerboard solid and a supefluid at larger densities until the total density is larger than the superfluid density. Now, with correlated hopping, the physical picture should be different. Indeed, with correlated hopping, the system gains an energy t′ when a particle hops along a diagonal to a next-nearest neighbor site, provided that there is a particle in a nearest neighboring site, as depicted in figure 4.15 (a). In a checkerboard solid, all the sites of one sub-lattice are occupied. If a particle is added to the system, it has to be in the empty sub-lattice where all its nearest neighboring sites are filled. In this configuration, the correlated hopping is equivalent to a standard next-nearest neighbor hopping with amplitude t′ and the system can lower its energy by letting the particle hop in the empty sub-lattice, as depicted in figure 4.15 (b). The particles hopping in the empty sub-lattice should then condense, leading to the supersolid phase sketched in figure 4.15 (c) with one sub-lattice completely filled (black circles) and the other sub-lattice superfluid (blurred circles). As it will be shown in the following sections, these predictions are indeed realized.
4.2 Supersolid phases
4.2.1
61
Semi-classical approximation
In order to map the whole phase diagram, a first approach consists in using a semi-classical approximation. This approximation can be done by first mapping the hard-core bosons operators onto spins 1/2 operators with the Matsubara-Matsuda transformation [74]: 1 − Srz 2 = Sr+ = Sr−
nr =
(4.6)
br b†r
(4.7) (4.8)
where Sr± = Srx ± Sry and nr = b†r br . After this transformation, which is exact, the Hamiltonian (4.1) becomes H =
X 1 N(V − µ) + (µ − 2V ) Srz 2 r X X V z z x x y y −t Sr Sr+δ + Sr Sr+δ + Sr Sr+δ + 2 r δ=±x,±y XX X y y x x −t′ Sr+δ Sr+δ ′ + Sr+δ Sr+δ′ δ=±x δ′ =±y
r
+2t′
XX X r
δ=±x δ′ =±y
y y x x Srz Sr+δ Sr+δ ′ + Sr+δ Sr+δ′ .
(4.9)
Without correlated hopping (t′ = 0), this Hamiltonian corresponds to a XXZ model in a magnetic field. The correlated hopping adds a next-nearest neighbor XX type interaction as well as a three-spin term. With this transformation, an empty site is replaced by a spin up and an occupied site is replaced by a spin down. Therefore the density n is replaced by the magnetization per site m = 1/2 − n and the chemical potential µ is replaced by a magnetic field B = µ − 2V . The solid order parameter becomes S(k 6= 0) =
1 X z z ik(r′ −r) 1 X ik(r′ −r) ′ n ie = . hn hS ′ S ie r r N 2 r,r′ N 2 r,r′ r r
(4.10)
Because the classical and semi-classical approximations explicitly break the U(1) gauge symmetry (z axis rotation of all the spins) except for insulating phases, there is no need to study the asymptotic behavior of the two-point correlation functions hb†r1 br2 i and hbr i can be used as a superfluid order parameter. With the Matsubara-Matsuda transformation it becomes hbr i = hSr+ i. Its modulus is given by the length of the spin projection onto the xy plane p |hbr i| = hSrx i2 + hSry i2 (4.11)
while its phase is given by the angle between the x axis and the spin projection onto the xy plane. The bosonic phases can be easily translated in spin notation. As shown in figure 4.16, an insulating state with checkerboard order (a) becomes a Néel state (b), a superfluid state (c) becomes a ferromagnetic state with all the spins having a non-zero
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Correlated hopping, a minimal model
Figure 4.16: The bosonic phases can be easily translated in spin notation. A checkerboard solid (a) becomes a Néel state (b). A superfluid (c) becomes a ferromagnetic state with a non-zero projection onto the xy plane (d). A supersolid (e) becomes a state with Néel order but with the spins pointing up which have a non-zero projection onto the xy plane (f). In this sketch, black circles denotes completely filled sites, blurred circles denotes partially filled sites with a non-zero superfluid order parameter and black arrows denotes expectation value of the spin operators.
projection onto the xy plane (d). Finally, a supersolid phase with checkerboard order (e) becomes a state with a Néel order but with all the spins pointing up which have a non-zero projection onto the xy plane (f). Note that depending on the density, the spins pointing down in the supersolid could also have non-zero xy component. A first approximation consists in treating the spin operators Sr as classical vectors of length 1/2 [55]. Within this approximation, the ground state is given by the arrangement of spins {Srclass } which minimizes the classical Hamiltonian (see appendix D). The minimization is then performed numerically on a four site square cluster with periodic boundary conditions to allow for broken translational symmetry. The next approximation consists in evaluating the quantum fluctuations around the classical ground state with a standard linear spin wave approach (see appendix E). We consider a N site square lattice with periodic boundary conditions and suppose that the periodicity of the classical ground state, as well as the expectation values of observables in
4.2 Supersolid phases
63
Figure 4.17: (a) Quantum fluctuations can lift the classical ground state degeneracy. (b) Quantum fluctuations can lower the energy of a classical excited state in such a way that it becomes a ground state. the ground state, can be described by a four site square unit cell. The first step is to find a classical ground state {Srclass } on the four site cluster with periodic boundary conditions defined by the unit cell. On each site r, the spin operator Sr is then replaced by a new spin operator Sr′ which is obtained from Sr by applying the rotation which transforms Srclass into 21 ez . After this transformation, the classical ground state corresponds to the ferromagnetic state with all the spins pointing in the z direction. The resulting Sr′ are then mapped onto bosonic operators ar by using the Holstein-Primakoff transformation: Sr′z = S − a†r ar s √ √ a†r ar ar ≃ 2Sar 2S 1 − Sr′+ = 2S s √ a†r ar √ ′− † Sr = ≃ 2Sa†r . 2Sar 1 − 2S
(4.12) (4.13) (4.14)
The linear spin-wave approximation consists in supposing that the quantum fluctuations around the classical ground state are small enough (ha†r ar i ≪ 2S in the ground state) and in replacing the square root by its lowest order expansion. The last step of the linear spin wave approximation is to neglect all the terms in the Hamiltonian which contains more than two bosonic operators. The resulting Hamiltonian is quadratic in bosonic operators and can be diagonalized by using a Fourier transformation and a Bogoliubov transformation. It is then possible to evaluate the ground state expectation value of any observable, and in particular the order parameters, the magnetization per site and the energy. If the classical ground state is unique, or if its degeneracy is due to a symmetry of the Hamiltonian, this procedure is in principle sufficient. However, if the classical ground state is degenerate and the degeneracy is not related to a symmetry, one can expect quantum fluctuations to lift this degeneracy (figure 4.17 (a)). One could also expect the
64
Correlated hopping, a minimal model
quantum fluctuations to change the energy of a higher energy classical state in such a way that it could become the ground state (figure 4.17 (b)). It is however not necessary to check every possible classical state, as only the classical states which are stationary points of the classical energy can give a semi-classical ground state. Therefore in principle, the semi-classical treatment should be performed for every state which is a stationary point of the classical energy, the resulting semi-classical energies should be compared, and the minimum should be extracted. In practice, a numerical method is used to find several (∼ 100) different classical states which are local minima of the classical energy. For each of these states the semi-classical treatment is performed and only the results which minimize the semi-classical energy are kept as ground states. µ − V and µ − t′ phase diagrams The semi-classical results are shown in figure 4.18. The first column presents the density n as a function of the chemical potential µ and the correlated hopping t′ at fixed nearest neighbor repulsion V . The corresponding µ − t′ phase diagrams at fixed nearest neighbor repulsion V are shown in the second column. Each row corresponds to a different nearest neighbor repulsion. When V = 0 (first line of figure 4.18), the phase diagram is dominated by a superfluid phase. There is a phase separation region at low n and large t′ , in agreement with the exact diagonalization and quantum Monte Carlo results discussed previously. The superfluid phase corresponds to a ferromagnetic state, with a non-zero component in the xy plane, as depicted in figure 4.16 (c) and (d). For 0 < V < 2, a supersolid phase appears in the large µ and large t′ region of the phase diagram and extends towards lower t′ when V increases, as shown in the second line of figure 4.18. In the supersolid phase, the state is close to a Néel state, but the spins acquire a ferromagnetically ordered non-zero component in the xy plane. This state is similar to the state depicted in figure 4.16 (e) and (f), except that the spins pointing down have also a non-zero xy component. For V > 2, as shown in the last two lines of figure 4.18, a n = 1/2 insulating (solid) phase appears from t′ = 0 when V = 2 and extends towards larger t′ when V increases. In this insulating phase, which appears as a plateau in the n versus µ curves, the system is in the Néel state shown in figure 4.16 (a) and (b). At V = 2, the supersolid region reach the limit t′ = 0, and becomes larger when V increase. The order parameters are shown in figure 4.19 for two reprentative cases t′ = 0.9 V = 2.1 (left panel) and t′ = 0.8 V = 3 (right panel). Figure 4.20 presents essentially the same information as figure 4.18, however it is useful in order to better understand the whole phase diagram. It presents the same quantities as figure 4.18 but as a function of the nearest neighbor repulsion V and at fixed correlated hopping t′ . These phase diagrams present two interesting features. Firstly, the large V and large t′ region is dominated by a very large supersolid phase that appears for densities above 1/2. Secondly, this supersolid phase is stable even at very small V , where the checkerboard solid is not stabilized. This feature is strongest at t′ = 1, where the semi-classical approximation predicts a stable supersolid phase for all V > 0 while the solid phase is stabilized only when V is larger than a critical value VcSS ≃ 2.3 (see figure 4.20). In other words, the
4.2 Supersolid phases
Figure 4.18: Semi-classical results obtained on a N = 150 × 150 lattice. Each row corresponds to a given nearest neighbor repulsion V . The first column presents the density n as a function of the chemical potential µ and the correlated hopping t′ at fixed V . The second column presents the µ − t′ phase diagrams at fixed V . White regions denotes phase separation, darker regions corresponds to superfluid phases, supersolid phases and solid (insulating) phases, as shown in the second column.
65
66
Correlated hopping, a minimal model
Figure 4.19: (a) Density n, (b) static structure factor S(π, π) and (c) superfluid order parameter |hbr i| with r in the A sub-lattice (plain lines) and B sub-lattice (dashed lines) as a function of the chemical potential for two representative cases. Results obtained with a semi-classical calculation. Left panel: supersolid without neighboring solid at t′ = 0.9 and V = 2.1. Right panel: t′ = 0.8 and V = 3.
correlated hopping can stabilize large supersolid phases. But this effect is so strong that for a given parameter set (t′ , V ), the correlated hopping can stabilize a supersolid phase even when there is no neighboring solid phase as µ is changed. Note that correlated hopping appears to be crucial for this physics to be realized. Indeed, the same model where a standard next-nearest neighbor hopping replaces the correlated hopping can also stabilize supersolid phases, but in that case the supersolid phases are never stabilized without an adjacent solid phase (see section 4.3.2). Upon increasing the chemical potential µ, the semi-classical approximation predicts first order transitions from superfluid to solid and from superfluid to supersolid, while the transitions from supersolid to superfluid and solid to supersolid are second order. Note that the second order transition from solid to superfluid for V > 2 and t′ > 0 becomes a first order transition from solid to superfluid when t′ = 0 (figure 4.18). An attentive reader may have noticed that the paired superfluid phase predicted by exact diagonalization and quantum Monte Carlo in section 4.1 is not present in the phase diagram. This has to be understood as a limitation of the semi-classical approximation which cannot describe that type of phase.
4.2 Supersolid phases
Figure 4.20: Semi-classical results obtained on a N = 150 × 150 lattice. Each row corresponds to a given nearest correlated hopping t′ . The first column presents the density n as a function of the chemical potential µ and the nearest neighbor repulsion V at fixed t′ . The second column presents the µ−V phase diagrams at fixed t′ . White regions denotes phase separation, darker regions corresponds to superfluid phases, supersolid phases and solid (insulating) phases, as shown in the second column.
67
68
Correlated hopping, a minimal model
Figure 4.21: Maximum number of Holstein-Primakoff bosons maxr {ha†r ar i} obtained with the semi-classical approximation on a N = 150 × 150 lattice as a function of the chemical potential µ and the correlated hopping t′ for V = 3.0. The color coding indicates which phase is realized for the corresponding parameters V, t′ and µ. Validity of the semi-classical approximation and comparison with the classical approximation The main approximation in the semi-classical treatment of the model consists in replacing the square root appearing in the Holstein-Primakoff transformation by its lowest order expansion around a†r ar /(2S) = 0 s a†r ar ≃ 1. (4.15) 1− 2S As a thumb rule, this approximation is expected to be qualitatively correct provided that the ground state expectation value ha†r ar i is small compared to 2S. This expectation value has been evaluated with the other quantities and it satisfies ha†r ar i < 0.2 for any V, t′ , µ and r, which is small compared to 2S, even for spins 1/2. Interestingly, ha†r ar i is larger around the phase transitions between two phases with and without solid order, as shown in figure 4.21 which presents maxr {ha†r ar i} as a function of µ and t′ for V = 3.0. Therefore, one may expect that the phase transitions predicted by the semi-classical approximation are slightly shifted with respect to the transitions obtained with an exact treatment of the model. In order to determine the phase diagram for more complicated models, it can be necessary to use only the classical approximation instead of the semi-classical approximation. It is therefore interesting to know what differences should be expected. Close to a phase transition, within the classical approximation, upon increasing the chemical potential µ the energy of the first excited state crosses the ground state energy at a critical value µc
4.2 Supersolid phases
69
Figure 4.22: Sketch of the classical (plain lines) and semi-classical (dashed lines) energies versus the chemical potential µ for two low energy states close to a phase transition. (a) Level crossing at a second order transition. The quantum fluctuations change the energies of the two state by the same amount. (b) Level crossing at a first order transition. The change in energy due to quantum fluctuations is not the same for the two states. and becomes the new ground state. If the transition is continuous, the two states have to be identical at the transition point and the effect of the quantum fluctuations will be the same for both states. In particular their energy will change by the same amount and the transition will occur at the same µc in the semi-classical approximation (see figure 4.22 (a)). By contrast, at a first order transition, both states are different. Therefore, the two energies will in general not change by the same amount and the intersection will occur at another µc in the semi-classical approximation (see figure 4.22 (b)). In addition to this effect, one should also expect renormalisation of all the measured quantities, due to the quantum fluctuations. These effects can be seen in figure 4.23, which presents the classical and semi-classical density as a function of the chemical potential for V = 3 and t′ = 0.5. It clearly shows a renormalisation of the density upon adding the quantum fluctuation. The phase transitions also behave as expected, with the second order transition occurring at the same value of the chemical potential for both the classical and semi-classical approximation and the first order transitions occurring at slightly different chemical potentials. In conclusion, as long as we are only interested in phase diagrams, the classical approximation will give the same results as the semi-classical approximation, except at first order transitions where one should expect small differences.
4.2.2
Quantum Monte Carlo
The semi-classical approximation is really useful to map out the entire phase diagram and indicates the regions where the different phases can be found. However, it is a strong approximation, therefore it is interesting to check these predictions with a more reliable method. This section presents quantum Monte Carlo results obtained by K. P. Schmidt with the algorithm described in section 4.1.4. As it is not my work, I will only briefly
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Correlated hopping, a minimal model
Figure 4.23: Comparison of the classical (dashed lines) and semi-classical (plain lines) density n as a function of the chemical potential for V = 3 and t′ = 0.5. The vertical dashed lines denote classical phase transitions while the vertical plain lines denote semi-classical phase transitions. present the main results. All the details can be found in reference [66]. The goals of this quantum Monte Carlo calculation are three-fold. Firstly, these results are used to confirm the stability of the supersolid phase predicted by the semi-classical approximation, even in the absence of a neighboring solid. Secondly, the predictions regarding the order of the phase transitions are checked. In addition, quantum Monte Carlo is also used to study the finite temperature transitions out of the supersolid phase. Finally, the comparison between quantum Monte Carlo and semi-classical results can give an indication about the reliability of the semi-classical approximation. The various phases are determined by studying the density n, the solid order parameter, which is given by the static structure factor at the wave vector (π, π) S(k = (π, π)) =
1 X ′ hnr′ nr ieik(r −r) 2 N ′
(4.16)
r,r
and the superfluid order parameter, given by the superfluid stiffness ρs =
1 (Wx2 + Wy2 ). 2βN
(4.17)
Here Wx and Wy are the total winding numbers in x and y directions. All the simulations are done on a N = L × L sites square lattices with up to 24 × 24 sites. The zerotemperatures properties are obtained at an inverse temperature β = 2L. Figure 4.24 presents the zero-temperature phase diagram for t′ = 0.95 as a function of 1/V and µ/V while simulation results for V = 2.2 (1/V = 0.45) are shown in the left panel of figure 4.25. The intermediate V region is dominated by a large supersolid phase appearing at densities above 1/2, which extends far below the critical value VcCB =
4.2 Supersolid phases
71
Figure 4.24: Zero-temperature phase diagram for t′ = 0.95 as a function of 1/V and µ/V . White squares (black circles) denote first (second) order phase transitions deduced from quantum Monte Carlo. White dashed lines are semi-classical results. The other lines are only guide to the eye.
2.38 (1/VcCB = 0.42) for the development of the checkerboard solid, down to VcSS = 1.74 (1/VcSS = 0.57). The evolution of the supersolid phase from the limit t′ = 0 is shown in figure 4.26 which presents the zero-temperature phase diagram for V = 2.8 as a function of t′ and µ. Simulation results for t′ = 0.75 are shown in the right panel of figure 4.25. Although semi-classical predictions clearly overestimates the supersolid regions, both phase diagrams show that quantum Monte Carlo and the semi-classical approximation agree on the overall structure of the phase diagram. In particular, the presence of the large supersolid region, which can exist without a neighboring solid phase is correctly predicted by the semi-classical approximation. According to quantum Monte Carlo simulations (figure 4.26), the phase transition between the solid and the supersolid falls into the conventional superfluid-insulator universality class, as expected since the gapped excitations of the solid order are not expected to influence the nature of the quantum phase transition, and in agreement with the results recently reported for a spin model [75]. By contrast, the transition between the supersolid and the superfluid is first order below t′ ≃ 0.25 and seems to be continuous above. For the sizes available, the extracted critical exponents are consistent with the continuous transition being in the 3D Ising universality class. This might be a finite size effect though: The long-wavelength gapless excitations of the superfluid phase are expected to change the universality class of this quantum phase transition [76], and could therefore give rise to a crossover phenomenon at large length scales. Interestingly, the semi-classical
72
Correlated hopping, a minimal model
Figure 4.25: (a) Density n, (b) static structure factor S(π, π) and (c) superfluid stiffness ρ as a function of the chemical potential for two representative cases. Left panel: supersolid without neighboring solid at t′ = 0.95 and V = 2.2 (1/V = 0.455). Right panel: t′ = 0.75 and V = 2.8 (1/V = 0.36). Due to remarkably small size effects, the results obtained with the various lattice sizes are almost indistinguishable.
approximation agrees on the first or second order nature of the transitions, except for the first order transition from solid to supersolid below t′ ≃ 0.25, where the semi-classical approximation predicts a second order transition for all t′ > 0. Quantum Monte Carlo can be used to determine the finite temperature phase transitions out of the supersolid phase. In general one expects two melting transitions for a supersolid: a Kosterlitz-Thouless (KT) transition when the superfluid stiffness vanishes, and another one when the solid order melts, whose universality class depends on the type of order [77]. This question has already been addressed, first for hard-core bosons on the triangular lattice [78], then for a spin model with anisotropic exchange integrals [76]. In both cases, two phase transitions have indeed been observed. In the case of the spin model, closer to the present case since the melting of the solid is in the Ising universality class, the KT transition has been found to lie always below the Ising transition, suggesting that the supersolid needs a solid phase to develop. In the present case, we expect the situation to be quite different since at zero temperature a supersolid can exist without a solid. The quantum Monte Carlo finite temperature phase diagrams are shown 4.27 for two
4.2 Supersolid phases
Figure 4.26: Zero-temperature phase diagram for V = 2.8 as a function of t′ and µ. White squares (black circles) denote first (second) order phase transitions deduced from quantum Monte Carlo. White dashed lines are semiclassical results. The other lines are only guide to the eye.
Figure 4.27: Finite temperature phase diagrams as a function of the chemical potential µ presenting the melting of the supersolid phase. (a) t′ = 0.75 and V = 2.8: both solid and supersolid phases are stable at zero-temperature. (b) t′ = 0.95 and V = 2.2: at zero temperature the supersolid phase is stable without an adjacent solid phase. Black squares and white circle are quantum Monte Carlo results while the lines are only guides to the eye.
73
74
Correlated hopping, a minimal model
representative cases. The case where both solid and supersolid phases are stable at zerotemperature (t′ = 0.75 and V = 2.8) is shown in figure 4.27 (a), while figure 4.27 (b) presents the case where the supersolid phase is present without an adjacent solid phase (t′ = 0.95 and V = 2.2). The first phase diagram indeed shows two transition lines which goes smoothly to T = 0 upon approaching either the solid phase (KT) or the superfluid phase (Ising). They cross in the middle of the supersolid phase, defining a region close to the solid where the KT transition is below the Ising transition, as in reference [76], but also a region close to the superfluid where the Ising transition is below the KT transition. This is even more dramatic for the second case in figure 4.27 (b), where the melting of the Ising order occurs entirely inside the superfluid phase.
4.2.3
Summary
By using a semi-classical treatment of the model, we could map out the whole phase diagram of the model (4.1). The main outcome of this analysis is that a large supersolid region is stabilized when correlated hopping dominates the kinetic energy. This has to be contrasted with the limit where only simple hopping is present where no supersolid phase is stabilized. This effect is so strong that the supersolid phase is stabilized even when the nearest neighbor repulsion is too small to stabilize a solid phase. Regarding the phase transitions, the semi-classical approach predicts first order transitions from superfluid to both solid and supersolid upon increasing the chemical potential, while the transitions from supersolid to superfluid and solid to supersolid are second order. These predictions have been confirmed by quantum Monte Carlo simulations, except for the transition from solid to supersolid below t′ ≃ 0.25, which is first order according to the quantum Monte Carlo results. Another important conclusion of this analysis is that the classical and semi-classical results are in qualitative agreement with quantum Monte Carlo results. Firstly, both classical and semi-classical phase diagrams are really similar, differing only slightly at the first order transitions. The main difference between these two approximations is a small renormalization of all the measured quantities, which is not so important if we are only interested in determining the phase diagram. Secondly, although the classical and semiclassical approximations overestimate the supersolid regions, they can correctly predict the different phases obtained with quantum Monte Carlo. In particular, the predictions concerning the solid phases are in good agreement with quantum Monte Carlo, although also slightly overestimated. Concerning the phase transitions, classical and semi-classical results are not perfect but in most cases, they can predict whether a phase transition is first or second order. This is particularly interesting for more elaborate models, which cannot be investigated with quantum Monte Carlo due to the minus sign problem. For these models, one could expect the classical approximation to give qualitatively correct results. Regarding the finite temperature phase transitions out of the supersolid, quantum Monte Carlo predicts in general two distinct phase transitions, a Kosterlitz-Thouless (KT) transition corresponding to the disappearance of the superfluid order and an Ising transition when the solid order melts. Upon increasing the temperature, the weakest order
4.3 Generalizations of the model
75
melts first. At a given value of the chemical potential where at zero temperature the system is in a supersolid phase close to a solid phase, the superfluid order is weaker and the KT transition occurs at lower temperature than the Ising one. In contrast, when the system is closer to a superfluid phase, the solid order is weaker and the Ising transition occurs at lower temperature than the KT one. All these predictions show that correlated hopping strongly favors supersolid phases, which can even be stabilized without an adjacent solid. We suspect that correlated hopping is crucial for this physics to be realized, but it could as well be a consequence of the next-nearest neighbor hopping term contained in the correlated hopping. In order to answer this question, one should study the same model where correlated hopping is replaced by a next-nearest neighbor hopping and compare the phase diagrams. This is the subject of section 4.3.2.
4.3
Generalizations of the model
Following the approach which consist in studying a simple model in order to understand some peculiar features of the effective Hamiltonian for the Shastry-Sutherland model, several generalizations of the model (4.1) can be done. Firstly, the various kinetic terms in the effective Hamiltonian have different signs. An obvious generalization consists in studying model (4.1) but with negative t′ (see section 4.3.1). Secondly, in order to check if the correlated hopping is crucial for the realization of a supersolid phase without an adjacent solid phase, we will study the model (4.1) where correlated hopping is replaced by a next-nearest neighbor hopping (see section 4.3.2). This section concentrates on the zero-temperature phase diagrams of these models obtained by a classical approximation (see appendix D). In general, all the calculations are performed numerically on all non-equivalent four sites clusters with periodic boundary conditions, although in some cases we checked for higher periodicities by doing the calculation on larger clusters having up to eight sites.
4.3.1
Negative t′
In the previous section, the phase diagram of Hamiltonian (4.1) has been thoroughly investigated but only for the case of negative hopping and correlated hopping amplitudes (positive t and t′ ). But there is no reason to restrict the investigation of Hamiltonian (4.1) to positive t and t′ . This might be even very interesting. On the one hand because it induces frustration among the kinetic terms and on the other hand because it is the general situation for realistic systems like for example the Shastry-Sutherland discussed in the next chapter. It is not necessary to investigate the four different sign combinations though, as the cases with t < 0 are directly related to the cases with t > 0 by the transformation −br if r ∈ A br → (4.18) br if r ∈ B where A and B are the two sub-lattices of the square lattice. Indeed, applying this transformation to Hamiltonian (4.1) leaves all the terms invariant, except the hopping
76
Correlated hopping, a minimal model
term which changes sign. Transformation (4.18) therefore maps Hamiltonian (4.1) with negative t onto itself but with positive t. Note that the observables are in general not invariant under this transformation, and they have to be transformed between the two cases. For the model investigated previously (t, t′ > 0), both the hopping and the correlated hopping act as ferromagnetic couplings between the xy components of the spins (see section 4.2.1). In any superfluid or supersolid phase, the system can easily minimize the kinetic energy by ordering the xy components of the spins ferromagnetically. In terms of bosons, it corresponds to a phase with all the particles having the same phase. By contrast, when t > 0 and t′ < 0, the coupling between the xy components of nearest neighbor spins is still ferromagnetic, but becomes antiferromagnetic for next-nearest neighbor spins. Although in the limits t ≫ t′ and t ≪ t′ the system can still find an ordering which minimizes the kinetic energy, this is not true anymore when t ≃ t′ . As a consequence, one should expect the phases with non-zero xy components of the spins (superfluid and supersolid) to have higher energies than in the case t, t′ > 0, while the energy of the solid phase should not change. This should result in a phase diagram with less superfluid and supersolid regions than in the unfrustrated case. In order to check these predictions, a classical approximation was used to build the zero-temperature phase diagram of Hamiltonian (4.1) with t > 0 and t′ < 0. All the calculations were performed on four-site clusters, but because the frustration of the model could lead to larger periodicities in the superfluid and supersolid phases, a calculation with clusters having up to eight sites was done in each different phase. Similarly to the previous case, the energy scale is fixed by t + |t′ | = 1. The resulting phase diagrams as a function of the correlated hopping t′ and the chemical potential µ for various values of the nearest neighbor repulsion V are shown in figure 4.28, while the structure of the various superfluid and supersolid phases are sketched in figure 4.29. The density and the order parameters are shown in figure 4.30 as a function of the chemical potential for two reprentative cases t′ = −0.6 V = 1 (left panel) and t′ = −0.6 V = 2 (right panel). When V = 0, the phase diagram is dominated by superfluid phases. However, because of the competition between hopping and correlated hopping, the phase diagram is already more interesting than in the unfrustrated case (t, t′ > 0). In the limit where simple hopping dominates (t′ ≃ 0), the system stabilizes a superfluid phase (SF 2) with a ferromagnetic ordering of the xy components of the spins (i.e. all particles have the same phase), corresponding to the superfluid phase obtained for the unfrustrated system. In the limit where correlated hopping dominates (t′ ≃ −1), the next-nearest neighbor antiferromagnetic interaction between the xy components of the spins can be minimized by having a Néel ordering of the xy components of the spins in each square sub-lattice (SF 1). The directions of the spins in the two sub-lattices are independent. This can be understood, by thinking of the simple hopping as a ferromagnetic nearest neighbor interactions between the xy components of the spins. For each spin Sr , the xy components of the two spins Sr+x y y x x and Sr+y are in opposite direction. Therefore Srx Sr+x + Sry Sr+x = −(Srx Sr+y + Sry Sr+y ) and the contributions in x and y directions simply cancel each other. As the simple hopping is the only coupling in the xy plane between the two sub-lattices, they become effectively decoupled.
4.3 Generalizations of the model
77
Figure 4.28: Zero-temperature phase diagrams as a function of the correlated hopping t′ and the chemical potential µ for various nearest neighbor repulsion V . White (black) circles corresponds to first (second) order transitions obtained with the classical approximation. The lines are only guides to the eyes .
When V increases, a first supersolid phase is stabilized by the nearest neighbor repulsion at densities n > 1/2. This phase (SS 1) appears in the region of transition between the superfluid (SF 1) and the n = 1 insulating phase and extends towards smaller values of the chemical potential as V increase. As depicted in figure 4.29, this supersolid phase is based on a checkerboard solid order with one sub-lattice completely filled and the other which is partially filled and presents a Néel ordering of the xy components of the spins. This phase clearly minimizes the correlated hopping which acts as a next-nearest neigh-
78
Correlated hopping, a minimal model
Figure 4.29: Structure of the various superfluid (SF) and supersolid (SS) phases. The density nr at a given site r is represented by the intensity of circle, which goes from white for nr = 0 to black for nr = 1. When nr 6= 0, 1, the particles acquire a superfluid component whose phase is represented as the angle between the black arrow and the x axis. In terms of spins, the arrows corresponds to the xy components, while the color of the circle corresponds to the z component.
Figure 4.30: (a) Density n, (b) static structure factor S(π, π) and (c) superfluid order parameter |hbr i| with r in the A sub-lattice (plain lines) and B sub-lattice (dashed lines) as a function of the chemical potential for two representative cases. Results obtained with a classical calculation. Left panel: supersolid without neighboring solid at t′ = −0.6 and V = 1. Right panel: t′ = −0.6 and V = 2.
4.3 Generalizations of the model
79
bor antiferromagnetic interactions between the xy components of the spins. At larger V , another supersolid phase (SS 2) appears in the middle of the phase diagram at densities n < 1/2. This phase is also based on a checkerboard solid order but the xy components of the spins are ferromagnetically ordered. Between V = 1 and V = 1.5 a n = 1/2 solid phase with checkerboard order is stabilized in the middle of the phase diagram. As V continues to increase, the superfluid phases are slowly replaced by solid and supersolid (SS 1) phases. By comparing these results to those of the unfrustrated model (t, t′ > 0), one can observe several differences. As expected the superfluid regions are smaller at intermediate values of the correlated hopping (t′ ≃ −0.5) while the solid regions are larger and appear at a smaller value of the nearest neighbor repulsion V . In contrast to what was expected, the supersolid phase (SS 1) is larger and can also be stabilized even when the nearest neighbor repulsion is not large enough to stabilize the solid phase. Although it contradicts our expectations, this can be easily understood by looking at the structure of the supersolid depicted in figure 4.29. Since one of the sub-lattices is completely filled, the spins of this sub-lattice have no xy components and the simple hopping term has no effect on this state. The correlated hopping term can therefore be fully minimized by having the other sub-lattice with a Néel ordering of the xy components. In conclusion, only the superfluid phases suffer from the frustration of the system and become unstable, leaving a phase diagram dominated by supersolid and solid phases. In comparison to the unfrustrated model (t, t′ > 0), this phase diagram is much more complex, with several different phases characterized by interesting superfluid orders. This is clearly a consequence of the frustration induced by the competition between simple hopping and correlated hopping. Regarding the validity of the results, this frustration could become problematic. Indeed, it could stabilize other types of phases, which cannot be described by the classical approximation. It would be therefore interesting to compare these results to the predictions of other methods. Quantum Monte Carlo suffers from the minus sign problem with this model, but one could use exact diagonalizations. The results given here can anyway be useful to know which types of phase could be expected and in what region of the parameter space to look for them.
4.3.2
Next-nearest neighbor hopping
The previous sections have shown that model (4.1) can stabilize a supersolid phase even when there is no adjacent solid phase. This effect was clearly a consequence of the correlated hopping. However, correlated hopping might not be crucial for this effect, which could as well be a consequence of the next-nearest neighbor hopping term contained in the correlated hopping. In order to answer this question, one has to study model (4.1) where correlated hopping is replaced by a next-nearest neighbor hopping with amplitude t2 : X X X H = −µ nr + V nr nr+δ r
−t
X X r
δ=±x,±y
r
δ=x,y
b†r br+δ
− t2
XX X r
δ=±x δ′ =±y
b†r br+δ+δ′ .
(4.19)
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Correlated hopping, a minimal model
In terms of the spins used in the classical approximation (defined in Eqs. (4.6),(4.7) and (4.8)), the nearest neighbor hopping becomes a ferromagnetic or antiferromagnetic interaction between the xy components of next-nearest neighbor spins when t2 > 0 or t2 < 0 respectively. In the following we concentrate on the two cases t > 0, t2 > 0 and t > 0, t2 < 0. The cases with t < 0 are not treated because they can directly be obtained from the cases with t > 0 by using the transformation (4.18). The energy scale will be fixed by t + |t2 | = 1. Note that the unfrustrated (ferromagnetic) side of this model has been studied by Chen and collaborators [79] by quantum Monte Carlo. This enables us to compare the numerically unbiased results with the semi-classical approach we are applying. Additionally, we can also investigate with our approach the frustrated side similarly to the case of correlated hopping studied in the last section. Case t2 > 0 The zero-temperature classical phase diagram is shown in figure 4.31 as a function of the next-nearest neighbor hopping t2 and the chemical potential µ for various values of the nearest neighbor repulsion V . The density and the order parameters are shown in figure 4.32 as a function of the chemical potential for t2 = 0.5 and V = 3. When the nearest neighbor repulsion is smaller than 2 (not shown), the phase diagram displays only a superfluid phase with a ferromagnetic ordering of the xy components of the spins (i.e. all particles have the same phase), as depicted in figure 4.29 (SF 2). When V = 2, a n = 1/2 checkerboard solid phase appears at µ = 4 which is stable for all values of the next-nearest neighbor hopping t2 > 0. Simultaneously, a supersolid phase appears at t2 = 1 and it is stable for at all densities except n = 1/2. This phase has a checkerboard solid order and the xy components of the spins are ferromagnetically ordered, as depicted in figure 4.29 (SS 2). When V increases, the main effect is that the solid and supersolid phases slowly replace the superfluid phase. Note that as far as we can tell, these results seem consistent with the quantum Monte Carlo results by Chen and collaborators. By comparing these results to those of the model with correlated hopping (discussed in section 4.2.1), one can observe several similarities but also some interesting differences. The phases stabilized by both models are the same, with a solid order which is always based on a checkerboard pattern and a superfluid order which is always uniform. This is not surprising as the solid order is a consequence of the nearest neighbor repulsion which is the same for both models and the uniform superfluid order is a consequence of the kinetic terms which, for both models, act as ferromagnetic interactions for the xy components of the spins. Another similarity is the strong tendency to stabilize large supersolid regions when the kinetic energy is dominated by either correlated hopping or next-nearest neighbor hopping. Finally, the phase transitions at densities n > 1/2 are second order in both models. Regarding the differences between the two models, the supersolid phases stabilized by correlated hopping appear only at densities n > 1/2, while they appear at any densities with next-nearest neighbor hopping. This can be understood by realizing that the correlated hopping corresponds to a next-nearest neighbor hopping if a neighboring site is occupied, while it has no effect if the neighboring sites are empty. Therefore at high den-
4.3 Generalizations of the model
81
Figure 4.31: Zero-temperature phase diagrams obtained with a classical calculation as a function of the next-nearest hopping t2 > 0 and the chemical potential µ for various nearest neighbor repulsion V . All the transitions are second order, except at t2 = 0 where the transitions between superfluid and solid are first order. sities in a checkerboard supersolid correlated hopping and next-nearest neighbor hopping have the same effect, while at low density, the correlated hopping has no effect. This argument also explains why the phase transitions differ for densities n < 1/2. They are second order with a finite next-nearest neighbor hopping and first order with correlated hopping. Finally, the most important difference is that correlated hopping can stabilize a supersolid phase even when the nearest neighbor repulsion is too small to stabilize a solid phase, while with next-nearest neighbor hopping the supersolid phases are only stabilized with an adjacent solid phase. Case t2 < 0 Figure 4.33 presents the classical phase diagram as a function of the next-nearest neighbor hopping t2 and the chemical potential µ for various values of the nearest neighbor repulsion V . The density and the order parameters are shown in figure 4.35 as a function of the chemical potential for the two cases t′ = −0.6 V = 2 (left panel) and t′ = −0.3 and V = 2 (right panel). When V = 0, the phase diagram is dominated by two superfluid phases. In the region where nearest neighbor hopping dominates, the system stabilizes a superfluid phase with a ferromagnetic ordering of the xy components of the spins, corresponding
82
Correlated hopping, a minimal model
Figure 4.32: (a) Density n, (b) static structure factor S(π, π) and (c) superfluid order parameter |hbr i| with r in the A sub-lattice (plain lines) and B sub-lattice (dashed lines) as a function of the chemical potential for V = 3 and t2 = 0.5. Results obtained with a classical calculation. to the superfluid phase obtained with t2 > 0 and depicted in figure 4.29 (SF 2). This phase clearly minimizes the nearest neighbor hopping which acts as ferromagnetic coupling between the xy components of nearest neighbor spins. In the other limit, where nextnearest neighbor hopping dominates, the system stabilizes a superfluid phase with a Néel ordering of the xy components of the spins in each sub-lattice, as depicted in figure 4.29 (SF 1). In this phase the directions of the spins in the two sub-lattices are independent. This phase minimizes the next-nearest neighbor hopping which acts as antiferromagnetic coupling between the xy components of next-nearest neighbor spins. When V > 0.7, a solid phase with checkerboard order appears at density n = 1/2 for intermediate values of t2 . Simultaneously, the system realizes two supersolid phases, separated by the solid phase. The first one, which appears at densities n > 1/2, has a checkerboard solid order with one sub-lattice completely filled and the other which is partially filled and presents a Néel ordering of the xy components of the spins, as depicted in figure 4.29 (SS 1). The second supersolid phase is obtained from the first one by the particle-hole transformation. It has also a checkerboard solid order but one sub-lattice is completely empty and the other is partially filled with a Néel ordering in the xy plane. When V increase, the size of both solid and supersolid regions increases, and they slowly replace the superfluid phases. In order to complement the understanding of the whole phase diagram, figure 4.34 presents the zero-temperature phase diagram as a function of the nearest neighbor repulsion V and the chemical potential µ for a next-nearest neighbor hoping t2 = −0.35
4.3 Generalizations of the model
83
Figure 4.33: Zero-temperature phase diagrams as a function of the nextnearest neighbor hopping t2 < 0 and the chemical potential µ for various nearest neighbor repulsion V . White (black) circles corresponds to first (second) order transitions obtained with the classical approximation. The lines are only guides to the eyes .
(a) and t2 = −0.25 (b). Note that the information is the same as in figure 4.33. A comparison to the model with correlated hopping (see section 4.3.1) leads to essentially the same conclusions as for the unfrustrated models t2 > 0 and t′ > 0. Both systems stabilize the same type of phases and have a strong tendency to stabilize supersolid phases when the kinetic energy is dominated by t2 or t′ . The main result is that this model does not stabilize a supersolid phase without a neighboring solid phase.
84
Correlated hopping, a minimal model
Figure 4.34: Zero-temperature phase diagrams as a function of the nearest neighbor repulsion V and the chemical potential µ at fixed next-nearest neighbor hoping t2 = −0.35 (a) and t2 = −0.25 (b).
Figure 4.35: (a) Density n, (b) static structure factor S(π, π) and (c) superfluid order parameter |hbr i| with r in the A sub-lattice (plain lines) and B sub-lattice (dashed lines) as a function of the chemical potential for two representative cases. Left panel: t′ = −0.6 and V = 2. Right panel: t′ = −0.3 and V = 2. Results obtained with a classical calculation.
4.4 Conclusion
4.4
85
Conclusion
In this chapter we studied a model of hard-core bosons on the square lattice with correlated hopping. In the first part of the chapter, we focused on the low density limit of the phase diagram. We used exact diagonalizations and quantum Monte Carlo simulations to establish the existence of an unconventional pairing phase in a region of the phase diagram where correlated hopping dominates the kinetic energy. Both correlated hopping and a nearest neighbor repulsion are crucial for the realization of this phase. The correlated hopping acts as an attractive interaction between the particles. Therefore, if the nearest neighbor repulsion is too small, a macroscopic portion of the particles tend to glue together and the system is unstable against phase separation. If the nearest neighbor repulsion is too large, the bound states between pairs of particles are destroyed and a standard superfluid is realized. The major outcome of this part is that there is a finite window of nearest neighbor repulsion where the correlated hopping can form stable pairs of particles which condense and the repulsion is large enough to prevent phase separation. In the next part of this chapter, we studied the whole phase diagram with a semiclassical approximation and quantum Monte Carlo simulations, focusing on the possible realization of a supersolid phase. The main result of this analysis is that correlated hopping strongly favors large supersolid phases. This effect is so strong that the supersolid can be stabilized even when the nearest neighbor repulsion is too small to stabilize a solid phase. By studying a similar model where correlated hopping was replaced by a next-nearest neighbor hopping, we could establish that correlated hopping is crucial for this effect. Indeed, even if the two models have similar phase diagrams and stabilize large supersolid phases, only the model with correlated hopping can stabilize a supersolid phase without an adjacent solid phase. For both correlated hopping and next-nearest neighbor hopping models, we also derived the phase diagrams for the case of frustrated kinetic couplings (t2 < 0 and t′ < 0). For both models, the phase diagrams are richer than in the unfrustrated cases and show various competing superfluid and supersolid phases. The superfluid phases become less stable because of the frustration which leads to phase diagram mostly dominated by solid and supersolid phases. However, as in the unfrustrated case only the model with correlated hopping can stabilize a supersolid phase without a neighboring solid phase. Thanks to the quantum Monte Carlo simulations, we could also check that the semiclassical approximation can correctly predict the different phases and produce a qualitatively correct phase diagram in the case of unfrustrated kinetic couplings. It is particularly efficient at describing the solid phases but it can also, in most cases, predict whether a phase transition is first or second order. The main problem seems to be that this approximation overestimates the size of the supersolid phases. The solid phases are also slightly overestimated. Regarding the experimental implications of these results, whenever correlated hopping dominates the kinetic energy, one can expect to find supersolid phases, even in regions where no plateau has been detected. In SrCu2 (BO3 )2 , such a domain exists above the 1/8 plateau, in a field range accessible to NMR, a technique well suited to detect translational symmetry breaking. However, one should stress that the efficiency of the correlated
86
Correlated hopping, a minimal model
hopping is strongly related to the geometry of the underlying solid order. In this simple model, only a checkerboard solid order is stabilized and it is perfect for this correlated hopping which can then make additional particles hop along the diagonals in the empty sub-lattice. In a more complex model, with longer range interactions and various correlated hopping terms, one should expect different types of solid order, possibly more complex. In these geometries, it is not sure that the various correlated hopping terms can be so efficient and stabilize large supersolid phases. We also found that correlated hopping can stabilize a small region of paired supersolid phase at low densities. Although not easily stabilized, this phase may be realized in SrCu2 (BO3 )2 at low magnetization, in agreement with the two-triplet bound state proposed by Momoi and Totsuka [53].
CHAPTER
5
The Shastry-Sutherland model In this chapter, we want to study the magnetic properties in the presence of a magnetic field of the spin-1/2 Heisenberg model on the Shastry Sutherland lattice, also known as Shastry-Sutherland model, which is defined by the Hamiltonian X X X Srz (5.1) Sr · Sr′ − B H = J′ Sr · Sr′ + J
≪r,r′ ≫
r
where the ≪ r, r ′ ≫ bonds build an array of orthogonal dimers while the < r, r ′ > bonds correspond to inter-dimer couplings (see figure 5.1). We consider only the antiferromagnetic case J, J ′ > 0 and we restrict to the values of the couplings J ′ /J < (J ′ /J)c1 ∼ 0.7 corresponding to the singlet phase in the absence of a magnetic field (see chapter 3). This chapter starts by a derivation of an effective model describing the low-energy degrees of freedom for the Shastry-Sutherland model in a magnetic field. This is done by first reformulating the Hamiltonian in terms of triplet operators in section 5.1.1, and by using perturbative continuous unitary transformations in section 5.1.2. The resulting effective Hamiltonian is then described in section 5.1.3. In section 5.2, we use a classical approximation to derive the zero-temperature phase diagram of the effective Hamiltonian. The results are then compared to previous calculations on the Shastry-Sutherland model and experimental implications are discussed
5.1 5.1.1
Effective Hamiltonian Triplet operators
In the region of the phase diagram we are interested in, the low energy physics is best described in terms of singlets and triplets (see chapter 3). It would be therefore useful if the system could be described in terms of particles, corresponding to the triplets, created on top of a vacuum corresponding to the state with all the dimers in the singlet state [9,80].
88
The Shastry-Sutherland model
Figure 5.1: (a) The Shastry-Sutherland lattice can be seen as a set of orthogonal dimer bonds (plain black lines) connected by inter-dimer bonds (dashed grey lines). (b) The sites are numbered from left to right in horizontal dimers and from bottom to top in vertical dimers.
In order to do this, one needs to introduce some formal definitions. We consider N dimers on a square lattice. The spin operators of the two spins 1/2 of the dimer sitting at site r of the square lattice are denoted by S1r and S2r . Note that the sites are numbered from left to right in horizontal dimers and from bottom to top in vertical dimers, as depicted in figure 5.1. For a given dimer at site r, the basis of common eigenvectors of z z S1r and S2r with eigenvalues σ1 and σ2 is denoted by {|σ1 σ2 ir |σ1 , σ2 =↑, ↓}, while the basis of the singlet and triplets is denoted by
1 |sir = √ (| ↑↓ir − | ↑↓ir ) 2 |t−1 ir = | ↓↓ir 1 |t0 ir = √ (| ↑↓ir + | ↑↓ir) 2 |t1 ir = | ↑↑ir.
(5.2) (5.3) (5.4) (5.5)
For the dimer at site r, we define the creation operators of triplets t†−1r , t†0r and t†1r as well as the vacuum |0ir in the following way: |0ir = |sir |t−1 ir = t†−1r |0ir |t0 ir = t†0r |0ir |t1 ir = t†1r |0ir.
(5.6) (5.7) (5.8) (5.9)
5.1 Effective Hamiltonian
89
The spin operators can then be written in terms of triplet operators: 1 + S1r = √ −t†1r + t−1r + t†1r t0r + t†0r t−1r 2 1 † † † − S1r = √ −t1r + t−1r + t0r t1r + t−1r t0r 2 1 z t0r + t†0r + t†1r t1r − t†−1r t−1r S1r = 2 1 + S2r = √ t†1r − t−1r + t†1r t0r + t†0r t−1r 2 1 − S2r = √ t1r − t†−1r + t†0r t1r + t†−1r t0r 2 1 † † † z −t0r − t0r + t1r t1r − t−1r t−1r . S2r = 2
(5.10) (5.11) (5.12) (5.13) (5.14) (5.15)
These spin operators will satisfy the usual commutation relations, provided the triplet operators satisfy the following commutation relations ∀r, r ′ and ∀α, β = 0, ±1 ! ! X [tαr , t†βr′ ] = δr,r′ δαβ 1 − t†γr tγr − t†βr tαr (5.16) γ
[tαr , tβr′ ] = 0
(5.17)
as well as the hard-core constraint tαr tβr = 0 ∀α, β = 0, ±1.
(5.18)
With these commutation relations, the triplet operators are neither bosons nor fermions nor hard-core bosons. However, when we restrict to the subspace with the dimers only in the singlet or triplet |t1 i states, the operators t1r become hard-core bosons. If we denote by A the sub-lattice of the horizontal dimers and B the sub-lattice of the vertical dimers, Hamiltonian (5.1) can be rewritten as X X z z H = −B (S1r + S2r )+J S1r · S2r r
+J
′
+J
′
X r∈A
X r∈B
r
{S2r · (S1r+x + S2r+x ) + (S1r + S2r ) · S1r+y }
{(S1r + S2r ) · S1r+x + S2r · (S1r+y + S2r+y )}
(5.19)
where r is now a site of the dimer square lattice. With the help of Eqs (5.10)-(5.18), the magnetization at site r can be written in terms of triplet operators z z S1r + S2r = t†1r t1r − t†−1r t−1r
as well as the intra-dimer interaction 3 S1r · S2r = − + t†0r t0r + t†1r t1r + t†−1r t−1r 4
(5.20)
(5.21)
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The Shastry-Sutherland model
and the inter-dimer interactions (r 6= r ′ ) S1r · (S1r′ + S2r′ ) =
1n† t ′ t0r t1r′ + t†0r t†1r′ t1r′ − t†−1r′ t0r t−1r′ − t†0r t†−1r′ t−1r′ 2 1r +t†0r′ t−1r t1r′ + t†−1r t†1r′ t0r′ + t†−1r′ t−1r t0r′ + t†−1r t†0r′ t−1r′
(5.22)
−t†1r′ t1r t0r′ − t†1r t†0r′ t1r′ − t†0r′ t1r t−1r′ − t†1r t†−1r′ t0r′
+t†1r t†1r′ t1r t1r′ + t†−1r t†−1r′ t−1r t−1r′ − t†−1r t†1r′ t−1r t1r′ − t†1r t†−1r′ t1r t−1r′ +t†1r t†0r′ t0r t1r′ + t†1r t†−1r′ t0r t0r′ + t†0r t†0r′ t−1r t1r′ + t†0r t†−1r′ t−1r t0r′
+t†0r t†1r′ t1r t0r′ + t†0r t†0r′ t1r t−1r′ + t†−1r t†1r′ t0r t0r′ + t†−1r t†0r′ t0r t−1r′ and S2r · (S1r′ + S2r′ ) =
1n † −t1r′ t0r t1r′ − t†0r t†1r′ t1r′ + t†−1r′ t0r t−1r′ + t†0r t†−1r′ t−1r′ 2 −t†0r′ t−1r t1r′ − t†−1r t†1r′ t0r′ − t†−1r′ t−1r t0r′ − t†−1r t†0r′ t−1r′
o
(5.23)
+t†1r′ t1r t0r′ + t†1r t†0r′ t1r′ + t†0r′ t1r t−1r′ + t†1r t†−1r′ t0r′
+t†1r t†1r′ t1r t1r′ + t†−1r t†−1r′ t−1r t−1r′ − t†−1r t†1r′ t−1r t1r′ − t†1r t†−1r′ t1r t−1r′ +t†1r t†0r′ t0r t1r′ + t†1r t†−1r′ t0r t0r′ + t†0r t†0r′ t−1r t1r′ + t†0r t†−1r′ t−1r t0r′ +t†0r t†1r′ t1r t0r′
+
t†0r t†0r′ t1r t−1r′
+
t†−1r t†1r′ t0r t0r′
+
t†−1r t†0r′ t0r t−1r′
o
.
With this operation, the Shastry-Sutherland Hamiltonian (5.1) describing spins 1/2 on the Shastry-Sutherland lattice is replaced by a Hamiltonian describing triplet particles on the dimer square lattice. Clearly, both Hamiltonians have the same symmetries. Because of the orthogonal arrangement of the dimers, both Hamiltonian are invariant under a translation of two dimer lattice spacings in both x and y directions. The z axis rotational symmetry of Hamiltonian (5.1) becomes a global U(1) gauge symmetry, since the rotation of all the spins by an angle θ about the z axis becomes the following global gauge transformation for the triplets t−1r → e−iθ t−1r t0r → t0r t1r → eiθ t1r .
5.1.2
(5.24) (5.25) (5.26)
Perturbative continuous unitary transformations
The goal of this section is to extract the relevant degrees of freedom by deriving an effective Hamiltonian for the low energy subspace in the limit of weakly coupled dimers (J ′ ≪ J). This consists of the 2N states having zero or one t1 triplet on each dimer site: Q nsubspace ′ 1r r t1r |0i with n1r = 0, 1 (see chapter 3). When J = 0, the effective Hamiltonian can be easily obtained by projecting the Shastry-Sutherland Hamiltonian on the low energy subspace, which is achieved by removing all the terms containing t−1r or t0r operators.
5.1 Effective Hamiltonian
91
However, when J ′ > 0 the problem is not so simple, as the inter-dimer couplings connect the states of the low-energy subspace to states containing also t0 and t−1 triplets. A solution consists in using a unitary transformation defined by a unitary operator U which transforms the Hamiltonian H into an effective Hamiltonian Heff = U † HU. This transformation should satisfy the constraints that the resulting effective Hamiltonian is U(1) invariant (z axis rotations) and conserves the total number triplet. Supposing that such a transformation exists, the effective Hamiltonian will not connect the low-energy states having only t1 triplets to other states having t0 or t−1 triplets. One can therefore study only the relevant degrees of freedom by projecting the Hamiltonian onto the lowenergy subspace. Although no known method can actually determine this unitary transformation exactly for the Shastry-Sutherland Hamiltonian, some methods can do this approximately. This is the case of the perturbative continuous unitary transformations (PCUT), which will be used in this work (see appendix F). For a given operator A, PCUT can be used to determine perturbatively and up to high-order in J ′ /J the transformed operator U † AU. In particular, the general form of the effective Hamiltonian obtained by PCUT and projected onto the low-energy subspace reads X X X Heff = U † HU = −µ b†r br + (5.27) hr1 ,··· ,rn b†r1 · · · b†rn/2 brn/2+1 · · · brn r
n=2,4,6,··· r1 ,··· ,rn
where the notation br ≡ t1r is used for convenience and the coefficients hr1 ,··· ,rn are obtained as a series in J ′ /J. Note that since the total magnetization is invariant under the transformation, the magnetic field B simply becomes a chemical potential µ = B. In this work, we have kept all the terms with up to n = 6 operators as well as the four-body interactions (nr1 nr2 nr3 nr4 ) which appear up to order 8 in J ′ /J. The coefficients hr1 ,··· ,rn have been evaluated up to order 12 and were then extrapolated using Padé extrapolants. The final value used for the coefficients was the average of all the valid extrapolants, while the error bars were estimated by the difference between smallest and largest extrapolant. For the two-body interactions (nr1 nr2 ), which dominate the physics at low densities (see below), we have kept more terms, namely those that appear up to order 10. Their coefficients were evaluated up to order 15 and then extrapolated with DlogPadé extrapolants. The final value was again chosen as the average of all the valid extrapolants and the error bars estimated by the difference between smallest and largest extrapolant. The resulting Hamiltonian contains more than 15000 processes. If the expansion is stopped at order 3, this Hamiltonian corresponds exactly to the Hamiltonian derived by Momoi and Totsuka [53] with standard perturbation theory. PCUT can also be used to evaluate other observables [81, 82]. Relevant observables z z are the magnetization on each site S1r and S2r as well as the total magnetization. In z z order to evaluate the transformed S1r and S2r we only kept the dominant contributions which are the terms linear in the density nr = b†r br X (1) z U † S1r U ≃ (5.28) sr,r′ nr+r′ r′
z U † S2r U ≃
X r′
(2)
sr,r′ nr+r′ .
(5.29)
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The Shastry-Sutherland model
Moreover, we only kept the terms at a distance |r ′| smaller than three dimer lattice (1) (2) spacing. The coefficients sr,r′ and sr,r′ were evaluated up to order 10 in J ′ /J and then extrapolated using DlogPadé extrapolants. We are ultimately interested in evaluating ground state expectation values of various observables. In order to do this, one can use the following property, which is valid for any unitary transformation hψ0 |O|ψ0 i = hψ0,eff |Oeff|ψ0,eff i (5.30) where O is an operator, Oeff = U † OU is the effective operator, |ψ0 i is the ground state of the Hamiltonian H and |ψ0,eff i = U † |ψ0 i is the ground state of the effective Hamiltonian. Therefore, in order to evaluate the expectation value of O in the ground state of H, one can simply evaluate the expectation value of Oeff in the ground state of Heff .
5.1.3
Description of the effective model
Hamiltonian It is not possible to describe all the terms of the effective Hamiltonian (5.27). However, we will try to present its most important features. This Hamiltonian contains ten different types of processes. Apart from the chemical potential term (nr ), the dominant terms are the two-body interactions (nr1 nr2 ), the three-body interactions (nr1 nr2 nr3 ) and the four-body interactions (nr1 nr2 nr3 nr4 ). The kinetic terms consist of the usual hopping (b†r1 br2 ) which have by far the smallest coefficients, correlated hopping (nr1 b†r2 br3 ) already discussed in chapter 4, double correlated hopping (nr1 nr2 b†r3 br4 ) which lets a particle hop from r4 to r3 only if the sites r1 and r2 are both occupied, pair hopping (b†r1 b†r2 br3 br4 ) which lets pairs of particles hop from r3 and r4 to r1 and r2 , correlated pair hopping (nr1 b†r2 b†r3 br4 br5 ) which lets the pair of particles at sites r4 and r5 hop to r2 and r3 provided site r1 is occupied. The last type of kinetic processes (b†r1 b†r2 b†r3 br4 br5 br6 ) lets the three particles at sites r4 , r5 and r6 hop to sites r1 , r2 and r3 . We start with the usual two-body interactions (nr1 nr2 ) and for convenience we will rewrite the corresponding coefficients hr1 ,r2 ,r1 ,r2 as Vn . As depicted in figure 5.2 (a) for the two-body interactions appearing up to order 8, we define Vn = hr1 ,r2 ,r1 ,r2 where r1 is the position of the thick dimer and r2 is the position of the dimer labeled Vn . Note that although the coefficients V3 and V3′ both correspond to third neighbor interactions, they are not equivalent. The same is true for V7 and V7′ . This is a consequence of the symmetries of the underlying Shastry-Sutherland lattice. The coefficients Vn obtained by averaging the valid DlogPadé extrapolants of the 15-th order expansion are shown in figure 5.2 (b) as a function of J ′ /J. The lowest order (J ′ /J)m appearing in the series is written on each curve as O(m). There is a clear hierarchy in the coefficients which are sorted according to the order of appearance. Interestingly, the coefficients do not decrease monotonically with the distance between the two sites involved in the two-body interactions. The largest coefficient is V1 , which is not followed by V2 as usual, but by V3 . There is also a strong anisotropy which can be seen when comparing V3 and V3′ . At low J ′ /J, the coefficients decrease strongly as a function of the order of appearance and the coefficients smaller than V4 may be neglected. This is not true for larger J ′ /J where the terms V3′ , V5 and
5.1 Effective Hamiltonian
93
Figure 5.2: (a) Definition of the most important two-body interaction. Vn is the coefficient of the two body interaction between the thick dimer and the dimer labeled Vn . (b) Coefficients of the two-body interactions as a function of J ′ /J obtained by averaging the valid DlogPadé extrapolants of the 15-th order expansion. Inset: DlogPadé extrapolants (solid lines) as well as the bare series (dashed lines) for V3′ and V5 . The labels O(m) indicates at what order m in J ′ /J the corresponding coefficient appears.
V7 appearing at order 6 become important and, as we shall see below, contribute to the formation of the low density plateaus. In order to get an idea about the precision of these results, one can compare the various DlogPadé extrapolations and the bare series. This is shown in figure 5.3 (a) for the coefficients V1 and V3 and in the inset of figure 5.2 (b) for V3′ and V5 as a function of J ′ /J. Except for V1 , the various extrapolations and the bare series are basically indistinguishable below J ′ /J = 0.5. Beyond that value and up to J ′ /J ∼ 0.6, various Padé extrapolants still give consistent results for these coefficients. Up to J ′ /J = 0.5, the various extrapolations for the coefficient V1 give consistent results. However, the various predictions quickly diverge beyond that point. The next type of terms are the three-body interactions (nr1 nr2 nr3 ). For convenience, the coefficients hr1 ,r2 ,r3 ,r1 ,r2 ,r3 are rewritten as Wn . As depicted in figure 5.4 (a) for the terms appearing up to order 4, the coefficient of the three-body interaction acting on the sites r1 , r2 and r3 shown as thick black dimer will be denoted by Wn = hr1 ,r2 ,r3 ,r1 ,r2 ,r3 . Note that processes related by symmetry to the ones shown in figure 5.4 (a) have the same coefficient. On the right of each term, O(m) indicates the order m at which the term appears. The absolute value of the coefficients Wn obtained by averaging the valid Padé extrapolants of the 12-th order expansion are shown in figure 5.4 (b) as a function of J ′ /J. Note that these coefficients can be positive (plain lines) or negative (dashed lines). This figure shows two interesting features. Firstly, several three-body interactions are attractive. This is the case for the coefficient W1 which is by far the largest of these
94
The Shastry-Sutherland model
Figure 5.3: (a) Various DlogPadé extrapolants (plain lines) as well as the bare series up to order 15 (dashed lines) for V1 (black) and V3 (grey) as a function of J ′ /J. For V3 the curves cannot be distinguished. (b) Various Padé extrapolants (plain lines) as well as the bare series up to order 12 (dashed lines) for |W1 | (grey) and W2 (black) as a function of J ′ /J. For W2 the various extrapolants cannot be distinguished. coefficients. Secondly, at low J ′ /J the coefficients decrease strongly as a function of the order of appearance and are smaller than the two-body interactions. However, at larger J ′ /J they become as large as the two-body interaction, and will have some impact at not too low densities. A comparison of the various Padé extrapolants and the bare series is shown in figure 5.3 (b) for W1 and W2 . The various W2 extrapolants cannot be distinguished up to J ′ /J = 0.6. This also true for W1 up to J ′ /J ≃ 0.4, however the extrapolation becomes less accurate up to J ′ /J ≃ 0.5 where it becomes really hard to extrapolate. Note that this happens only for very local interaction, which should only be important for high density phases. The last type of interactions which are kept in this effective Hamiltonian are the fourbody interactions (nr1 nr2 nr3 nr4 ). Their coefficients are denoted by Xn , as depicted in figure 5.5 (a) for the terms appearing at order 4. Processes related by symmetry to the ones shown in figure 5.5 (a) have the same coefficient. Figure 5.5 (b) shows the absolute value of the coefficients Xn obtained by averaging the valid Padé extrapolants of the 12-th order expansion. Up to J ′ /J ∼ 0.5 the four-body interactions which appear only at order 4 are smaller than the three-body interactions. This is not true anymore above J ′ /J ∼ 0.5 where they become really important. Note that again certain very local contributions are hard to extrapolate. Next we consider the kinetic processes. As already stated by several authors [27,49,53], the coefficient of the usual hopping (b†r1 br2 ) is extremely small. The only hopping term
5.1 Effective Hamiltonian
95
Figure 5.4: (a) Definition of the three-body interaction appearing up to order 4. The name of the coefficient is shown on the left, while the order m where each term appear is shown on the right. (b) Absolute value of the coefficients for the three-body interactions appearing up to order 4 as a function of J ′ /J obtained by averaging the valid Padé extrapolants of the 12-th order expansion. Dashed lines (plain lines) corresponds to negative (positive) coefficients Wn . appearing up to order 8 is a next nearest hopping which appears only at order 6. However, the kinetic energy is not small, as other types of kinetic processes appear at lower order. The most important type of such processes are the correlated hopping (nr1 b†r2 br3 ) which appear already at order 2. The correlated hopping processes appearing up to order 4 are depicted in figure 5.6 (a). The position r1 of the density operator is denoted by a thick black dimer, while the position r2 (r3 ) of the creation (annihilation) operator is denoted by a thick grey (white) dimer. Note that because the coefficients are the same for a given process and its hermitian conjugate, the positions r2 and r3 can be interchanged. When applying a lattice symmetry to one of these processes, the absolute value of the coefficient does not change but the sign of the coefficient can change. If the creation and destruction operators act on two dimers having the same orientation (horizontal or vertical), the sign will not change. This is the case for tc1 , tc6 and tc7 . If the two dimers have different orientations (tc2 , tc3 , tc4 , tc5), then the sign will change when applying a mirror about the x axis or y axis as well as a rotation by an angle π/2. The coefficients of the correlated hopping appearing up to order 4 are shown in figure 5.6 (b). These coefficients are smaller than those of the two-body interactions, but similar to the three-body interactions, with two dominant processes tc1 , tc2 . The other types of kinetic processes kept in the effective Hamiltonian are the double
96
The Shastry-Sutherland model
Figure 5.5: (a) Definition of the four-body interaction appearing up to order 4. (b) Absolute value of the coefficients for the four-body interactions appearing up to order 4 as a function of J ′ /J obtained by averaging the valid Padé extrapolants of the 12-th order expansion. Dashed lines (plain lines) corresponds to negative (positive) coefficients Xn . correlated hopping (nr1 nr2 b†r3 br4 ), which appears at order 3 and whose dominant processes are shown in figure 5.7, the pair hopping (b†r1 b†r2 br3 br4 ) appearing at order 4 and shown in figure 5.8 and the correlated pair hopping (nr1 b†r2 b†r3 br4 br5 ) appearing at order 4 and shown in figure 5.9. The last type of kinetic process (b†r1 b†r2 b†r3 br4 br5 br6 ) appears only at order 6 and is not shown. The positions of the density operators are denoted by thick black dimers, while the positions of the creation (annihilation) operators are denoted by thick grey (white) dimers. The coefficients do not change if grey and black dimers are interchanged. The exchange of the grey and white dimers (hermitian conjugation) does not change the coefficient. However, when applying a mirror about the x axis or y axis as well as a rotation by an angle π/2 to one of these processes, the sign of the coefficient will change if there is an odd number of white and grey horizontal dimers. If this number is even, the sign will not change. To summarize, the effective Hamiltonian is dominated by the interaction processes. As a consequence, one should expect a phase diagram dominated by solid phases. The relative importance of the various interaction terms follows simple trends. Firstly, the two-body interactions have by far the largest coefficients. Secondly, the magnitude of the coefficients decreases when the distance between the sites or the number of sites involved in the interaction increases. However, although this effect is really strong at low J ′ /J, it decreases when J ′ /J increases towards the phase transition. Therefore around J ′ /J ∼ 0.5,
5.1 Effective Hamiltonian
97
Figure 5.6: (a) Definition of the correlated hopping appearing up to order 4. (b) Absolute value of the coefficients for the of the correlated hopping appearing up to order 4 as a function of J ′ /J obtained by averaging the valid Padé extrapolants of the 12-th order expansion. Dashed lines (plain lines) corresponds to negative (positive) coefficients tcn . the three and four-body interactions become important, as well as the longer-range interactions. Note that above J ′ /J ∼ 0.5, some extrapolations become problematic and the results should not be trusted. At low J ′ /J where only short range two-body interactions are important, one should only expect some plateaus at densities around n ∼ 0.5 where the distance between the particles is small. However, at larger J ′ /J the longer range interactions which become important could stabilize high-commensurability plateaus at low density. The three and four-body interactions should also become important, but mainly for intermediate and high density, where the distance between the particles is small. By considering the relative importance of the two, three and four-body interactions at large J ′ /J, one should expect the five-body interactions to be also important. However, these interactions should quickly become negligible as the distance between the sites involved increases, and therefore have an effect only at high densities. Concerning the kinetic energy, it is clearly dominated by correlated hopping, while the usual hopping almost vanishes. The vanishing hopping explains the localized triplet excitations seen as an almost flat lowest branch of excitations with inelastic neutron scattering at zero magnetic field, while one could expect the correlated hopping to stabilize a low density (magnetization) phase with a condensation of two-particle bound states [53]. Another interesting feature is that the coefficients of the kinetic terms can be positive or negative depending on the geometry of the process. A first consequence is that the
98
The Shastry-Sutherland model
Figure 5.7: (a) Dominant double correlated hopping processes (nr1 nr2 b†r3 br4 ) appearing up to order 3. (b) Absolute value of the coefficients as a function of J ′ /J obtained by averaging the valid Padé extrapolants of the 12-th order expansion. Dashed lines (plain lines) corresponds to negative (positive) coefficients tcn .
Figure 5.8: (a) Pair hopping processes (b†r1 b†r2 br3 br4 ) appearing up to order 4. (b) Absolute value of the coefficients as a function of J ′ /J obtained by averaging the valid Padé extrapolants of the 12-th order expansion. Dashed lines (plain lines) corresponds to negative (positive) coefficients tcn . corresponding processes should be frustrated, which should increase the energy of the state with a superfluid component and therefore contribute to the stabilization of plateaus (see
5.1 Effective Hamiltonian
99
Figure 5.9: (a) Dominating correlated pair hopping processes † † (nr1 br2 br3 br4 br5 ) appearing up to order 4. (b) Absolute value of the ′ coefficients as a function of J /J obtained by averaging the valid Padé extrapolants of the 12-th order expansion. Dashed lines (plain lines) corresponds to negative (positive) coefficients tcn . section 4.3). A second consequence is that quantum Monte Carlo simulations will be problematic for this system.
Local magnetization z Apart from the Hamiltonian, PCUT was also used to evaluate the transformed S1r and z S2r in the form z U † S1r U ≃ z U † S2r U ≃
(1)
X
(1)
(5.31)
(2)
(5.32)
sr,r′ nr+r′
r′
X
sr,r′ nr+r′ .
r′
(2)
The coefficients sr,r′ and sr,r′ have been evaluated up to order 10 and then extrapolated using DlogPadé extrapolants. In the following we will illustrate the results up to order 1. z z If r denotes a horizontal dimer site, the transformed S1r and S2r are given by 1 nr − 2 1 z nr + U † S2r U ≃ 2 z U † S1r U ≃
J′ nr−x + 2J J′ nr−x − 2J
J′ nr+x 2J J′ nr+x 2J
(5.33) (5.34)
100
The Shastry-Sutherland model
Figure 5.10: (a) A state |ψeff i with a particle at site r0 . White (Black) z dimer denotes empty (occupied) sites. (b) Local magnetization hψ|Sir |ψi. Full (empty) circles corresponds to negative (positive) magnetization. The radius of the circles is proportional to the magnetization amplitude. while if r denotes a vertical dimer site 1 J′ J′ nr − nr−y + nr+y (5.35) 2 2J 2J 1 J′ J′ z U † S2r U ≃ nr + nr−y − nr+y . (5.36) 2 2J 2J As an example, suppose that we want to evaluate the local magnetization for a state of the effective model having only one particle at site r0 corresponding to a horizontal dimer |ψeff i = b†r0 |0i, as depicted in figure 5.10 (a). Then, if we denote by |ψi = U|ψeff i the state after the reverse unitary transformation, the local magnetization in the ShastrySutherland model are given by J′ if r = r0 − y 2J 1 if r = r0 z z 2 ′ (5.37) hψ|S1r |ψi = hψeff |U † S1r U|ψeff i = J − 2J if r = r0 + y 0 otherwise J′ if r = r0 − y − 1 2J if r = r0 z z 2′ hψ|S2r |ψi = hψeff |U † S2r U|ψeff i = (5.38) J if r = r0 + y 2J 0 otherwise. z U ≃ U † S1r
The resulting local magnetization are shown in figure 5.10 (b). Interestingly, the isolated particle of the effective model becomes a triplet, dressed by two up-down neighboring dimers, which is in agreement with the interpretations of Cu NMR on SrCu2 (BO3 )2 [32].
5.2 5.2.1
Zero-temperature phase diagram Method
Because of the complexity of the effective Hamiltonian as well as the expected highcommensurability structures at low-density, one needs an efficient method to map out the
5.2 Zero-temperature phase diagram
101
phase diagram. We decided to use a classical approximation (see appendix D), because it is known to give a good qualitative description of the phase diagram and is particularly efficient at describing the solid phases. As in section 4.2.1, the first step consists in mapping the hard-core boson operators onto spins 1/2 operators with the MatsubaraMatsuda transformation [74]: 1 − Srz 2 = Sr+ = Sr−
nr =
(5.39)
br b†r
(5.40) (5.41)
where Sr± = Srx ± Sry and nr = b†r br . These spin operators should not be confused with the spin operators of the initial Shastry-Sutherland model. Indeed, while in the ShastrySutherland model there are two spins 1/2 per dimer, after the Matsubara-Matsuda transformation there is only one spin per dimer. A Matsubara-Matsuda spin down (up) corresponds to the presence (absence) of a particle in the effective model, which can be seen as a triplet t1 dressed by a polarization cloud (see the end of section 5.1.3) in the initial Shastry-Sutherland model. After this transformation one can use the classical approximation which consists in treating the spin operators Sr as a classical vector of length 1/2. The ground state is then given by the arrangement of spins which minimizes the classical Hamiltonian (see appendix D). In order to allow for high-commensurability breaking of the translational symmetry, the minimization is performed numerically on all possible non-equivalent clusters with periodic boundary conditions having up to 32 sites. The ground state energies of the various clusters are then compared, and only the states which have the minimum energy are kept as ground states. In order to check that 32 site clusters are sufficient, for J ′ /J = 0.5 we also applied the same procedure but with up to 64 sites clusters. The possible phases are identified using the solid order parameter S(k 6= 0) =
1 X 1 X z z ik(r′ −r) ik(r′ −r) ′ nr ie = hn hS ′ S ie r N 2 r,r′ N 2 r,r′ r r
(5.42)
and the superfluid order parameter hbr i = hSr+ i. Its modulus is given by the length of the spin projection onto the xy plane p (5.43) |hbr i| = hSrx i2 + hSry i2
while its phase is given by the angle between the x axis and the spin projection onto the xy plane. For a discussion of the various phases in terms of the Matsubara-Matsuda spins, please refer to section 4.2.1.
5.2.2
Results
In the following, we will concentrate on the parameter range 0 6 J ′ /J 6 0.5, because of the extrapolation problems above J ′ /J ∼ 0.5, and we will restrict to densities n 6 0.5, because at higher densities the neglected interactions acting on more than four particles
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Figure 5.11: Density n as a function of the chemical potential µ for J ′ /J = 0.3. (a) Comparison of the results obtained by keeping all the term appearing up to order 5,6,7 and 8 in the effective Hamiltonian. (b) Comparison of the results obtained by keeping terms with up to 4, 6 and 8 bosonic operators in the effective Hamiltonian.
should also become important. At low J ′ /J, the various terms in the effective Hamiltonian should quickly vanish as the order at which they appear increases and one should expect our high-order effective Hamiltonian to have well converged. In order to characterize the convergence of the perturbative expansion at J ′ /J = 0.3, figure 5.11 (a) presents the density curves as a function of the chemical potential obtained by keeping all the terms appearing up to order 5, 6, 7 and 8 in the effective Hamiltonian. Note that the coefficients are still evaluated up to order 12 (15 for the two-body interactions) and extrapolated. Except around the 2/5 plateau, all density curves lie on top of each other, which indicates that the effective Hamiltonian has converged and that the results can be trusted. On the other hand, the size of 2/5 plateau decreases when the order increases, which strongly suggest that it would not be stabilized with a higher order calculation. Another important question is whether it is sufficient to keep only the terms with up to eight bosonic operators in the effective Hamiltonian (5.27). In order to answer this question, figure 5.11 (b) presents the density as a function of the chemical potential for J ′ /J = 0.3 obtained by keeping only the terms with up to 4, 6 and 8 bosonic operators in the effective Hamiltonian. The curves are again on top of each other, except around the 2/5 plateau. As a consequence, for J ′ /J = 0.3 the effective Hamiltonian contains sufficiently high-order terms to describe the physics of the Shastry-Sutherland model.
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This is of course also true for lower J ′ /J = 0.3 where the higher order terms decrease even faster. The same procedure is applied at J ′ /J = 0.5, where the higher order terms in the effective Hamiltonian should become more important. Figure 5.12 (a) and (b) present the density curves as a function of the chemical potential obtained by keeping all the terms appearing up to order 5, 6, 7 and 8 in the effective Hamiltonian, while figure 5.12 (c) and (d) presents the density as a function of the chemical potential for J ′ /J = 0.3 obtained by keeping only the terms with up to 4, 6 and 8 bosonic operators in the effective Hamiltonian. Figure 5.12 (b) and (d) are a zoom on the lower density features of figures 5.12 (a) and (c) respectively. The convergence is clearly not as good as in the J ′ /J = 0.3 case, which shows that the high-order expansion is required in this region of the phase diagram, but also that three and four-body interactions are really important. It is nevertheless possible to extract some stable plateaus (1/9, 2/15,1/6, 2/9, 1/3 and 1/2). Interestingly, except for the results obtained by keeping only the terms appearing up to order 5, all the curves are on top of each other below the 1/6 plateau. This has two important consequences. Firstly, it shows that the three and four-body interactions have no effect on the low-density plateaus. Secondly, the terms of the effective Hamiltonian appearing at order 6 are determinant for the low-density physics. This procedure can be repeated for various J ′ /J ratios in order to build the whole phase diagram which is shown in figure 5.13. Note that the density versus chemical potential curves directly correspond to magnetization in unit of the saturation value versus magnetic field curves in the initial Shastry-Sutherland model (5.1). Indeed, because the total magnetization is invariant under the PCUT transformation, the density of particle is equal to the magnetization in unit of the saturation value and the chemical potential is equal to the magnetic field B. The phase diagram is dominated by a series of plateaus (solid phases), at 1/3 and 1/2 already at very small J ′ /J, then by plateaus at 2/9, 1/6, 1/9 and 2/15 at larger J ′ /J. Note that this calculation clearly rules out the possibility of a 1/8 plateau and does not predict a 1/4 plateau. However, the possibility of a 1/4 plateau cannot be excluded since it would appear in a region where the perturbative approach has not converged. The classical calculation predicts first order transitions between the plateaus below 1/3 at low J ′ /J and between the plateaus below 1/6 at larger J ′ /J. The nature of the other transitions cannot be decided on the basis of the present calculation and one could expect smaller plateaus to be stabilized in these regions. Above the 1/2 plateau and at low enough J ′ /J (not shown) where the perturbative expansion should have converged, the classical calculation predicts large supersolid phases in agreement with the third order calculation performed by Momoi and Totsuka [53]. Apart from the density curves, the classical calculation also gives access to the structure of the plateaus. Figure 5.14 presents the arrangements of the particles in the effective model, while the corresponding local magnetization in the Shastry-Sutherland model (see the end of section 5.1.3) are shown in figure 5.15. As already suggested at the end of section 5.1.3 in the case of an isolated particle, each particle of the effective model corresponds to a triplet dressed by two up and down neighboring dimers in the Shastry-Sutherland model, which is in agreement with the interpretation of Cu NMR on SrCu2 (BO3 )2 [32].
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Figure 5.12: Density n as a function of the chemical potential µ for J ′ /J = 0.5. (a),(b) Comparison of the results obtained by keeping all the term appearing up to order 5,6,7 and 8 in the effective Hamiltonian. (c),(d) Comparison of the results obtained by keeping terms with up to 4, 6 and 8 bosonic operators in the effective Hamiltonian. (b) and (d) are a zoom on the lower density features of figures 5.11 (a) and (c) respectively. Stable plateaus are labelled in black, while the plateaus which may disappear by keeping more terms in the Hamiltonian are labeled in grey.
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Figure 5.13: (a) Phase diagram as a function of J ′ /J and the chemical potential µ. White regions correspond to region where the perturbative approach has not converged. (b) Zoom on the low-density phases. (c) Density n as a function of J ′ /J and the chemical potential µ. Dashed lines correspond to regions where the perturbative approach has not converged. The density versus chemical potential curves corresponds to the total magnetization in unit of the saturation value versus the magnetic field B curves in the initial ShastrySutherland model (5.1).
5.2.3
A minimal model for the low-density plateaus
Interestingly, the low-density plateaus up to 1/6 can be understood with a much simpler model. Firstly, within the classical approximation the kinetic terms do not contribute in
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The Shastry-Sutherland model
Figure 5.14: Structure of the main plateaus. White (Black) dimer denotes empty (occupied) sites. The line shows the unit cell compatible with the periodicity of the state. For the 2/15 plateau, two structures have the same energy within the error bars of the method. the solid phase. This can be understood by remembering that the kinetic terms transforms into xy plane couplings in terms of the Matsubara-Matsuda spin. But in a solid phase, all the spin are parallel to the z axis and have no xy plane components. Therefore all the kinetic terms give zero on states without supersolid order. Secondly, none of the three and four-body interactions kept in the effective Hamiltonian can have an effect on these plateaus because of the large distances between the particles. This can also be seen in figure 5.12 (d) where the density curves obtained by keeping terms with up to 4, 6 and 8 bosonic operators in the effective Hamiltonian are exactly on top of each other up to the 1/6 plateau. Finally, the only terms of the effective Hamiltonian which have an effect on these plateaus are the two-body interactions. But there is more. By observing the structure of the plateaus depicted in figure 5.14, one can easily obtain an expression for their classical energy up to order 8 in J ′ /J: E0 = 0
(5.44) 1 2 E1/9 = − µ + V6 (5.45) 9 9 1 2 (5.46) E2/15 = − µ + (V3′ + 2V6 + V7 + 2V8 ) 15 15 1 1 E1/6 = − µ + (V3′ + 2V7 ) (5.47) 6 6 where En is the energy of the plateau at density n and the Vm are the coefficients of the twobody interaction defined in figure 5.2 (a). These energies do not explicitly depend on the
5.2 Zero-temperature phase diagram
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Figure 5.15: Local magnetization (S z ) of the main plateaus at J ′ /J = 0.5. Full (empty) circle correspond to magnetization along (opposite to) the magnetic field. The radius of the circles is proportional to the magnetization amplitude. The line shows the unit cell compatible with the periodicity of the state. coefficients of the two-body interaction appearing up to order 4 (V1 , V2 , V3 and V4 ) which can as well be set to infinity. These energies depend only on the coefficients of the two-
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The Shastry-Sutherland model
Figure 5.16: Energy of the 1/9, rhomboid 1/8, 2/15 and 1/6 plateaus as a function of the chemical potential µ at J ′ /J = 0.5. The error bars obtained by comparing various DlogPadé extrapolations of the energies are smaller than the line width. body interactions appearing at order 6 (V3′ and V7 ) and 8 (V6 and V8 ). As a consequence, the two-body interactions appearing at lower order (V1 , V2 , V3 and V4 ) can as well be set to infinity, while the coefficients appearing at order 8 play almost no role and can be neglected. In conclusion, within the classical approximation, a minimal model containing only the two-body interactions appearing up to order 8 with V1 = V2 = V3 = V4 = ∞ will produce exactly the same phase diagram up to density n = 1/6 and the same model but without the interactions V6 and V8 will give a really good approximation. An interesting consequence of these results is that the low-density physics is determined by the terms appearing at order 6 in the effective Hamiltonian, which means that a high-order expansion is indeed required in order to have a sufficient precision for these terms. This minimal model can be used to study the stability of these plateaus versus the two possible structures proposed for the 1/8 plateau by Miyahara and Ueda [49]. The energies of these structures are given by 1 E1/8,rhomboid = − µ + 8 1 E1/8,square = − µ + 8
1 (V5 + V7 ) 8 1 V5 . 4
(5.48) (5.49)
The resulting energies for the 1/9, 1/8, 2/15 and 1/6 plateaus as a function of the chemical potential for J ′ /J = 0.5 are shown in figure 5.16. For the 1/8 plateau, only the structure with the lowest energy is shown. Clearly, even by including the error bars estimated by comparing the various DlogPadé extrapolants of the energies, the 1/8 plateau is not stabilized.
5.2 Zero-temperature phase diagram
5.2.4
109
Comparison with other theoretical works
Before we started this project, the best results achieved by deriving and studying an effective Hamiltonian for the Shastry-Sutherland model were obtained by Momoi and Totsuka [53]. They used perturbation theory to derive an effective Hamiltonian up to third order in J ′ /J. This effective Hamiltonian corresponds exactly to the Hamiltonian we get by stopping the expansion at third order in J ′ /J. It contains the two-body interactions V1 , V2 and V3 (figure 5.2), as well as the three-body interactions W1 and W2 (figure 5.4). The kinetic energy is dominated by correlated hopping (tc1 and tc2 in figure 5.7) but also has several double correlated hopping processes (figure 5.7). The phase diagram of the effective model was then obtained with the same classical approximation as the one we used. The phase diagram contains two plateaus at 1/2 and 1/3 with the same checkerboard and stripe structures as the one we predict. Above each plateau, there is a supersolid phase with the same solid order as the corresponding plateau. Below the 1/3 plateau, they obtain structures with larger periodicities but do not give any details. In order to check the strong coupling expansion, they also used exact diagonalizations directly on the Shastry-Sutherland model, but only on a rectangular 2 × 6 dimers cluster. Finally they showed that a phase characterized by a condensation of two-triplet bound states is stabilized at very low magnetization. This phase is similar to the paired superfluid phase we discussed in section 4.1 and is a consequence of the correlated hopping which acts as an attractive interaction between the triplets. Their phase diagram strongly differs from ours, and since we used the same classical approximation, it is clearly a consequence of the third order expansion which is not sufficient. Regarding the exact diagonalization results, the differences can be explained by their choice of clusters which cannot stabilize any of the structure that we propose, except for the 1/3 and 1/2 plateaus (see figures 5.14 and 5.15). In order to determine the possible structures inside the 1/3, 1/4 and 1/8 plateau, Miyahara and Ueda [49] used a more phenomenological approach based on an effective Hamiltonian with only two-body interactions. They obtained the two-body interactions V1 , V2 and V3 with perturbation theory up to third order in J ′ /J. For the longer range interactions they simply used the Yukawa form V (r) =
V0 −r/ξ e . r
(5.50)
The resulting effective Hamiltonian was then solved classically on square clusters having up to 36 dimers and rectangular clusters with 24 and 32 dimers. Note that since this effective model contains only interactions, the classical and quantum calculations are equivalent. In addition to this calculation, they also solved directly the Shastry-Sutherland model with exact diagonalizations on rectangular clusters having 8, 10 and 12 dimers. The free parameters V0 and ξ of the effective model were determined by comparing the magnetization curves obtained by the classical calculation and by exact diagonalizations of the Shastry-Sutherland model. Because this work was devoted to SrCu2 (BO3 )2 , they focused on the case J ′ /J = 0.65. Let us first discuss the results obtained with the classical treatment of the effective model. The magnetization curve as a function of the chemical potential shows plateaus at 1/2, 1/3, 1/4, 1/6, 1/8 as well as several other smaller
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The Shastry-Sutherland model
plateaus. However, these results can hardly be predictive. Firstly, because the third order perturbative calculation of the coefficients V1 , V2 and V3 is clearly not sufficient for the large ratio J ′ /J = 0.65. Secondly, the interactions relevant for the low-magnetization plateaus were obtained by a phenomenological approach which cannot reproduce the nonmonotonic behavior of the coefficients. Finally, although they used large clusters, the 2/15 plateau that we predict cannot be stabilized in any of their square of rectangular clusters. Regarding the exact diagonalizations of the Shastry-Sutherland model, although exact in principle, this approach strongly suffers from the choice of clusters. They only used square clusters with 8 and 10 dimers and a rectangular cluster with 12 dimers, which are only compatible with the 1/2 and 1/3 plateaus that we predict while our 1/9, 2/15, 1/6 and 2/9 plateaus cannot be stabilized (see figures 5.14 and 5.15). Therefore they cannot claim that these plateaus do not appear in the Shastry-Sutherland model. Another interesting approach was proposed by Miyahara and collaborators [50] who directly evaluated ground state properties of the Shastry-Sutherland model with exact diagonalizations. The main goal of this work was to determine spin density profiles (local magnetization hSrz i) for possible ground states inside the various plateaus. In the Shastry-Sutherland model, the translational symmetry was expected to be broken inside the various plateaus (except 1/2) and therefore the ground state has a finite degeneracy due to the translational symmetry. In that case, a local quantity such as the magnetization hSrz i can have an arbitrary value, depending on the linear combination of the ground states which is used. For example, a uniform linear combination will lead to a uniform magnetization. In the real material however, one observes the broken translational symmetry and this could be due to a pinning by impurities or to a lattice distortion. In this work, they decided to overcome this problem by coupling the system to phonons. The main limitation of this approach is again the size of the clusters available for the exact diagonalizations. They had to restrict to cluster having 8 and 12 dimers, and therefore did not claim to be able to discuss the stability of the various plateaus. In particular, these clusters were not compatible with our 1/9, 2/15 and 2/9 structure. However, the comparison with our results is interesting as the structures they obtained for the 1/2 and 1/3 plateaus are in good agreement with what we present in figure 5.15. They also proposed two possible 1/8 structures as well as a 1/4 structure. Similarly to our results and except for the 1/2 plateau, the building brick of these structures was always a dimer with large magnetization along the magnetic field, interpreted as a triplet t1 , dressed by a polarization cloud with Friedel-like oscillations. Instead of using an effective model for the triplet particles, another approach is to use a Chern-Simons treatment of the Shastry-Sutherland model. This approach, first used by Misguich and collaborators [51], is based on a generalization of the Jordan-Wigner transformation to two-dimensional lattices to map the spin operators of the ShastrySutherland model onto spinless fermions [83–85]: P
′
Sr+ = c†r ei r ′ 6=r arg(r,r )nr ′ P ′ Sr− = e−i r ′ 6=r arg(r,r )nr ′ cr 1 Srz = nr − 2
(5.51) (5.52) (5.53)
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111
Figure 5.17: (a) Hofstadter spectrum for the Shastry-Sutherland lattice at J = J ′ = 1. Exy is the kinetic energy of the fermions. The lower thick grey lines denotes the Fermi energy. (b) Magnetization M in unit of the saturation value Msat as a function of the magnetic field B for J ′ /J = 0.4 obtained with Chern-Simons theory (black curve) and comparison with experiments (dashed grey curve). Figures reproduced from Misguich and collaborators [51]. where c†r (cr ) is a creation (annihilation) operator of fermions at site r of the Shastry Sutherland lattice, nr = c†r cr and arg(r, r ′) is the angle between r − r ′ and an arbitrary direction in the lattice. After this transformation, the Shastry-Sutherland model becomes a model of interacting fermions in the presence of a fictitious magnetic field, and the real magnetic field B of the Shastry-Sutherland model becomes a chemical potential for the fermions X 1 H = −B nr − (5.54) 2 r X 1 † iAτ (r) 1 1 † −iAτ (r) + Jr,τ c e cr+τ + cr+τ e nr+τ − cr + nr − 2 r 2 2 r,τ where the vector potential Aτ (r) of the fictitious magnetic field is given by X Aτ (r) = (arg(r ′′ , r) − arg(r ′′ , r + τ )) nr′′
(5.55)
r′′ 6=r,r+τ
and Jr,τ = J when r and r + τ form a dimer bond and Jr,τ = J ′ when r and r + τ form an inter-dimer bond. Although this approach maps the Shastry-Sutherland model onto a
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The Shastry-Sutherland model
fermionic model, the resulting problem is not easier to solve because of the complicated non-local interactions between the fermions. In order to solve this problem, Misguich and collaborators used a mean-field approximation, in which the operators nr in the vector potential were replaced by their ground state expectation value hnr i, while the Ising term was treated with the mean-field decoupling nr nr′ = hnr inr′ + hnr′ inr − hnr ihnr′ i. In addition, they considered only the uniform case hnr i = hni. The flux φ of the fictitious magnetic field per square plaquette is then tied to the density of fermions and thus to the magnetization of the Shastry-Sutherland model by φ ≡ Ay (r) − Ay (r + x) + Ax (r + y) − Ax (r) = 2πhni.
(5.56)
Within this approximation, the Ising part of the Hamiltonian simply renormalizes the chemical potential. The kinetic part of the Hamiltonian becomes a problem of free fermions hopping in a uniform fictitious magnetic field which can be solved for rational values of φ/(2π) = p/q. For a given q, the spectrum is made of q sub-bands, leading to a fractal spectrum as a function of φ which is known as the Hofstadter butterfly [86] (see figure 5.17 (a)). The density curve as a function of the chemical potential is determined in the following way: For a given density hni one directly obtains the magnetic flux φ = 2πhni. For this value of φ, the Hofstadter spectrum of the kinetic part of the Hamiltonian is computed and the bands are filled until the density of fermions corresponds to hni. The corresponding energy E(hni) (with B = 0) is then stored. By repeating this operation for various values of the density hni, one obtains the energy as a function of hni. The density as a function of the chemical potential (B) is then obtained by a Legendre transformation. The resulting magnetization curve for J ′ /J = 0.4 is shown in figure 5.17 (b) along with the experimental results from Onizuka and collaborators [39]. Interestingly, although the approximation assumes a uniform density, there are plateaus at 1/2, 1/3. These plateaus appear whenever the Hofstadter spectrum of the kinetic energy is gapped above the Fermi level, similarly to what happens in the integer quantum Hall effect. The arrangement of plateaus is therefore a consequence of the xy plane couplings of the Shastry-Sutherland spins which transform into the kinetic part of the fermionic Hamiltonian. Another interesting feature of these results is the overall good agreement with the experimental results, although there are some problems at lower magnetization. In particular, this calculation does not predict any plateau below 1/3, although a 1/4 plateau is stabilized for slightly larger values of J ′ /J. They also determined the phase diagram for 0.25 < J ′ /J < ∞. The general behavior is similar to what we have in the sense that the low J ′ /J limit is dominated by a 1/2 plateaus and upon increasing the ratio J ′ /J, first a 1/3 plateau appears, followed by a smaller 1/4 plateau. However, their results differ from our results since for J ′ /J 6 0.5 they only have 1/2, 1/3 and 1/4 plateaus and the phase diagram contains large regions of apparently gapless phases. This could be a consequence of the uniform mean-field approximation which neglects the density-density interactions by transforming it into a renormalization of the chemical potential. Note that they also claim to observe plateaus at fractions 1/n for n > 5 around J ′ /J ∼ 0.67, but do not discuss it, possibly because the agreement between their magnetization curve and the experiments is not good for these
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Figure 5.18: Magnetization as a function of the magnetic field for J ′ /J ≃ 0.45 obtained with a uniform approximation (dashed green curve), a non-uniform approximation (red curve) and by experimental measurement of SrCu2 (BO3 )2 (dark blue curve). Figure reproduced from Sebastian and collaborators [40]. values of the J ′ /J ratio. This approach has also been used recently by Sebastian and collaborators [40]. They extended this calculation to the non-uniform case where the local density hnr i was allowed to relax on each site of a rectangular super unit cell containing up to 144 dimer sites. By contrast to the uniform approximation, this approximation does not neglect the densitydensity interactions appearing in the Ising part of the Hamiltonian, and should therefore better describe the phases with broken translational symmetry. An optimal value of the ratio J ′ /J ≃ 0.45 was obtained by comparing the theoretical results to an experimental magnetization curve as function of the magnetic field obtained by torque measurements. The resulting magnetization curve is shown in figure 5.18, which presents a comparison between uniform (dashed green), non-uniform (red) calculations and experimental SrCu2 (BO3 )2 measurements (blue). The uniform approximation should be equivalent to the calculation done by Misguich and collaborators. However it predicts several plateaus below 1/4 which were not discussed by Misguich and collaborators. On the basis of these
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The Shastry-Sutherland model
articles, it not possible to know if the results obtained by Sebastian and collaborators are in agreement with those obtained by Misguich and collaborators. The most interesting results are given by the non-uniform approximation, which predicts plateaus at 1/q with q = 2, 3, · · · , 9 and at 2/9. There are obvious similarities between these results and ours. Firstly, all the plateaus that we predict, except the 2/15, are stabilized in their calculation. Secondly, the magnetization curve is dominated by plateaus above the 1/9 plateau. This is particularly interesting because our classical approximation should give really good results when the system is in a solid phase (plateau). In particular, in the limit where the system is dominated by interactions and has no kinetic energy, the classical and quantum ground states are identical. But there are also some differences. They predict a phase with finite magnetization but without plateaus in a large range of magnetic field below the 1/9 plateau, similar to what is observed experimentally. They do not discuss the nature of this phase, but it could correspond to the phase with condensation of two-triplet bound states proposed by Momoi and Totsuka [53]. As already discussed in section 4.2.1, the classical approximation cannot describe that type of phase and replaces it with a phase having zero-magnetization. One could also imagine a pairing phase at low magnetization, followed by a simple superfluid phase at larger magnetization which could be underestimated by our classical approximation. Finally, it could also be an artefact of their method whose reliability is not known. They also predict 1/8 and 1/7 plateaus instead of our 2/15 plateau. This can a consequence of their choice of super unit cells which are not compatible with the periodicity of our 2/15 plateau. Finally, some aspects of their results, for instance a well developed plateau at 1/5, are definitely ruled out by our analysis and must be considered as artefacts of their method. Within the non-uniform approximation, they also had a direct access to the local magnetization hSrz i = hnr i − 1/2 inside the various plateaus. Although the 1/2 and 1/3 structures that they propose are in agreement with our results (figure 5.15) and with previous results, the structure of the other plateaus strongly differs, and looks like not totally converged results. It could be that the algorithm used for the numerical minimization of the energy was stuck in a local minimum and could not reach the global minimum. Very recently, Abendschein and Capponi [54] proposed an approach based on a nonperturbative effective Hamiltonian which was solved with exact diagonalizations. They used the non-perturbative contractor renormalization (CORE) technique to derive the effective Hamiltonian for the low-energy subspace which consists of the states having only singlet and triplet t1 . The resulting Hamiltonian is as complicated as our Hamiltonian and also contains ∼ 104 processes. Although non-perturbative, this approach also has some limitations: In order to evaluate the coefficients of the various terms appearing in the Hamiltonian, the actual calculation was done on a 3 × 3 dimer cluster. A consequence is that only the processes which fit into this cluster are taken into account in the effective Hamiltonian. For the two-body interactions, it means that only V1 , V2 , · · · , V5 appears in the Hamiltonian and the terms V6 , V7 , V8 , · · · are neglected (see figure 5.2). In addition, the truncation to the 3 × 3 cluster introduces an error in the various coefficients which should have an additional contribution from larger clusters. The contribution from larger
5.2 Zero-temperature phase diagram
115
Figure 5.19: Coefficients Vn of the two-body interactions as a function of J ′ /J obtained with CORE (plain lines) and with our perturbative calculations (dashed lines). CORE results reproduced from Abendschein and Capponi [54]. clusters should decrease when increasing the cluster size. Consequently, for local processes like V1 there are almost no differences between the results obtained with 2 × 2 and 3 × 3 clusters, while for longer range processes, one should expect larger error bars. Unfortunately, it is not possible to give an estimate for these error bars. Figure 5.19 presents a comparison of the coefficients of the two-body interactions obtained with our perturbative calculation (dashed lines) and obtained with the CORE technique (plain lines). Up to J ′ /J ∼ 0.5, there is a good agreement between the two approaches. Above J ′ /J ∼ 0.5, there is an abrupt change in the larger coefficients (V1 and V3 ) obtained with the nonperturbative approach which is not in agreement with the perturbative results. However, this happens in the region close to the transition towards the intermediate phase where the perturbative results should not be trusted. Interestingly, the other coefficients are in qualitative agreement up to J ′ /J ∼ 0.65. This effective Hamiltonian was then solved with exact diagonalizations on clusters having N = 32, 36, 64 and 72 dimers. They also used exact diagonalizations directly on the Shastry-Sutherland model, but could only do it on N = 32 and N = 36 clusters because of the larger Hilbert space. The resulting magnetization curves at J ′ /J = 0.5 and phase diagram are shown in figure 5.20. Let us concentrate on the results below the 1/2 plateau. For the clusters with N = 32 and N = 64 dimers (figures 5.20 (a) and (b)), the magnetization curve displays several steps and in particular 1/8 and 1/4 plateaus whose sizes and location do not change when changing the cluster size. However, these clusters can only stabilize 2/N plateaus and therefore cannot stabilize the 1/3 plateaus which is known to appear in this system. For this reason, they also used clusters of size N = 36 and N = 72 (figures 5.20 (c) and (d)) and in that case the magnetization curve displays stable 1/9, 1/6, 2/9 and 1/3 plateaus. Finally they also checked the stability of the various plateaus by doing a more systematic study of the scaling on various square and rectangular clusters. They conclude that for J ′ /J = 0.5, stable plateaus appear at
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The Shastry-Sutherland model
Figure 5.20: (a) Magnetization m/msat as a function of the magnetic field h for J ′ /J = 0.5 obtained by exact diagonalizations of the Shastry-Sutherland model on a N = 32 cluster (black lines) and exact diagonalizations of the effective Hamiltonian on N = 32 and N = 64 clusters (grey lines). The magnetization m/msat and magnetic field h are respectively equal to our density n and chemical potential µ. (c) Same quantities but obtained on N = 36 and N = 72 clusters. (b)-(d) Phase diagram obtained by exact diagonalizations of the effective Hamiltonian on a N = 32 and N = 36 cluster respectively. Figures reproduced from Abendschein and Capponi [54]. fractions 1/9, 1/8, 1/6, 2/9, 1/4, 1/3 and 1/2. There are several similarities between these results and ours. Firstly, their phase diagram is dominated by plateaus, which confirms our choice of the classical approximation. Secondly, apart from the 1/8 and 1/4 plateaus, we predict the same series of plateaus. Moreover, the 1/4 plateau is not in contradiction with our results as it appears in a region where our approximation has not converged. Finally, the overall behavior of the phase diagrams shown in figures 5.20 (b) and (d) is also in good agreement with our predictions (figure 5.13). The most striking difference with our results is the absence of the 2/15 plateau which is replaced by a 1/8 plateau. We think about two possible explanations. It could again be explained by their choice of clusters which cannot stabilize our 2/15 structure. But there is also another possibility. They neglected the two-body interactions V6 and V7 which are crucial for the physics at these low magnetizations. In order to see the effect of this
5.2 Zero-temperature phase diagram
117
approximation, we determined the magnetization curve as a function of the magnetic field with the minimal model described in section 5.2.3 and V6 = V7 = 0. Interestingly, neither the 1/8 nor the 2/15 were stabilized and the magnetization directly jump from the 1/9 plateau to the 1/6 plateau. This result is not in contradiction with what they obtained though, since exact diagonalization cannot be used on clusters large enough to stabilize simultaneously the 1/9, 1/8 and 1/6 plateaus. To summarize, all the theoretical works agree on the presence of plateaus at 1/2, 1/3 which appears at low J ′ /J. We do not predict a plateau at 1/4, however it is predicted by all the other theoretical works and is not in contradiction with our results since it should appear in a region where our approach is not reliable. Concerning the physics below the 1/4 plateau, a coherent picture starts to emerge with the works by Sebastian and collaborators, Abendschein and Capponi as well as our work. Around J ′ /J ∼ 0.5, we all agree on the presence of plateaus at 1/9, 1/6, 2/9 and we all predict a phase diagram dominated by plateaus above the 1/9 plateau. The main difference is that we predict a plateau at 2/15 instead of the 1/8 plateau predicted by the two other groups and we claim that this is due to their choice of clusters which cannot stabilize the 2/15 structure.
5.2.5
Comparison with SrCu2(BO3)2 experiments
The first observation of a 1/8 and a 1/4 plateaus in the magnetization curve of SrCu2 (BO3 )2 was done in 1999 by Kageyama and collaborators [30]. By using a pulsed magnetic field, they were able to measure the magnetization as a function of the magnetic field up to 45 T and down to temperature T = 0.5K for a polycrystalline SrCu2 (BO3 )2 sample. They observed a continuous transition from the singlet gapped ground state to the gapless magnetic state occurring around Hc1 ≃ 20T. The 1/8 plateau was observed between Hc2 = 27.9T and Hc3 = 29.8T followed by a continuous transition to the 1/4 plateaus stabilized between Hc4 = 37.0T and Hc5 = 41T. One year after the publication of these results, Onizuka and collaborators [39] published the first observation of the 1/3 plateau predicted by Momoi and collaborators [53] as well as Miyahara and collaborators [49]. They measured the magnetization curve of a single SrCu2 (BO3 )2 crystal in pulsed magnetic field up to 53T with the field H k c (c is perpendicular to the Cu planes) and up to 57T with H ⊥ c. The resulting magnetization curve is reproduced in figure 5.21. For both H ⊥ c and H k c, they measured the 1/8, 1/4 and 1/3 plateau, with a possible first order transition between the 1/4 and 1/3 plateau. However, below the 1/8 plateau, the direction of the magnetic field becomes important: For H k c there seems to be a first order transition between the low magnetization gapless phase and the 1/8 plateau, while for H ⊥ c they observed an anomaly around 1/10. They excluded the possibility of a 1/10 plateau because it is not seen with H k c, but could not explain this anomaly. In 2002, Kodama and collaborators [32] published the first experimental evidence of a broken translational symmetry in the 1/8 plateau. They used Cu nuclear magnetic resonance (NMR) of a single SrCu2 (BO3 )2 crystal at a temperature T = 35mK and in a field H k c up to 28T. Each single Cu ion should give six peaks in the NMR spectra at distinct frequencies which depend on the local magnetization hS z i of the Cu ion along the
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The Shastry-Sutherland model
Figure 5.21: SrCu2 (BO3 )2 magnetization as a function of the magnetic field. Figure reproduced from Onizuka and collaborators [39]. axis parallel to the magnetic field. In a uniform phase, the contribution from all the Cu ions should appear at the same frequency and one should observe only six peaks in the NMR spectra. That is indeed what happens at a magnetic field H = 26T corresponding to the gapless phase below the 1/8 plateau. By contrast, at H = 27.6T where the 1/8 plateau is stable, they observed the appearance of many more peaks, which indicates the presence of a magnetic superstructure. In order to fit this spectrum, they needed at least 11 distinct Cu sites, with approximately 1/8 of the Cu ions having a magnetization hS z i = 0.30 and approximately 1/8 of the Cu ions having a magnetization hS z i = 0.20. They also measured the NMR spectra for various values of the magnetic field around the transition towards the 1/8 plateau and observed a superposition of the two phases, which can be explained if the transition is first order but broadened by some disorder in the sample. Note that these results suggest that there are no magnetization plateaus below 1/8. Thanks to the work by Miyahara and collaborators [50] discussed previously, where the local magnetization was evaluated for various possible plateaus, they could derive a theoretical NMR spectra and compare it with the experimental results at H = 27.6T inside the 1/8 plateau. Although not perfect, the best fit was obtained with the rhomboid 1/8 structure. One of problem was that this structure had only 8 distinct Cu sites, while
5.2 Zero-temperature phase diagram
119
at least 11 distinct sites are needed to reproduce the spectra. However, Miyahara and collaborators were able to show that in the presence of an inter-layer coupling, the number of distinct Cu sites in the 1/8 rhomboid structure should be 16. Instead of a 1/8 plateau, we predict a 2/15 plateau. This does not seem to be in contradiction with the NMR measurement. Indeed, the local magnetization in the 2/15 plateau has the same type of structure as in the 1/8 plateaus, with a building brick consisting of one dimer with large magnetization along the magnetic field, interpreted as a triplet t1 , dressed by a polarization cloud with Friedel-like oscillations. Interestingly, the values of the local magnetization around the triplets are almost the same in the various low-magnetization plateaus, and would therefore produce similar NMR spectra. The only problem with the 2/15 is that they observed approximately 1/8 of Cu sites having a local magnetization hS z i = 0.30 and approximately 1/8 with hS z i = 0.20. It would be interesting to know if the approximate value 1/8 would be compatible with 2/15. In 2007, another measurement of the magnetization curve was performed by Sebastian and collaborators [40] who claimed to have observed plateaus at 1/q with q = 2, · · · , 9 and 2/9. They used torque measurements in a pulsed magnetic field H k c up to 85T and with temperature down to 29mK. The resulting magnetization curve indeed presents several small anomalies consistent with plateaus at 1/q with q = 2, · · · , 9 and 2/9 that they could explain theoretically with the Chern-Simons treatment described previously. Although not totally in agreement with our predictions, this result is interesting as it suggests the possibility of additional plateaus between the 1/8 and 1/4 plateaus and in particular the 1/9, 1/6 and 2/9 plateaus which are stabilized in our calculation. However, the value of the magnetization in the various plateaus is still highly controversial. Indeed the torque measurements are performed in the following way: The sample is fixed on a cantilever which is placed in a homogeneous magnetic field. As a consequence, the sample is submitted to the torque τ = M × H where M is the magnetization of the sample and H the applied magnetic field. The torque is transmitted to the cantilever which can rotate around an axis perpendicular to H. Therefore the torque is equal to m⊥ H where m⊥ is the component of the magnetization perpendicular to the field and the rotation axis. Sebastian and collaborators made the assumption that the variation of m⊥ as a function of H was the same as that of the longitudinal magnetization we are interested in. However, this is not true anymore in the presence of intra-dimer Dzyaloshinskii-Moriya interaction, which has been shown to contribute to the torque by adding a uniform transverse magnetization component [87]. As a consequence, although torque measurements can correctly capture phase transitions, it cannot give the true variation of the longitudinal magnetization and the value of the magnetization measured in the plateaus by Sebastian and collaborators may not be accurate. In 2008, two interesting articles focusing onto the low-magnetization plateaus were published. Takigawa and collaborators [41] performed B nuclear magnetic resonance measurements in order to study the phases above the 1/8 plateau. This was done with magnetic field H k c from 26.5T to 31T and various temperatures down to T = 0.06K. They started by analysing the behavior of the NMR spectra at T = 0.19K for various values of the magnetic field from 26.5T (below the 1/8 plateau) to 28.7T (above the 1/8 plateau). These results confirmed the uniform phase below the 1/8 as well as the first
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The Shastry-Sutherland model
Figure 5.22: SrCu2 (BO3 )2 phase diagram as a function of the magnetic field H and the temperature T . NMR measurements are depicted by solid circles and triangles. This figure also include results obtained by torque measurements [42] (open circles and squares) as well as specific heat data (open triangles). Figure reproduced from Takigawa and collaborators [41].
order transition towards the 1/8 plateau observed by Kodama and collaborators [32]. Above the 1/8 plateau, although the magnetization increases with the magnetic field, the NMR spectra still show the characteristic behavior which indicates the presence of highly polarized triplets with Friedel-like oscillations and suggests additional symmetry breaking with respect to the 1/8 plateau. In addition, upon increasing the magnetic field, the spectral shape changes, which shows that the structure of the local magnetization evolves. These results would be compatible with a supersolid phase where a fraction of the triplets condense while the other triplets stay localized in a super-lattice. Additional measurements were performed for various values of the magnetic field and temperature in order to extract a phase diagram as a function of the magnetic field and the temperature which is reproduced in figure 5.22. This figure also presents results obtained by torque measurements [42] which will be discussed later. At low temperature and upon increasing the magnetic field the system goes through a uniform phase, the 1/8 plateau, a first intermediate phase I1 and a second intermediate phase I2 . At larger temperature, the system is in the uniform phase. These measurements are consistent with first order transitions between these phases. The phase I1 is the possible supersolid phase discussed above. In the phase I2 , the results are consistent with a localization of highly polarized triplets surrounded by Friedel-like oscillations. However, this phase is characterized by a broadened and featureless spectrum, which suggest an increased disorder or an incommensurate structure. In addition to these phases, they also observed the sudden disappearance of some peaks in the spectra just above the 1/8 plateau (28.7T) and upon lowering the temperature between 0.19K and 0.16K. This results suggest a possible phase transition towards a low temperature phase. At the same time, Levy and collaborators [42] performed torque measurements in a homogeneous field H k c up to 31T and temperature down to 50mK in order to determine the phase diagram as function of the magnetic field and the temperature (reproduced in figure 5.23 (a)). This phase diagram confirms the NMR measurements by Takigawa and
5.2 Zero-temperature phase diagram
121
Figure 5.23: (a) SrCu2 (BO3 )2 phase diagram as a function of the magnetic field H and the temperature T obtained by torque measurements. (b) Longitudinal magnetization (black lines) and torque divided by the magnetic field (red lines) as a function of the magnetic field. Black triangles denote the longitudinal magnetization measured by NMR. Figure reproduced from Levy and collaborators [42]. collaborators [41]: Upon increasing the magnetic field, they observe the transition from the uniform phase to the 1/8 plateau, followed by a transition towards an intermediate phase I1 starting around 28.3T. This phase ends up around 29.5T and is followed by another phase I2 . Interestingly, the phase transition at 29.5T presents a strong hysteresis below 200mK, which could be compatible with the new low-temperature phase suggested by the NMR measurements at 28.7K. In addition, they also measured for the first time the pure longitudinal magnetization up to 31T at 60mK. This was done by placing the sample in a strong field gradient in which the torque component becomes negligible with respect to the force F = −M µ0 ∇ · H where M is the magnetization of the sample and H the applied magnetic field. The resulting longitudinal magnetization as a function of the magnetic field is shown in figure 5.23 (b) along with a torque measurement. The longitudinal magnetization is flat within the 1/8 plateau, as expected. Interestingly, it is also flat in the adjacent phase (between 28.4T and 29.5T) with only a small increase when approaching the phase transition at 29.5T. Note that the magnetization between 28.4T and 29.5T does not correspond to any simple p/q ratio. Thanks to these results, it is also possible to compare the real longitudinal magnetization with the torque divided by the magnetic field which is generally used to determine the magnetization. As it can be seen in figure 5.23 (b), both predictions strongly differ, and this could be a consequence of the intra-dimer Dzyaloshinskii-Moriya interaction, which contributes to the torque by adding a uniform transverse magnetization component. In this article, they also criticized the results obtained by Sebastian and collaborators
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The Shastry-Sutherland model
[40] who claimed to have observed 1/9 and 1/7 plateaus around the 1/8 plateau. Their reasoning is the following: By reducing by 2% the transition lines reported by Sebastian and collaborators, one can recover the transitions measured by Levy and collaborators. The latter values agree with NMR measurements, which are known to be accurate in the determination of the magnetic field. As a consequence, the incorrect values reported by Sebastian and collaborators could be due to an incorrect field scale. In addition, according to pulsed field measurements by Onizuka and collaborators [39], the plateau measured by Levy and collaborators between 26.7T and 28.3T can only correspond to the 1/8 plateau. Therefore, the phase above the 1/8 plateau cannot be a 1/7 plateau since the increase in the longitudinal magnetization measured by Levy and collaborators is too small. If we forget about the results from Sebastian and collaborators, a coherent picture starts to emerge from all these experiments concerning the low-temperature properties of SrCu2 (BO3 )2 . The magnetization curve as a function of the chemical potential displays plateaus at 1/8, 1/4 and 1/3 of the saturation value. Below the 1/8 plateau, there is a uniform gapless phase extending from ∼ 20T to ∼ 27T which is followed by a possible first order transition to the 1/8 plateau. Upon increasing the magnetic field, there is a possible first order phase transition from the 1/8 plateau to a phase characterized by an almost flat magnetization and which ends up around 29.5T . The NMR measurements in the 1/8 plateau and the two phases above are consistent with a breaking of the translational symmetry due the localization of highly polarized triplets dressed by a polarization cloud. Although not free of criticism, the results from Sebastian and collaborators are also interesting since they suggest that the structure of the magnetization curve below the 1/3 plateau may be much more complicated than initially expected. The theoretical predictions for the Shastry-Sutherland model are not in agreement with these results. The most surprising aspect of our results is the absence of the 1/8 plateau which has been observed in SrCu2 (BO3 )2 . Taking for granted that the experimental identification of the 1/8 plateau is correct (a point which might actually deserve further investigation in view of the rather different absolute scales reported in pulsed and steady magnetic field experiments), we can think of two possible origins of this discrepancy. The first one is that the physics changes dramatically when J ′ /J approaches the critical value where in the absence of magnetic field the dimer product wave-function is no longer the ground state but is replaced by the intermediate phase with a possible plaquette order. This seems unlikely however in view of the first order nature of this transition. The other possibility is that the Shastry-Sutherland model is not appropriate for the description of the SrCu2 (BO3 )2 and would need additional couplings. According to a recent abinitio calculation by Mazurenko and collaborators [88], there are three types of residual couplings, all of the same order of magnitude (J/100): In plane Dzyaloshinskii-Moriya interactions (both inter- and intra-dimer), further neighbor in plane exchange couplings, and inter-plane exchange couplings. Although small, these residual couplings are of the same order as the couplings which dominate the physics in the low-magnetization plateaus, and could therefore drastically change the low-magnetization physics. Apart from the plateau structure, our results also differ with SrCu2 (BO3 )2 measurements concerning the behavior below the 1/9 plateau. Indeed, we predict a first order transition between zero magnetization and the 1/9 plateau, while the magnetization curve
5.3 Conclusion
123
of SrCu2 (BO3 )2 shows a large region of finite magnetization appearing around 20T. We can think of two possible explanations. Firstly one expects a phase with a condensation of two-triplet bound states to be stabilized in this region [27, 47, 48, 53]. However, as already discussed in section 4.2.1, the classical approximation cannot describe that type of phase and replaces it with a phase having zero-magnetization. Secondly, this could again be explained by residual couplings. In particular, one of the effects of the inter-dimer Dzyaloshinskii-Moriya interactions should be to add a nearest-neighbor hopping term in the effective Hamiltonian [27, 89, 90]. This term should be first order in the coupling constant, and therefore have an amplitude of the order J/100 which corresponds to the energy scale of the interactions dominating the physics of the low-magnetization plateaus. As a consequence, the system may stabilize a superfluid phase at low magnetization which could explain the behavior of the measured magnetization curve. Note that depending on the exact value of the resulting hopping term, one could also expect the superfluid phase to replace the 1/9 plateau.
5.3
Conclusion
In this chapter we studied the spin 1/2 Shastry-Sutherland model in the presence of a magnetic field. We started by deriving an effective Hamiltonian with perturbative continuous unitary transformations (PCUT) up to high-order in J ′ /J in order to extract the relevant low-energy degrees of freedom. By analysing the resulting Hamiltonian, we realized that it has several interesting features. Firstly, it is dominated by interactions which consist of the usual two body interactions but also of three and four-body interactions. Secondly, for the values of J ′ /J corresponding to SrCu2 (BO3 )2 , the longer-range interactions become important and cannot be neglected anymore. Thirdly, the usual hopping is extremely small and the kinetic energy is dominated by correlated hopping. After having derived the effective Hamiltonian, we determined its zero-temperature phase diagram for couplings J ′ /J 6 0.5 and densities n 6 0.5 by using a classical approximation. We found that the phase diagram is dominated by plateaus, which is a consequence of the dominant interactions. In the low J ′ /J limit where only short-range interactions are important, only 1/2 and 1/3 plateaus are stabilized. However, at larger J ′ /J, we found that the longer-range interactions stabilize additional high-commensurability plateaus at 1/9, 2/15, 1/6 and 2/9. We found that the physics of the low density plateaus (1/9, 2/15 and 1/6) is essentially determined by the terms of the effective Hamiltonian appearing at order 6, which confirms the necessity of the high-order perturbative expansion used in this work. We were also able to determine the local magnetization in the various plateaus and found that the building brick of the low density plateaus is a triplet dressed by a polarization cloud, in agreement with the interpretation of Cu NMR on SrCu2 (BO3 )2 [32]. However, these predictions are not in agreement with experimental SrCu2 (BO3 )2 measurements. The most probable explanation is that the Shastry-Sutherland model is not sufficient to describe the physics of SrCu2 (BO3 )2 , and additional couplings are necessary. Dzyaloshinskii-Moriya interactions, further neighbor in plane exchange couplings,
124
The Shastry-Sutherland model
and inter-plane exchange couplings are indeed present in SrCu2 (BO3 )2 but were neglected because they are small with respect to the intra and inter-dimer couplings. However, our results suggest that they could be important for the physics at low magnetization. On the theoretical side, one should therefore focus on the study of the ShastrySutherland model with these additional couplings and determine their effect on the lowmagnetization physics. It could also be interesting if the 2/15 plateau could be confirmed by other approaches, although it will not be possible with an approach using exact diagonalizations because of the large clusters required to stabilize this plateau. On the experimental side, our results suggest to have a closer look at the magnetization below 1/3, with two issues in mind: The absolute value of the magnetization at the plateau currently assumed to be 1/8, and the presence of 1/6 and 2/9 plateaus.
CHAPTER
6
General conclusion The identification of exotic states of quantum matter is an important issue in current research on strongly correlated quantum systems. Frustrated magnetic systems are really interesting for this issue, since the frustration usually induces large quantum fluctuations which prevents the system from stabilizing simple classical long-range magnetic order. By studying two specific examples of frustrated magnetic systems, we confirmed that frustration can lead to rich phase diagrams with competing exotic types of order. In the first part of this work, we studied the compass model which is a minimal model for the description of the orbital degrees of freedom in transition metal compounds. The system is expected to develop long-range orbital correlations in one spatial direction x (when Jx > Jz ) or z (when Jx < Jz ) with a possible first order transition at Jx = Jz where the correlations are long-range in both directions. However, this is not a simple classical long-range order. Indeed, although the direction of the pseudo-spins is fixed, the system is still free to flip the z component (x component) of all the pseudo-spins of one column (row) which leads to a nematic type of order. For transition metal compounds, this result shows that one should expect the orbital degrees of freedom to be essential for the lowtemperature physics. In particular, it will not be possible to describe the low-temperature phases by assuming a simple orbital ordering and studying only the magnetic degrees of freedom. However, the compass model is only a minimal model which has not been built in order to give predictive results for real transition metal compounds. The next step towards the understanding of orbital and magnetic ordering in transition metal compounds is to study more realistic models. In particular, the exact form of the couplings between the pseudo-spins is strongly related to the crystal structure of the compound and to the specific shape of the orbitals relevant for the magnetic degrees of freedom. Finally, the magnetic degrees of freedom should also be important and one may expect the coupling between spins and orbitals to generate a rich phase diagram. The second part of this work was devoted to the frustrated quantum antiferromagnet SrCu2 (BO3 )2 . We found that the zero-temperature phase diagram of the Shastry-
126
General conclusion
Sutherland model in a magnetic field, which is used to describe the magnetic properties of this material, is dominated by an interesting series of plateaus. The phases inside the various plateaus can be described by a crystallization of elementary excitations corresponding to triplets dressed by a polarization cloud. Although these results are really interesting for the understanding of the Shastry-Sutherland model, they are not in agreement with experimental SrCu2 (BO3 )2 measurements. In a near future, one should focus on this issue. On the experimental side, it would be interesting to perform new experimental measurement with our predictions in mind, while on the theoretical side one should study more realistic models consisting of a generalization of the Shastry-Sutherland model with additional longer-range in plane exchange couplings, inter-plane exchange couplings as well as Dzyaloshinskii-Moriya interactions. We also studied a minimal model in order to understand the effect of the correlated hopping which dominate the kinetic energy of the triplets in the Shastry-Sutherland model. Apart from a very rich phase diagram, the most interesting results was that correlated hopping strongly favor supersolid phases. As a consequence, we expect supersolid phases to appear naturally in materials characterized by a frustrated arrangement of dimers. Although recent NMR and torque measurements on SrCu2 (BO3 )2 are consistent with a supersolid of triplets above the 1/8 plateau, this remains to be firmly established.
Part III Appendix
APPENDIX
A
Green’s function Monte Carlo This appendix presents the Green’s function Monte Carlo (GFMC) method used to determine the zero-temperature properties of the compass model. The GFMC is a stochastic method which can be used to evaluate the ground state expectation values of observables for large finite size systems. This method is exact in the sense that it does not introduce any bias in the results and should give exact ground state properties after an infinite number of iterations. It is obviously not possible to do so, however the statistical error due to a finite number of iterations can be evaluated and gives a reliable estimate of the exact result. This appendix only presents an overview of the GFMC algorithm. More details can be found in the references [91–95].
A.1 A.1.1
Single walker GFMC Power method
Consider a system described by an Hamiltonian H and let {|ψi i}i=0,1,··· be a basis of the Hilbert space such that H|ψi i = Ei |ψi i with |ψ0 i corresponding to the ground state. Suppose that we have a trial state |ψT i such that hψT |ψ0 i = 6 0. This state can be expanded in the basis of the eigenvectors of H: X |ψT i = hψ0 |ψT i|ψ0 i + hψi |ψT i|ψi i. (A.1) i6=0
Next we define G = λ − H and suppose that it is possible to determine λ such that all eigenvalues of G are positive. Applying Gn onto the trial state give ! X λ − Ei n n n G |ψT i = (λ − E0 ) hψ0 |ψT i|ψ0 i + hψi |ψT i|ψi i . (A.2) λ − E0 i6=0
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Green’s function Monte Carlo
Because all eigenvalues λ − Ei of G are positive and E0 is the ground state energy of H, we have 0 < (λ − Ei )/(λ − E0 ) < 1 for all i and (A.3) converge to |ψ0 i. lim Gn |ψT i ∝ |ψ0 i.
n→∞
(A.3)
This result, which is known as the power method, is a powerful tool to filter the ground state out of a given trial state. In principle, a simple way to use it would be to store all components of a trial vector in a computer, apply several times G on this vector, and read the resulting state. However, for a system of N spins 1/2, the size of the Hilbert space increase as 2N , which is quickly too large to be stored in a computer. A solution to this problem is to use a stochastic approach.
A.1.2
Stochastic process
A walker (w, x) is defined by a state (or configuration) |xi and a weight w associated to this configuration. We suppose that |xi is an element of a basis of the Hilbert space {|xi}x=1,2,··· (for example, the common eigenstates of {Srz }). For simplicity, we assume that all the components hx|ψ0 i are real. We want to define a Markovian process for this walker, such that the probability distribution of the walker P (w, x) satisfies Z hx|ψ0 i = wP (w, x)dw. (A.4) This can be done by first decomposing Gx′ ,x = hx′ |λ − H|xi in two parts Gx′ ,x = px′ ,x bx
(A.5)
P with bx = P i Gi,x a normalization factor and px′ ,x a stochastic matrix which satisfies px′ ,x > 0 and x′ px′ ,x = 1. For a given walker (w, x), the new walker (w ′ , x′ ) is obtained by choosing x′ with probability px′ ,x and fixing w ′ = bx w. The evolution of the walkers defined here corresponds to a Markovian process with a transition probability px′ ,x δ(w ′ − bx w). If we denote by Pn (w, x) the probability density to have a walker (w, x) at the n-th iteration then the probability Pn+1 (w ′, x′ ) to have a walker (w ′, x′ ) at the next iteration is given by Z X ′ ′ (A.6) Pn+1 (w , x ) = dw px′ ,x δ(w ′ − bx w )Pn (w, x) |{z} x w ˜ Z X dw ˜ w˜ ′ = px′ ,x δ(w − w)P ˜ n ,x (A.7) bx x bx X px′ ,x w ′ = Pn ,x . (A.8) b b x x x
In order to check if (A.4) is satisfied, we look at the first moment of Pn Z Mn (x) = wPn (w, x)dw.
(A.9)
A.1 Single walker GFMC
131
Using (A.8) we have ′
Z
w ′ Pn+1 (w ′ , x′ )dw ′ Z X px′ ,x w ′ ′ Pn , x dw ′ = w b b x x x
Mn+1 (x ) =
(A.10) (A.11)
and with a change of variable w ′ = bx w ′
Mn+1 (x ) =
X
px′ ,x bx
x
=
X
Z
wPn (w, x)dw
(A.12)
px′ ,x bx Mn (x)
(A.13)
Gx′ ,x Mn (x).
(A.14)
Gx′ ,x Mn (x)
(A.15)
x
=
X x
Applying several times this result gives Mn+1 (x′ ) =
X x
=
XX x
Gx′ ,x Gx,x′′ Mn−1 (x′′ )
(A.16)
x′′
.. . X = (Gn+1 )x′ ,x M0 (x).
(A.17)
x
By comparing this result to (A.3) and identifying M0 (x) = hx|ψT i then we have ′
lim Mn (x ) = lim
n→∞
n→∞
Z
wPn (w, x)dw = hx|ψ0 i.
(A.18)
With limn→∞ Pn (w, x) = P (w, x) this result shows that (A.4) is satisfied. We can see here the origin of the famous sign problem in quantum Monte Carlo ′ methods. P In order to have a well defined Markovian process, the probability px ,x = Gx′ ,x / i Gi,x and therefore the matrix elements Gx′ ,x = λ − Hx′ ,x have to be positive for all x, x′ . While the diagonal matrix elements Gx,x = λ − Hx,x can always be chosen positive by tuning λ, this is not true for the off-diagonal matrix elements Gx′ ,x = −Hx′ ,x . As a consequence, the Green’s function Monte Carlo method can only be used when all the off-diagonal matrix elements Hx′ ,x are negative.
A.1.3 Assuming
Ground state energy P
x hx|ψ0 i
= 6 0, the ground state energy E0 can be obtained via
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Green’s function Monte Carlo
− E0 |ψ0 i x hx|ψ0 i P x hx|G|ψ0 i = P hx|ψ0 i P x ′ x,x′ Gx,x′ hx |ψ0 i P = . x hx|ψ0 i
λ − E0 =
Using (A.4), the energy becomes λ − E0
P
x hx|λ P
R Gx,x′ wP (w, x′)dw P R = wP (w, x)dw P Rx ′ x′ Rbx′ wP (w, x )dw P . = wP (w, x)dw x P
x,x′
Using the stochastic average h· · · i, defined by XZ hAi ≡ A(w ′ , x′ )P (w ′, x′ )dw ′
(A.19) (A.20) (A.21)
(A.22) (A.23)
(A.24)
x′
for any function A(w, x), the energy can be rewritten λ − E0 =
hbwi . hwi
(A.25)
The advantage of this formulation is that we know how to sample efficiently these stochastic average using Monte Carlo. Suppose that we have a set of walkers {(wi, xi )}i=1,2,··· distributed according to P (w, x). Then the stochastic average is given by N 1 X A(wi , xi ). hAi = lim N →∞ N i=1
(A.26)
But our Markov process is just a way of building such a set {(wi , xi )}i=1,2,··· . So the energy can be approximated by generating a set of walkers {(wi , xi )}i=1,2,··· using the procedure described previously and evaluating PN i=1 bxi wi . (A.27) λ − E0 ≃ P N i=1 wi
The results should be a good approximation provided that N is large enough. In addition, it is possible to evaluate the statistical error due to a finite N. Supposing that we have a set of N independent walkers {(wi , xi )}i=1,2,··· ,N , the statistical error on the energy can be estimated by s 1 1 hbx wi2 (bx w)2 √ − (A.28) w hwi2 N hwi
A.1 Single walker GFMC
133
P where hF (w, x)i = N1 i F (wi , xi ). However, because the walkers are obtained with a Markovian process, they are clearly not independent. Starting from a walker at (wi , xi ) one will need several iterations (say ncorr ) in order to get a walker (wi+ncorr , xi+ncorr ) which is not correlated with (wi , xi ). Supposing that ncorr is known, one can therefore build a set of independent walkers (Wn , Xn ) by splitting the set of N walkers into N/ncorr subsets of size ncorr and defining (Wn , Xn ) as the average of the (wi , xi ) onto the n-th subset. The correlation time ncorr can be estimated by studying the behavior of the statistical error as a function of ncorr . It should first increase and then reach a saturation value when ncorr becomes larger than the true correlation time. In principle, the evaluation of the energy described in this section should work but in practice it does not. Indeed, after each iteration the weight are changed according to w ′ = bx w. Supposing that the order of magnitude of bx is b, after N iterations it should scale like N −1 Y wN = w0 bxi ∼ bN −1 (A.29) i=1
which, for large N, should either diverge if b > 1 or vanish if b < 1. One way to circumvent this problem is to do several distinct small runs with l iterations each in order to accumulate statistics. Another easier solution is to do one long run with N ≫ l iterations and consider it as several superimposed runs of l iterations. Each configuration xn in the Markov chain can be seen as a starting configuration for a small run of l iterations. If the weight is set to 1 at the beginning of each small run, Q Q after l iteration it will be wn+l = li=1 bxn+i and we keep the walker (xn+l , wn+l = li=1 bxn+i ) for the statistics. If this operation is repeated at each iteration n, the energy (A.27) becomes P P Ql bx n bxn+l wn+l n bxn+l λ − E0 ≃ P = P Ql i=1 n+i . n wn+l i=1 bxn+i n
It can also be rewritten
with
P l n bxn Gn λ − E0 ≃ P l n Gn Gln =
l Y
(A.30)
(A.31)
bxn−i .
i=1
This approach works well provided that l is adapted to each system. It should be large enough for the algorithm to converge and small enough for the weights not to diverge.
A.1.4
Importance sampling
The convergence of the algorithm can be strongly improved by using a guiding function |ψG i, provided that |ψG i is close to the ground state. For any operator A we define A˜ by hx′ |ψG i A˜x′ ,x = Ax′ ,x hx|ψG i
(A.32)
134
Green’s function Monte Carlo
˜ by and for any state |ψi we define |ψi ˜ = hx|ψihx|ψG i. hx|ψi
(A.33)
With these notations, the power method (A.3) still holds ˜ n |ψ˜T i = |ψ˜0 i. lim G
n→∞
(A.34)
Just like in section A.1.2, one can build a Markovian process to generate a set walkers (w, x) distributed with probability P (w, x) which satisfy Z ˜ hx|ψ0 i = wP (w, x)dw. (A.35) ˜ x′ ,x = hx′ |G|xi ˜ To do this we decompose G in two parts ˜ x′ ,x = px′ ,x bx G
(A.36)
P ˜ with bx = P i Gi,x a normalization factor and px′ ,x a stochastic matrix which satisfies px′ ,x > 0 and x′ px′ ,x = 1. For a given walker (w, x), the new walker (w ′ , x′ ) is obtained by choosing x′ with probability px′ ,x and fixing w ′ = bx w. Exactly like in section A.1.2 we can show that the probability distribution of these walkers satisfies (A.35). If the guiding function is chosen such that all its components hx|ψG i are real, the ground state energy can be obtained by hψG |G|ψ0 i hψ |ψ i P G 0 ′ x,x′ hψG |xiGx,x′ hx |ψ0 i P = x hψG |xihx|ψ0 i P hx′ |ψG i ˜ hx′ |ψ˜0 i x,x′ hψG |xi hx|ψG i Gx,x′ hx′ |ψG i = P hx|ψ˜0 i x hψG |xi hx|ψG i P ′ ˜ ˜ x,x′ Gx,x′ hx |ψ0 i = . P hx|ψ˜0 i
λ − E0 =
(A.37) (A.38) (A.39)
(A.40)
x
Using (A.35) it can be rewritten in terms of stochastic averages λ − E0
R ˜ x,x′ wP (w, x′)dw G P R = wP (w, x)dw P Rx ′ x′ Rbx′ wP (w, x )dw P = wP (w, x)dw x hbwi . = hwi P
x,x′
(A.41) (A.42) (A.43)
A.1 Single walker GFMC
135
In order to see how the guiding function can improve the convergence, let us consider the limiting case where it is exactly equal to the ground state hx|ψG i = hx|ψ0 i: X ˜ x,x′ (A.44) G bx′ = x
=
X
Gx,x′
x
=
hx|ψG i hx′ |ψG i
(A.45)
X hψG |G|x′ i x
(A.46)
hx′ |ψG i
= λ − E0 .
(A.47)
In that specific case, there are no statistical fluctuations, and the stochastic averages are exact even at the first iteration. Since a guiding function close to the ground state can strongly improve the convergence, it is worth spending some time to find an optimal guiding function. For the compass model, we used a jastrow guiding function given by ! 1X vr,r′ σrz σrz ′ |Fx i (A.48) |ψG i = exp 2 r,r′ where |Fx i is the ferromagnetic state in the x direction such that σrz |Fx i = |Fx i for all r. The parameters vr,r′ are determined in order to minimize the energy E=
hψG |H|ψG i . hψG |ψG i
(A.49)
Supposing that E is determined for any set of parameters {vr,r′ }, the minimization can be performed with any simple numerical minimization algorithm. Due to large size of the Hilbert space which increase as 2N for a system of N spins 1/2, it is not possible to evaluate the energy exactly. However, one can again use a stochastic approach. If we denote by {|xi}x=1,2,··· a basis of the Hilbert space (for example, the eigenstates of {σrz }), the energy can be rewritten as P 2 x Ex |hx|ψG i| E= P (A.50) 2 x |hx|ψG i|
with the local energy Ex defined by
Ex =
X Hx,x′ hx′ |ψT i x′
hx|ψT i
.
(A.51)
The sum in (A.50) can be approximated by E≃
1 X Exi N i
(A.52)
136
Green’s function Monte Carlo
where {x0 , x1 , · · · , xN } is a set of N configurations distributed according to |hx|ψT i|2 . P (x) = P 2 x |hx|ψT i|
(A.53)
The set of configurations {xi }i=1,··· ,N can be generated with the Metropolis algorithm: For a given configuration xi , we propose a new trial configuration x′ . If the random number ξ ∈ [0, 1] drawn with uniform distribution satisfy 2 P (x′ ) hx′ |ψT i ξ< = P (x) hx|ψT i
(A.54)
then the trial configuration is accepted as a new configuration (xi+1 = x′ ). If the new configuration is not accepted, the old configuration is conserved (xi+1 = xi ).
A.2
Many walkers GFMC
The previous results can be generalized to the case with M walkers running in parallel. The set of walkers is denoted by (w, x) with w = (w1 , · · · , wM ) and x = (x1 , · · · , xM ). If each walker evolves independently according to the Markov process described in section A.1.4, the probability density to have the walkers (w, x) at iteration n is given by Pn (w, x) = Pn (w1 , x1 )Pn (w2 , x2 ) · · · Pn (wM , xM ) and the property (A.35) becomes Z Z ˜ hx|ψ0 i = dw1 · · · dwM
X
x1 ,··· ,xM
w1 δx,x1 + · · · + wM δx,xM P (w, x) M
(A.55)
(A.56)
where P (w, x) = limn→∞ Pn (w, x) is the probability distribution of the walkers.
A.2.1
Reconfiguration
Because the walkers are independent, this approach is essentially equivalent to the single walker case and does not bring anything new in terms of convergence. But we saw that one problem is that the weight of the walkers either diverges or vanishes after a small number of iterations. With many walkers, it is now possible to regularly remove the walkers with vanishing weight and replace them with copies of walkers with large weight which are the most important for the statistics. This process is called reconfiguration. To be more precise, the initial set of M walkers (w, x) is replaced by a new set of M walker (w ′, x′ ) having all the same weight wi = 1. The new states x′i are chosen randomly among the old configurations xk with a probability proportional to the old weight pk = Pwkwj , j thus keeping mainly the most important walkers. With this process, the probability
A.2 Many walkers GFMC
137
distribution Pn changes and becomes Pn′ with Z Z ′ ′ ′ Pn (w , x ) = dw1 · · · dwM G(w ′ , x′ ; w, x) =
X
G(w ′ , x′; w, x)Pn (w, x)
x1 ,··· ,xM
! P P ′ w δ j j xi ,xj j wj ′ P . δ wi − M j wj
M Y i=1
(A.57)
(A.58)
However, it is possible to show that the property (A.56) is still valid. It means that although this transformation introduce some correlations between the walkers, the bias is controlled and does not introduce errors in the evaluation of expectation values.
A.2.2
Ground state energy and stochastic average
In order to compute the energy , the walkers are initially set to random configurations and their weights are set to 1. Each walker iterates according to the Markovian process defined in section A.1.4 and every kb iterations the reconfiguration process is applied. Before every reconfiguration, the average weight is stored. M 1 X n w¯ = w M i=1 i n
(A.59)
as well as the energy of the walkers ¯bxn =
PM
n n i=1 bxi wi . n i=1 wi
PM
(A.60)
Here (win , xni ) are the weight and configuration of the walker i just before the n-th reconfiguration. After each reconfiguration, all the weights are set to 1. After several reconfigurations, the energy can be estimated by P ¯ l n bxn Gn λ − E0 ≃ P (A.61) l n Gn
with
Gln
=
l−1 Y
w ¯ n−j
(A.62)
j=0
A.2.3
Expectation value of diagonal operators (forward walking)
The algorithm used to evaluate the energy, can be directly used to evaluate expectation value of any operator O which is diagonal in the {|xi} basis: hx|O|x′ i = δx,x′ Ox,x . The expectation value is given by a similar expression P l hψ0 |O|ψ0 i n O x n Gn = P . (A.63) l hψ0 |ψ0 i n Gn
138
Green’s function Monte Carlo
With Oxn the weighted average of the operator O on the walkers measured just before the n-th reconfiguration PM n wi Oxn ,xn Oxn = i=1 PM ni i i=1 wi and Gln
Gln =
l−1 Y
w¯ n−j .
(A.64)
j=−k
Note that the product starts at j = −k whereas for the energy it started at j = 0. The parameters l and k has to be adapted for each system. They should be large enough for the algorithm to converge and small enough for the weights not to diverge.
APPENDIX
B
Exact diagonalization
The simplest method in order diagonalize an Hamiltonian H consists in evaluating all the matrix elements hx|H|x′i in a given basis {|xi} of the Hilbert space and using an efficient numerical library (for example LAPACK) to diagonalize the resulting matrix. It is then straightforward to evaluate the ground state expectation value of any observable. This the simplest exact diagonalization method. This method is interesting because it is exact (up to computer rounding precision), however it has a serious drawback since it can only be used for small lattice. Indeed, efficient diagonalization algorithms implemented in LAPACK on modern personal computers can only diagonalize matrices whose size do not exceed ∼ 5000 × 5000, and therefore the size of the Hilbert space should not exceed 5000. For a system with N spins 1/2 the size of the Hilbert space is 2N , which means that this approach can only diagonalize systems with up to 12 sites. Hopefully there are some possibilities in order to diagonalize larger systems. The first solution (described in section B.1) consists in using the symmetries of the system in order to build a block diagonal matrix having approximately as many blocks as the number of symmetries used. It is then possible to diagonalize each block separately, with the restriction that each block should be smaller than ∼ 5000 × 5000. For spins 1/2 systems with SU(2) and translational symmetries, one can then diagonalize systems having up to 16 sites on a personal computer. A second solution, which can be used together with the symmetries, is the Lanczos approximation (described in section B.2). This approximation is used to project the Hamiltonian onto a low-energy subspace which is much smaller than the initial Hilbert space. Because the size of the low-energy subspace can be really small (∼ 100 states) the main limitation comes from the memory of the computer since one still has to store vectors having as many elements as the size of the Hilbert space. By combining the use of the symmetries and the Lanczos approximation, it becomes possible to study the low energy physics for systems having up to 36 spins 1/2 on a personal computer.
140
B.1
Exact diagonalization
Symmetries
Consider a system having Ns symmetries {S1 , · · · , SNs } which commute with each other ([Si , Sj ] = 0 for all i, j) and form a group. Note that S is a symmetry of the system if it is unitary (SS † = S † S = 1) and commutes with the Hamiltonian ([H, S] = 0). It is then possible to find an orthonormal basis of the Hilbert space whose elements are simultaneously eigenvectors of all the operators {S1 , · · · , SNs }. Let us denotes the elements of this basis by |l, s1 , · · · , sNs i where si is the eigenvalue of the symmetry Si Si |l, s1 , · · · , sNs i = si |l, s1 , · · · , sNs i
(B.1)
and l = 1, 2, · · · span the states with degenerate eigenvalues (s1 , · · · , sNs ). Next we consider the state |ψi = H|l, s1, · · · , sNs i. Since all the symmetries commute with the Hamiltonian, we have Si |ψi = HSi |l, s1 , · · · , sNs i = si H|l, s1 , · · · , sNs i = si |ψi
(B.2) (B.3) (B.4)
for all i. Therefore |ψi is an eigenvector of all the symmetries with the same eigenvalues (s1 , · · · , sNs ) and can be written as the linear combination X H|l, s1 , · · · , sNs i = (B.5) al′ |l′ , s1 , · · · , sNs i. l′
As a consequence, the matrix elements of H between two states |l, s1 , · · · , sNs i and |l′ , s′1 , · · · , s′Ns i are given by X al′′ hl′ , s′1 , · · · , s′Ns |l′′ , s1 , · · · , sNs i (B.6) hl′ , s′1 , · · · , s′Ns |H|l, s1, · · · , sNs i = l′′
which is non-zero only if the two states |l, s1 , · · · , sNs i and |l′ , s′1 , · · · , s′Ns i have the same eigenvalues (s1 , · · · , sNs ) = (s′1 , · · · , s′Ns ), since the basis is orthogonal. This results means that the Hamiltonian is block diagonal with each block acting into a given symmetry sector. Note that a symmetry sector (s1 , · · · , sNs ) is defined as the subspace whose elements have eigenvalues (s1 , · · · , sNs ). In some specific cases, it can be easy to determine the basis of eigenvectors common to all the symmetries. For example, in the compass Q z model (see chapter 2) we chose to use the symmetries {P1 , P2 , · · · , PL }. Since Pj = i σi,j , by working in the basis of eigenvectors z common to all the σi,j , the Hamiltonian is directly block diagonal. However, in the general case it is not a trivial task. Hopefully, there is a recipe. Let {|xi}x=1,2,··· be an orthonormal basis of the Hilbert space. This basis can be separated in equivalence classes which are globally invariant under the action of the symmetries (i.e. {|xi, S1 |xi, · · · , SNs |xi} are in the same equivalence class). In each equivalence class labeled by l, we choose one representative which is denoted by |cl i. The orthonormal basis is then given by 1 X |l, s1 , · · · , sNs i = √ si Si |cl i (B.7) Al i
B.1 Symmetries
141
where Al = µl Ns is the normalization constant with Ns the number of symmetries and µl the number of symmetries which leave |cl i invariant.
Let us illustrate this by a simple example in the case where the symmetry group is given by the group of translations. We consider a 2 × 2 square lattice with periodic boundary conditions. The group of translation is given by {1, Tx , Ty , Tx Ty } and their eigenvalues are given by 1, eikx , eiky and ei(kx +ky ) respectively with kx , ky ∈ {0, π}. We choose the basis {|xi}x=1,2,··· ,16 to be the basis {|m1 , · · · , m4 i|m1 , · · · , m4 =↑, ↓} of eigenvectors common to all the operators Siz with eigenvalues mi =↑, ↓: Siz |m1 , · · · , m4 i = mi |m1 , · · · , m4 i.
(B.8)
The sites of the lattice are numbered according to 3 4 1 2 By applying the symmetry group onto the states of the basis, we obtain 7 equivalence classes: l l l l l l l
=1 =2 =3 =4 =5 =6 =7
: : : : : : :
{| ↑↑↑↑i} {| ↓↑↑↑i, | ↑↓↑↑i, | ↑↑↓↑i, | ↑↑↑↓i} {| ↓↓↑↑i, | ↑↑↓↓i} {| ↓↑↓↑i, | ↑↓↑↓i} {| ↓↑↑↓i, | ↑↓↓↑i} {| ↑↓↓↓i, | ↓↑↓↓i, | ↓↓↑↓i, | ↓↓↓↑i} {| ↓↓↓↓i}
and we chose the representative |c1 i = | ↑↑↑↑i, |c2 i = | ↓↑↑↑i, |c3 i = | ↓↓↑↑i, |c4 i = | ↓↑↓↑i, |c5 i = | ↓↑↑↓i, |c6 i = | ↑↓↓↓i, |c7 i = | ↓↓↓↓i. The µl are given by µ1 = 4, µ2 = 1, µ3 = 2, µ4 = 2, µ5 = 2, µ6 = 1 and µ7 = 4. The new basis is given by equation (B.7) which becomes 1 |l, kx , ky i = √ |cl i + eikx Tx |cl i + eiky Ty |cl i + ei(kx +ky ) Tx Ty |cl i Al
(B.9)
with l = 1, · · · , 7 and kx , ky = 0, π. This should in principle give 28 states, however 12 of them are zero which leave 16 states. The resulting basis contains seven states in the
142
Exact diagonalization
sector (kx , ky ) = (0, 0): |l = 1, kx = 0, ky = 0i = | ↑↑↑↑i 1 (| ↓↑↑↑i + | ↑↓↑↑i + | ↑↑↓↑i + | ↑↑↑↓i) |l = 2, kx = 0, ky = 0i = 2 1 |l = 3, kx = 0, ky = 0i = √ (| ↓↓↑↑i + | ↑↑↓↓i) 2 1 |l = 4, kx = 0, ky = 0i = √ (| ↓↑↓↑i + | ↑↓↑↓i) 2 1 |l = 5, kx = 0, ky = 0i = √ (| ↓↑↑↓i + | ↑↓↓↑i) 2 1 |l = 6, kx = 0, ky = 0i = (| ↑↓↓↓i + | ↓↑↓↓i + | ↓↓↑↓i + | ↓↓↓↑i) 2 |l = 7, kx = 0, ky = 0i = | ↓↓↓↓i three states in the sector (kx , ky ) = (π, 0): 1 (| ↓↑↑↑i − | ↑↓↑↑i + | ↑↑↓↑i − | ↑↑↑↓i) 2 1 |l = 4, kx = π, ky = 0i = √ (| ↓↑↓↑i − | ↑↓↑↓i) 2 1 (| ↑↓↓↓i − | ↓↑↓↓i + | ↓↓↑↓i − | ↓↓↓↑i) |l = 6, kx = π, ky = 0i = 2 |l = 2, kx = π, ky = 0i =
three states in the sector (kx , ky ) = (0, π) 1 (| ↓↑↑↑i + | ↑↓↑↑i − | ↑↑↓↑i − | ↑↑↑↓i) 2 1 |l = 3, kx = 0, ky = πi = √ (| ↓↓↑↑i − | ↑↑↓↓i) 2 1 |l = 6, kx = 0, ky = πi = (| ↑↓↓↓i + | ↓↑↓↓i − | ↓↓↑↓i − | ↓↓↓↑i) 2 |l = 2, kx = 0, ky = πi =
and three states in the sector (kx , ky ) = (π, π): 1 (| ↓↑↑↑i − | ↑↓↑↑i − | ↑↑↓↑i + | ↑↑↑↓i) 2 1 |l = 5, kx = π, ky = πi = √ (| ↓↑↑↓i − | ↑↓↓↑i) 2 1 |l = 6, kx = π, ky = πi = (| ↑↓↓↓i − | ↓↑↓↓i − | ↓↓↑↓i + | ↓↓↓↑i) . 2
|l = 2, kx = π, ky = πi =
Therefore in this basis, an Hamiltonian which is invariant by translation will be block diagonal with one block of size 7×7, and three blocks of size 3×3. Note that if in addition the Hamiltonian is invariant under the rotations about the z axis, the Hamiltonian will z conserve Stot . Thanks to our choice of the initial basis whose elements are eigenvectors
B.2 Lanczos approximation
143
of this symmetry, the resulting basis is directly a basis of eigenvectors common to the rotational and translational symmetries. Therefore the block in sector (kx , ky ) = (0, 0) z will be divided in five blocks of size 1, 1, 3, 1 and 1 in the sectors (Stot , kx , ky ) = (2, 0, 0), (1, 0, 0), (0, 0, 0), (−1, 0, 0), (−2, 0, 0) respectively. Similarly, each block in the sectors (kx , ky ) = (π, 0), (0, π) and (π, π) will be divided in three block of size one in the sectors z (Stot , kx , ky ) = (±1, 0, π), (0, 0, π), (±1, π, 0), (0, π, 0), (±1, π, π) and (0, π, π).
B.2
Lanczos approximation
The Lanczos algorithm is an iterative method to tridiagonalize a matrix H. This algorithm builds recursively a new set of orthonormal basis vectors in which the matrix H is tridiagonal. In the case where H is hermitian, the new basis {|ψn i}n=1,2,··· is generated using the recurrence relations βn+1 |ψn+1 i = H|ψn i − αn |ψn i − βn |ψn−1 i
(B.10)
where αn βn β1 |ψ0 i
= = = =
hψn |H|ψn i hψn |H|ψn−1i 0 0
(B.11) (B.12) (B.13) (B.14)
and the initial state |ψ1 i is usually chosen randomly. There are no constraints on the phase of the state |ψn+1 i which can be chosen such that βn+1 is real and positive. The algorithm terminates when |ψn+1 i = 0. Since the new states are generated by applying H recursively, the algorithm will only span a subspace of the Hilbert space corresponding to the symmetry sectors to which the initial state |ψ1 i belongs. For instance, if the initial state is a superposition of states in all symmetry sectors, the algorithm will span the whole Hilbert space. Whereas in the other extreme case where the initial state is only in one symmetry sector, the algorithm will span only this sector. It is therefore important to start by an analysis of the symmetries in order to build an initial state in the relevant symmetry sector. After n iterations, the projection of H onto the subspace generated by the Lanczos algorithm is tridiagonal, real and symmetric: α1 β2 0 · · · 0 .. .. . β2 α2 β3 . .. . H (n) = . 0 0 β α 3 3 . . . . . . . . . . . . βNH 0 · · · 0 βNH αNH
Although this is already an improvement since tridiagonal matrices can be much more efficiently diagonalized with numerical methods than general matrices, the algorithm has
144
Exact diagonalization
other interesting properties. Firstly, one does not need to store the whole matrix and all the new basis vectors in memory. It is sufficient to store the three vectors |ψn−1 i, |ψn i and |ψn+1 i as well as the non-zero matrix elements before and after the transformation. Secondly, the extreme eigenvalues of H (n) converge quickly to the extreme eigenvalues of H. It means that only a small number of iterations are needed to correctly describe the low energy physics. For instance, ∼ 100 basis vectors are sufficient to obtain a precision of the order of 10−8 .
APPENDIX
C
Perturbation theory
The main goal of this appendix is to explain the perturbative approach used to evaluate the energy gap between the lowest and highest energy states adiabatically connected to the ground state manifold of the Jz = 0 limit in the compass model (see section 2.4.2).
C.1
Second order perturbation theory
Consider a system described by an Hamiltonian H = H0 + V where V is a small perturbation which is a linear function of a parameter λ ≪ 1. We suppose that H0 can be exactly diagonalized and we denote by |ν, li the eigenstates having energy Eν H0 |ν, li = Eν |ν, li.
(C.1)
Here l is an additional quantum number which span the degenerate eigenspace. In addition we suppose that the set of eigenstates {|ν, li} form an orthonormal basis of the Hilbert space. Upon increasing λ from the limit λ = 0, one should expect the degeneracy of the ν-th eigenspace to be lifted and each state |ν, li to be continuously transformed into an eigenstate |ψν,l i of the full Hamiltonian having an energy εν,l H|ψν,l i = εν,l |ψν,l i.
(C.2)
146
Perturbation theory
Let us define the projectors Pν = the following properties
P
l
|ν, lihν, l| and Q = 1ν − Pν . These projectors have
[Pν , H0 ] [Qν , H0 ] Q2ν Pν2 Pν Qν = Qν Pν
= = = = =
0 0 Qν Pν 0.
(C.3) (C.4) (C.5) (C.6) (C.7)
Equation (C.2) can be rewritten as H(Pν + Qν )|ψν,l i = εν,l |ψν,l i.
(C.8)
By multiplying (C.8) by Pν from the left and using the properties (C.3)-(C.7) we have (Pν H0 Pν + Pν V Pν )Pν |ψν,l i + Pν V Qν Qν |ψν,l i = εν,l Pν |ψν,l i.
(C.9)
Similarly, by multiplying (C.8) by Qν from the left and using the properties (C.3)-(C.7) we have Qν V Pν Pν |ψν,l i + (Qν H0 Qν + Qν V Qν )Qν |ψν,l i = εν,l Qν Qν |ψν,l i. (C.10) One can then extract Qν |ψν,l i from equation (C.10) Qν |ψν,l i =
1 Qν V Pν Pν |ψν,l i (εν,l − H0 − Qν V Qν )
(C.11)
and insert it into equation (C.9) Pν H0 Pν + Pν V Pν + Pν V Qν
1 Qν V Pν Pν |ψν,l i = εν,l Pν |ψν,l i. (εν,l − H0 − Qν V Qν ) (C.12) With this last expression, we see that the projection of the eigenvectors |ψν,l i onto the ν-th eigenspace of H0 and the corresponding energies εν,l can be obtained as the eigenvectors and eigenvalues of an effective Hamiltonian 1 Qν V Pν . (C.13) Hν = Pν H0 + V + V Qν (εν,l − H0 − Qν V Qν ) Note however that this is not a simple eigenvalue problem since Hν is a function of the energies εν,l . The next step consists in expanding Hν in powers of V . By noticing that εν,l − H0 − Qν V Qν = (εν,l − H0 − Qν V Qν )(εν,l − H0 )−1 (εν,l − H0 ) = (1 − Qν V Qν (εν,l − H0 )−1 )(εν,l − H0 )
(C.14) (C.15)
we have 1 = (εν,l − H0 )−1 (1 − Qν V Qν (εν,l − H0 )−1 )−1 (εν,l − H0 − Qν V Qν ) ∞ X −1 = (εν,l − H0 ) (Qν V Qν (εν,l − H0 )−1 )n n=0
(C.16) (C.17)
C.1 Second order perturbation theory
147
and the effective Hamiltonian becomes ) ( ∞ X 1 1 n (Qν V Qν ) Qν V Pν . Heff,ν = Pν H0 + V + V Qν εν,l − H0 n=0 εν,l − H0
(C.18)
In addition Qν
X X X Pν ′ 1 1 Pν ′ Pν ′′ = Qν = εν,l − H0 εν,l − H0 ν ′′ 6=ν εν,l − Eν ′ ν ′ 6=ν ν ′ 6=ν
and the effective Hamiltonian becomes ( Heff,ν = Pν
H0 + V +
∞ X n=1
X
ν ′ 6=ν
V Pν ′ εν,l − Eν ′
!n ) V
Pν .
(C.19)
(C.20)
In order to remove the unknown energy εν,l from Heff,ν , we use a perturbative expansion in λ (0) (1) (2) Heff,ν = Heff,ν + Heff,ν + Heff,ν + · · · (C.21) (n)
with Heff,ν proportional to λn and similarly for the energies (0)
(1)
(2)
εν,l = εν,l + εν,l + εν,l + · · · (n)
with εν proportional to λn . This has to be done recursively. By comparing equations (C.20) and (C.21) one can directly extract the first term (0)
Heff,ν = Pν H0 Pν
(C.22)
(0)
and therefore εν,l = Eν . To the next order we have (1)
Heff,ν = Pν V Pν
(C.23) (0)
(1)
and one can extract the energy up to first order εν,l = εν,l + εν,l as the eigenvalues of (0) (1) Heff,ν + Heff,ν . For the second order in λ, only the third term in (C.20) with n = 1 will (0)
contribute with εν,l replaced by εν,l (2)
Heff,ν = Pν
X V Pν ′ V (0)
ν ′ 6=ν
εν,l − Eν ′
Pν . (0)
(C.24) (1)
(2)
One can then extract the energy up to second order εν,l = εν,l +εν,l +εν,l as the eigenvalues (0)
(1)
(2)
of Heff,ν + Heff,ν + Heff,ν . This procedure can in principle be used recursively to obtain larger order corrections, however it quickly becomes extremely complicated.
148
C.2
Perturbation theory
Gap in the ground state manifold
In the following we will concentrate on the specific case relevant for the evaluation of the gap in the compass model (see section 2.4.2). We want to determine the gap between lowest and highest eigenvalues of the effective Hamiltonian in the ground state manifold (ν = 0) of the unperturbed Hamiltonian. We consider the specific case where the matrix elements h0, l|V Pν1 V Pν2 V Pν3 · · · V |0, l′ i (C.25) between two different ground states |0, l′i and |0, li are zero if there is less than L operators V in the product. In addition, we suppose that the expectation value h0, l|V Pν1 V Pν2 V Pν3 · · · V |0, li
(C.26)
does not depend on l, as long as there is less than L operators V in the product. In order to evaluate the corrections to the energy ε0,l up to order L − 1, one can evaluate the matrix elements of the effective Hamiltonian (C.20) by taking into account all the terms which consist in a product with less than L operators V !n ) ( L−2 X X V Pν ′ V |0, li . (C.27) h0, l|Heff,0 |0, l′i = δl,l′ E0 + h0, l|V |0, li + h0, l| ε0,l − Eν ′ ′ n=1 ν 6=0
Because h0, l|V Pν1 V Pν2 V Pν3 · · · V |0, l′ i = 0 if l 6= l′ , all the non-diagonal matrix elements are zero. Moreover, since h0, l|V Pν1 V Pν2 V Pν3 · · · V |0, li does not depend on l, all the diagonal matrix elements are the same. Therefore the effective Hamiltonian is proportional to the matrix identity and has only one degenerate eigenvalue ε0,l . Solving this system in order to extract ε0,l and expand it up to order L − 1 in λ is a difficult task, if not impossible. However, since we are only interested in the gap between lowest and highest eigenvalue, it does not matter. The only useful result is that up to order L − 1 there is no lifting of the degeneracy and therefore no contribution to the gap. The first contribution to the gap appears with the term of the effective Hamiltonian (C.20) which has a product of L operators V , in which the energy ε0,l can be replaced by E0 since we are only interested in the lowest order contribution to the gap: P0
X
ν ′ 6=0 (L)
V Pν ′ E0 − Eν ′
!L−1
V P0 .
(C.28)
Although Heff,ν clearly contains other terms, this is the only term having a product of L operators V . Therefore, the lowest order contribution to the gap can be obtained as the difference between lowest and highest eigenvalue of the operator (C.28).
APPENDIX
D
Classical approximation D.1
Method
This appendix presents in more details the classical approximation used in chapters 4 and 5. We consider a system of spins on a lattice described by an Hamiltonian H. Let Sr be the spin operator at site r. The classical approximation consists in replacing the operators Sr by classical vectors of norm S: Sr = S sin(θr ) cos(φr )ex + S sin(θr ) sin(φr )ey + S cos(θr )ez .
(D.1)
The Hamiltonian becomes a function of the variable {θr , φr } and the ground state is the state which minimize the Hamiltonian. Since this minimization cannot be performed for an infinite lattice, one has to use the additional assumption that the classical ground state is periodic, with a periodicity defined by two translation vectors e1 and e2 (see figure D.1 (a)): Sr+e1 = Sr+e2 = Sr . (D.2) It is equivalent to study the system on a finite lattice defined by the vectors e1 and e2 with periodic boundary conditions (see figure D.1 (b)). For a given cluster, the minimization is performed by setting a random initial state {θr , φr} and using a Broyden-FletcherGoldfarb-Shanno algorithm [96] to find a local minimum of the Hamiltonian close to the initial conditions. For a cluster with N spins, this procedure is repeated ∼ 5N times and the solutions {θr , φr } having the minimum energy are kept as ground states. In order not to bias the results by selecting a specific cluster geometry, this procedure is repeated for all the non-equivalent clusters having less than a given number of spins (see section D.2), and the configurations {θr , φr} having the minimum energy are kept as ground state. Note that the classical approximation is free of finite size effects once cluster sizes commensurate with the size of the stabilized structures have been reached. Once a ground state {θr , φr} has been obtained, it is straightforward to evaluate any order parameter.
150
Classical approximation
Figure D.1: (a) We suppose a periodicity defined by the vectors e1 and e2 . (b) It is equivalent to study a finite size cluster defined by e1 and e2 with periodic boundary conditions. For instance
p
(Srx )2 + (Srx )2 and (for k 6= 0) S(k) =
1 X ik(r−r′ ) z z e Sr Sr . N 2 r,r′
Although this approximation is extremely simple and can be performed analytically on a small cluster for a simple Hamiltonian having only a small number of terms, this is not the case anymore for more complicated Hamiltonian like the effective Hamiltonian for the Shastry-Sutherland model or when the clusters contain a large number of spins. For instance, the evaluation of the Shastry-Sutherland model phase diagram in chapter 5 required several weeks of numerical calculation on the Greedy cluster at EPFL having ∼ 200 processors (greedy.epfl.ch). Provided that the Hamiltonian does not contain products of spin operators acting on the same site, this approximation is equivalent to the quantum variational approach on the spin 1/2 coherent states [53] |{φr , θr }i =
Y r
|φr, θr i
(D.3)
where |φr , θr i = eiφr /2 sin (θr /2) | ↓ir + e−iφr /2 cos (θr /2) | ↑ir.
(D.4)
Indeed, the expectation values of the spin operators on these states are equal to the components of the classical spins 1 sin (θr ) cos(φr ) 2 1 sin (θr ) sin(φr ) hφr , θr |Sry |φr , θr i = 2 1 hφr , θr |Srz |φr , θr i = cos (θr ) 2
hφr , θr |Srx |φr , θr i =
(D.5) (D.6) (D.7)
D.2 Finite size clusters
151
Figure D.2: (a) The periodicity of the lattice is given by two vectors ea and eb which define the unit cell. The lattice sites are denoted by circles. (b) A finite size cluster is defined by two vectors e1 and e2 . It contains all the sites (black circles) inside the parallelogram defined by the vectors e1 and e2 . In this example e1 = 3ea + eb and e2 = −4ea + eb .
D.2
Finite size clusters
This section presents the method used to generate the non-equivalent clusters for the classical approximation on a two-dimensional lattice. We consider the case where the lattice can be built by a periodic arrangement of a unit cell. The periodicity of the lattice is defined by two translation vectors ea and eb (see figure D.2 (a)). A finite size cluster with periodic boundary conditions can then be defined by two vectors e1 = a1 ea + b1 eb e2 = a2 ea + b2 eb
(D.8) (D.9)
where ai and bi are integral numbers (see figure D.2 (b)). The cluster consists of all the sites inside the parallelogram defined by the vectors e1 and e2 . The number of sites in the cluster is given by Nu · |a1 b2 − b1 a2 | where Nu is the number of sites per unit cells. Suppose that we want to generate all the possible clusters having a given number of sites. This is obviously impossible since there is an infinite number of such clusters. However, most of them will give the same physical system, and it should be possible to reduce the number of relevant clusters by using only those which are not equivalent. Let us first define what is meant by equivalent clusters. Consider a system defined by an Hamiltonian which is a function of operators Or defined on each site r of a lattice. Suppose that we impose a periodicity defined by the vectors e1 and e2 such that Or+e1 = Or+e2 = Or.
(D.10)
Now consider the same system but with a periodicity defined by the vectors e′1 and e′2 . If the system is invariant under the change of periodicity then the two clusters defined respectively by e1 , e2 and e′1 , e′2 are equivalent.
152
Classical approximation
Figure D.3: (a)-(b) Two clusters defined by the vectors e1 , e2 and e′1 , e′2 are equivalent if: (a) e′1 = ±e1 and e′2 = ±e2 or (b) e′1 = ±e2 and e′2 = ±e1 . (c) The cluster e1 , e2 and e′1 = e1 , e′2 = e2 − e1 are equivalent. All the clusters e′1 , e′2 generated from e1 , e2 by a symmetry of the lattice are clearly equivalent. In addition, there are also some obvious transformations which do not depend on the lattice. All the clusters generated from e1 , e2 by the transformations (e1 , e2 ) → (e′1 , e′2 ) with (see figure D.3 (a)) e′1 = ±e1 e′2 = ±e2
(D.11) (D.12)
e′1 = ±e2 e′2 = ±e1
(D.13) (D.14)
e′1 = e1 + me2 e′2 = e2 + ne1
(D.15) (D.16)
or (see figure D.3 (b))
or (see figure D.3 (c))
with m, n = 0, ±1, ±2, · · · are equivalent. Thanks to the last transformation, all the non-equivalent clusters having N sites (N/Nu unit cells) can be described by e1 = a1 ea + b1 eb e2 = a2 ea + b2 eb
(D.17) (D.18)
with |a1 |, |a2 |, |b1 |, |b2 | < N/Nu . Therefore, a simple algorithm to generate all the nonequivalent clusters having N sites is given by: 1. Generate all possible combinations of a1 , a2 , b1 , b2 with |a1 |, |a2 |, |b1 |, |b2 | < N/Nu . 2. Keep all the clusters having N sites (Nu · |a1 b2 − b1 a2 | = N).
D.2 Finite size clusters
153
Figure D.4: Non-equivalent clusters with up to 8 dimers for the ShastrySutherland lattice. 3. Apply the transformations defined previously to all the clusters and group them by equivalence classes. For the transformations (D.15)-(D.16), use only m and n such that |a1 |, |a2 |, |b1 |, |b2 | < N/Nu after the transformation. 4. Keep only one cluster from each equivalence class. As an example, the non-equivalent clusters with up to 8 dimers obtained for the ShastrySutherland lattice are shown in figure D.4.
154
Classical approximation
APPENDIX
E
Semi-classical approximation
This appendix presents the linear spin-wave approximation which is used to study the minimal model with correlated hopping in section 4.2. This approximation is used to evaluate the effect of small quantum fluctuations around a classical ground state for a system of spins on a lattice (see appendix D.2). By construction, this method can only describe systems whose ground state has long-range magnetic order.
E.1
Model
We consider a system of N spins S on a two-dimensional lattice. We suppose that the system is periodic (i.e. the ground state expectation value of any observable is periodic) with a periodicity defined by the two vectors e1 and e2 . In the following, the finite cluster enclosed in the parallelogram defined by these two vectors will be called a unit cell, although it can be much larger than the unit cell of the lattice since it will be chosen in order to correspond to the periodicity of the classical ground state. In addition, we consider only lattice having L1 × L2 unit cells generated by L1 copies of the unit cell along e1 and L2 copies along e2 (see figure E.1). We denote by Sj,n the spin operator at site Rj + rn , where Rj = x1 e1 + x2 e2 (with xi = 1, 2, · · · , Li ) is the position of the j-th unit cell and rn is the position of the n-th site inside the unit cell. Note that the lattice consists of Nu sub-lattices, labeled by n, where Nu is the number of sites per unit cell. We consider a generalized Heisenberg Hamiltonian in a magnetic field with an additional three-spin term which is required in order to describe the correlated hopping
156
Semi-classical approximation
Figure E.1: The lattice consists in L1 × L2 copies of a unit cell defined by the vectors e1 and e2 . studied in section 4.2: H =
Nu XX
+
n=1
j
+
z Bn Sj,n
Nu XX j
n=1
Nu X XX j
z Sj,n
(E.1)
n=1 τn
XX τn
t (0) Sj,n Jn,τ Sj,n,τn n
(1)
t ′. J Sj,n,τ ′ Sj,n,τn n n,τn ,τn
τn′
Here Nu is the number of sites inside the unit cell, Bn is the magnetic field along the z direction at site rn of the unit cell. We denote by τn (or τn′ ) the vectors between the n-th site of the unit cell and its neighbor, whereas Sj,n,τn is the spin operator at site Rj + rn + τn . Note that the site Rj + rn + τn is in general not in the n-th sub-lattice. The real matrix xx xy xz Jn,τn Jn,τ Jn,τ n n (0) yx yy yz Jn,τ Jn,τ = Jn,τ Jn,τ (E.2) n n n n zz zy zx Jn,τn Jn,τn Jn,τn defines the couplings between the spins Sj,n and Sj,n,τn , while the real matrix (1)
Jn,τn ,τn′
xy xx xz Jn,τ Jn,τ Jn,τ ′ ′ ′ n ,τn n ,τn n ,τn yx yy yz = Jn,τn ,τn′ Jn,τn ,τn′ Jn,τn ,τn′ zy zz zx Jn,τ Jn,τ Jn,τ ′ ′ ′ n ,τn n ,τn n ,τn
(E.3)
defines the couplings between the spins Sj,n,τn and Sj,n,τn′ in the three-spin term. The t notation Sj,n denotes the transposed vector: Sj,n
x Sj,n y t and Sj,n = = Sj,n z Sj,n
y x z Sj,n Sj,n Sj,n
.
(E.4)
E.2 Spins rotation
157
In addition we suppose that the two-spin (three-spin) operators in the Heisenberg term (1) (0) (three-spin term) act on different sites, i.e. Jn,τn = 0 if τn = 0 and Jn,τn ,τn′ = 0 if τn = 0 (1)
(0)
or τn′ = 0. Note that Bn , Jn,τn and Jn,τn ,τn′ do not depend on the unit cell j.
E.2
Spins rotation
Supposing that the classical ground state is given by class Sj,n = S sin(θn ) cos(φn )ex + S sin(θn ) sin(φn )ey + S cos(θn )ez
(E.5)
ej,n the first step consists in transforming the spin operators Sj,n into new spin operators S such that the classical ground state in terms of these new spins becomes the ferromagnetic state with all the spins in the z direction. This can be done with the rotation
where
ej,n Sj,n = Tn S
(E.6)
cos(θn ) cos(φn ) − sin(φn ) sin(θn ) cos(φn ) Tn = cos(θn ) sin(φn ) cos(φn ) sin(θn ) sin(φn ) . − sin(θn ) 0 cos(θn )
(E.7)
After the transformation, the Hamiltonian becomes H =
Nu XX j
+
n=1
ej,n ) + Bn (etz Tn S
Nu XX j
n=1
where
ej,n ) (etz Tn S
Nu X XX n=1 τn
j
XX τn
τn′
e et A(0) S S j,n n,τn j,n,τn
(1) et e S j,n,τn An,τn ,τn′ Sj,n,τn′
t (0) A(0) n,τn = (Tn ) Jn,τn Tn,τn (1)
(E.8)
(1)
An,τn ,τn′ = (Tn,τn )t Jn,τn ,τn′ Tn,τn′ 0 0 1 etz =
(E.9) (E.10) (E.11)
and Tn,τn = Tm where m denote the sub-lattice of the site rn + τn .
E.3
Holstein-Primakoff transformation
ej,n by bosonic operators aj,n and The next step consists in replacing the spin operators S † aj,n using the Holstein-Primakoff transformation q √ a†j,n aj,n + ˜ S = 2S 1 − j,n q 2S † aj,n √ a aj,n (E.12) − S˜j,n = 2Sa†j,n 1 − j,n2S ˜z Sj,n = S − a†j,n aj,n .
158
Semi-classical approximation
With this transformation, the classical ground state becomes the vacuum |0i while the excited states are obtained by adding particles to the vacuum. Assuming that the quantum ground state is close to the classical ground state (i.e. ha†j,n aj,n i ≪ 2S), one can expand the square root around a†j,n aj,n /(2S) = 0 and keep only the lowest order √ + S˜j,n ≃ √2Saj,n S˜− ≃ 2Sa†j,n ˜j,n z Sj,n ≃ S − a†j,n aj,n .
(E.13)
This approximation is clearly good in the limit S → ∞. However it can also be good for small S, provided that the quantum fluctuations around the ground state are small (ha†j,naj,n i ≪ 2S). Replacing the spin operators by the bosonic operators and keeping only terms with less than two bosonic operators leads to Nu XX
(
1 X (1) † Ωn,τn aj,n,τn aj,n,τn 2 τn j n=1 X X (3) X † (2) Ωn,τn ,τn′ a†j,n,τn aj,n,τn′ Ωn,τn aj,n aj,n,τn + +
H = Ec +
† Ω(0) n aj,n aj,n +
+
X τn
τn′
τn
τn
† † ∆(2) n,τn aj,n aj,n,τn +
XX τn
with
Ec
τn′
(3) ∆n,τn ,τn′ a†j,n,τn a†j,n,τn′ + h.c.
Nu Nu X X 1 X NS S+ Bn (Tn )zz + A(0) (S + 1) n,τn zz 2 n=1 Nu n=1 τn N u XX NS 2 3 X (1) + S+ (Tn )zz An,τn ,τn′ Nu 2 n=1 τ zz ′
N = Nu
n
(E.14)
(E.15)
τn
S X (0) 1 An,τn zz B (T ) − Ω(0) = − n n zz n 2 2 τ n S2 X X (1) − (Tn )zz An,τn ,τn′ 2 τ zz τn′ n X S (1) (0) 2 (1) (Tn )zz An,τn ,τn′ Ωn,τn = − An,τn zz − S 2 zz ′ τn
(E.16)
(E.17)
E.4 Fourier transformation Ω(2) n,τn =
159
o S n (0) (0) (0) + i A − i A An,τn xx + A(0) n,τn yx n,τn xy n,τn yy 2 n X S2 (1) + ((Tn )zx + i(Tn )zy ) × An,τn ,τn′ 2 xz τn′ (1) (1) (1) −i (An,τn ,τn′ + An,τn′ ,τn − i (An,τn′ ,τn yz
(3) Ωn,τn ,τn′
zx
S2 (1) (1) = (Tn )zz An,τn ,τn′ + An,τn ,τn′ 2 xx yy (1) (1) −i An,τn ,τn′ + i An,τn ,τn′ xy
(E.18)
zy
(E.19)
yx
o S n (0) (0) (0) An,τn xx − A(0) (E.20) + i A + i A n,τn yy n,τn xy n,τn yx 2 X S2 ((Tn )zx + i(Tn )zy ) + 2 ′ τ n (1) (1) (1) (1) × An,τn ,τn′ + i (An,τn ,τn′ + An,τn′ ,τn + i (An,τn′ ,τn xz yz zx zy 2 S (1) (1) = (Tn )zz An,τn ,τn′ − An,τn ,τn′ (E.21) 2 xx yy (1) (1) +i An,τn ,τn′ + i An,τn ,τn′ .
∆(2) n,τn =
(3)
∆n,τn ,τn′
xy
yx
class Note that the terms linear in the bosonic operators disappear when Sj,n is a stationary point of the classical Hamiltonian.
E.4
Fourier transformation
In order to use the periodicity of the lattice, we perform the Fourier transformation √ Nu X ik(Rj +rn ) e ak,n (E.22) aj,n = √ N k where
with
2π ∗ 2π ∗ k ∈ n1 e1 + n2 e2 |ni = 0, 1, · · · , Li L1 L2 1 e2 × ez (e1 × e2 ) · ez 1 = ez × e1 (e1 × e2 ) · ez
e∗1 = e∗2
160
Semi-classical approximation
and ez is a unit vector perpendicular to e1 and e2 . The resulting Hamiltonian can be written in matrix form X † H = Ec + AkMkAk (E.23) k
with
(+−)
Mk = A†k =
Ak
(++)
Mk Mk (−−) (−+) Mk Mk
!
(E.24)
a†k,1 · · · a†k,Nu a−k,1 · · · a−k,Nu ak,1 .. . ak,Nu = † a −k,1 .. .
(E.25)
(E.26)
a†−k,Nu
and (+−) Mk
n,m
=
Ω(0) n δn,m
+ (−+)
Mk
n,m
(++) Mk
(−−)
Mk
n,m
=
1 2 1 2
X
ikτn ˜ Ω(2) δn+τn ,m + n,τn e
τn
l=1
X
Ω(2) m,τm
τl′
τl
∗
e−ikτm δ˜n,m+τm
!
∗ ′ (3) (3) e−ik(τl −τl ) δ˜l+τl ,n δ˜l+τl′ ,m Ωl,τl ,τ ′ + Ωl,τ ′ ,τl l
l
(E.28)
m,n
X
ikτn ˜ δn+τn ,m + ∆(2) n,τn e
τn
X
−ikτm ˜ δm+τm ,n ∆(2) m,τm e
τm
Nu X X X
n,m
(E.27)
τm
Nu X X X
(+−) M−k
1 = 2
(1)
Ωl,τl δ˜l+τl ,n δn,m
τl
l=1
+
+
Nu X X
1 + 2 l=1 τ l ∗ (−−) = M−k
τl′
(3) ∆l,τl ,τ ′ l
+
(3) ∆l,τ ′ ,τl l
!
(E.29)
′ e−ik(τl −τl ) δ˜l+τl ,n δ˜l+τl′ ,m
n,m
where we used the notation 1 if rn and rl + τl are on the same sub-lattice δ˜l+τl ,n = 0 otherwise
(E.30)
(E.31)
E.5 Bogoliubov transformation
E.5
161
Bogoliubov transformation
In order to diagonalize the Hamiltonian, we introduce new operators bk,n and b†k,n obtained by a linear transformation of the operators ak,n and a†k,n bk,1 .. . bk,Nu (E.32) Ak = SkBk with Bk = † b −k,1 .. . b†−k,Nu
where Sk is a Nu × Nu complex matrix. This transformation should be such that the operators bk,n and b†k,n satisfy the bosonic commutation relations and that the Hamiltonian expressed in terms of the bk,n and b†k,n is diagonal. The first constraint will be satisfied if SkΛSk† = Λ
(E.33)
with Λ=
0 INu 0 −INu
(E.34)
and INu is the Nu × Nu identity matrix. Indeed, the bosonic commutation relations for the operators ak,n and a†k,n can be rewritten as [(Ak)n , (A†k)m ] = Λn,m for all n, m = 1, · · · , 2Nu
(E.35)
and similarly for the operators bk,n and b†k,n [(Bk)n , (Bk† )m ] = Λn,m for all n, m = 1, · · · , 2Nu .
(E.36)
Supposing that the operators ak,n and a†k,n satisfy the bosonic commutation relations and that SkΛSk† = Λ, we have [(Bk)n , (Bk† )m ] = [(Sk† Ak)n , (A†kSk)m ] 2Nu X (Sk† )n,p (Sk)q,m [(Ak)p , (A†k)q ] = | {z } p,q=1
=Λp,q
(Sk† ΛSk)n,m
= = Λn,m
where we used SkΛSk† = Λ ⇒ Λ = Sk−1 Λ(Sk† )−1 ⇒ Λ = (Sk−1 Λ(Sk† )−1 )−1 = Sk† ΛSk. The second constraint will be satisfied if Sk† MkSk = Dk
(E.37)
162
Semi-classical approximation
with Dk a diagonal matrix. Indeed, in that case X † X † † X † H = Ec + AkMkAk = Ec + BkSkMkSkBk = Ec + BkDkBk. k
k
(E.38)
k
This suggest to choose Sk as the matrix whose columns are eigenvectors of Mk and Dk as the matrix whose diagonal elements are the corresponding eigenvalues: MkSk = DkSk.
(E.39)
However, this requires Sk to be unitary and it is in general not possible to find a matrix Sk† satisfying simultaneously Sk−1 = Sk† , SkΛSk† = Λ and MkSk = DkSk. Hopefully, Del Maestro and Gingras [97] proposed a solution. The idea is to define the non-hermitian matrix Pk = −2ΛMk. (E.40) If we denote by S˜k the matrix whose column are eigenvectors of Pk and Dk the matrix whose diagonal elements are the corresponding eigenvalues, then PkS˜k = S˜kDk. (E.41) In addition, suppose that the eigenvectors and eigenvalues are sorted in such a way that (Dk)n,n 6 (Dk)m,m for all n < m. Although in general the matrix S˜k does not satisfy S˜kΛS˜k† = Λ, it is possible to form new linear combinations of the eigenvectors within a given eigenspace such that the new eigenvector matrix, denoted by Sk, satisfies SkΛSk† = Λ [97]. This is done in the following way. Let dj be the degeneracy of the j-th eigenspace. The linear combination of the eigenvectors can be performed by multiplying the matrix S˜k by a block diagonal matrix dB consisting of blocks dBj of size dj × dj for each eigenspace: Sk = S˜kdB. (E.42) We want dB to be such that (S˜kdB)Λ(S˜kdB)† = Λ, which can be rearranged as dBΛdB † = (S˜k† ΛS˜k)−1 . The solution is given by q (0) dBj = Vj Λj ∆j Vj−1 (E.43) (0)
where dBj is the dj × dj block of dB corresponding to the j-th eigenspace. Vj and ∆j are † the dj × dj eigenvectors and eigenvalues matrix of the matrix (S˜k,j Λj S˜k,j )−1 . S˜k,j is the 2Nu × dj matrix whose columns are the dj eigenvectors corresponding to the j-eigenspace of Pk, while Λj is the dj × dj block of Λ corresponding to the j-eigenspace. All the details can be found in the article by Del Meastro and Gingras [97]. Finally, since for a well defined system Mk should have only positive eigenvalues, Pk = −2ΛMk has half of its eigenvalues positive and the other half negative. Therefore −ω1 (k) ··· 0 .. .. . . −ωNu (k) Dk = (E.44) ω (−k) 1 .. .. . . 0 ··· ωNu (−k)
E.5 Bogoliubov transformation
163
with ωn (k) > 0 for all n and k and the Hamiltonian becomes X † AkMk Ak H = Ec + k
= Ec +
X
Bk† Sk† MkSkBk
k
1X † † BkSkΛPkSkBk 2 k 1X † † BkSkΛSkDkBk = Ec − 2 k 1X † = Ec − BkΛDkBk 2 k = Ec −
N
= Ec +
N
u u XX 1 XX ωn (k) + ωn (k)b†k,n bk,n . 2 n=1 n=1
k
k
It is now possible to evaluate expectation values of various operators on the ground state, which corresponds to the vacuum of the bosons bk,n . Let us start with Nu X −ik(Rj +rn ) ik′ (Rj′ +rn′ ) † e e hak,n ak′ ,n′ i ha†j,n aj ′ ,n′ i = N k,k′ Nu X −ik(Rj +rn ) ik′ (Rj′ +rn′ ) = e e h(A†k)n (Ak′ )n′ i N ′ k,k
2N Nu X Xu −ik(Rj +rn ) ik′ (Rj′ +rn′ ) † e e (Sk)m,n (Sk′ )n′ ,m′ h(Bk† )m (Bk′ )m′ i = N ′ ′ k,k m,m =1
= Similarly
Nu Nu X X e−ik(Rj +rn ) eik(Rj′ +rn′ ) (Sk)∗n,Nu +m (Sk)n′ ,Nu +m . N k m=1
(E.45)
Nu X −ik(Rj +rn ) −ik′ (Rj′ +rn′ ) † † e e hak,n ak′ ,n′ i N k,k′ Nu X −ik(Rj +rn ) ik′ (Rj′ +rn′ ) † † = e e hak,n a−k′ ,n′ i N k,k′ Nu X −ik(Rj +rn ) ik′ (Rj′ +rn′ ) e e h(A†k)n (Ak′ )Nu +n′ i = N k,k′
ha†j,n a†j ′ ,n′ i =
=
2N Nu X Xu −ik(Rj +rn ) ik′ (Rj′ +rn′ ) † e e (Sk)m,n (Sk′ )Nu +n′ ,m′ h(Bk† )m (Bk′ )m′ i N k,k′ m,m′ =1
Nu Nu X X = e−ik(Rj +rn ) eik(Rj′ +rn′ ) (Sk)∗n,Nu +m (Sk)Nu +n′ ,Nu +m . N k m=1
(E.46)
164
Semi-classical approximation
The number of Holstein-Primakoff bosons at site Rj + rn is then given by ha†j,n aj,n i =
Nu Nu X X |(Sk)n,Nu +m |2 . N k m=1
(E.47)
The ground state expectation values of the spin operators are given by (α = x, y, z) α hSj,n i = (Tn )αz (S − ha†j,n aj,n i)
(E.48)
while (for Rj + rn 6= Rj ′ + rn′ and α, β = x, y, z) α hSj,n Sjβ′,n′ i =
S (E.49) (((Tn )αx + i(Tn )αy )((Tn′ )βx − i(Tn′ )βy )) ha†j,n aj ′ ,n′ i 2 S + (((Tn )αx − i(Tn )αy )((Tn′ )βx + i(Tn′ )βy )) ha†j ′ ,n′ aj,n i 2 S + (((Tn )αx + i(Tn )αy )((Tn′ )βx + i(Tn′ )βy )) ha†j,n a†j ′ ,n′ i 2 S + (((Tn )αx − i(Tn )αy )((Tn′ )βx − i(Tn′ )βy )) haj,n aj ′ ,n′ i 2 +(Tn )αz (Tn′ )βz (S 2 − Sha†j,n aj,n i − Sha†j ′,n′ aj ′ ,n′ i + ha†j,n aj,n a†j ′ ,n′ aj ′ ,n′ i).
In order to evaluate the term ha†j,naj,n a†j ′ ,n′ aj ′ ,n′ i, we use Wick’s theorem a†j,n aj,n a†j ′ ,n′ aj ′ ,n′ = : a†j,n aj,n a†j ′ ,n′ aj ′ ,n′ : +ha†j,n aj,n i : +ha†j,n aj ′ ,n′ i
a†j ′ ,n′ aj ′ ,n′ : aj,n a†j ′ ,n′ +haj,n aj ′ ,n′ i : a†j,n a†j ′ ,n′ +ha†j,n aj,n iha†j ′,n′ aj ′ ,n′ i +ha†j,n aj ′ ,n′ ihaj,na†j ′ ,n′ i
(E.50) : +ha†j,n a†j ′ ,n′ i : aj,n aj ′,n′ : +haj,n a†j ′ ,n′ i : a†j,n aj ′,n′ : +ha†j ′ ,n′ aj ′ ,n′ i : a†j,n aj,n + ha†j,n a†j ′ ,n′ ihaj,n aj ′ ,n′ i
: : :
where h· · · i is the expectation value on the ground state which is the vacuum of the operators bk,n and : · · · : denotes the normal ordering of the operators bk,n . Since the ground state expectation values of a normal ordered term is zero, we have ha†j,n aj,n a†j ′ ,n′ aj ′ ,n′ i = ha†j,n aj,n iha†j ′,n′ aj ′ ,n′ i + ha†j,n a†j ′ ,n′ ihaj,n aj ′ ,n′ i +ha†j,n aj ′ ,n′ ihaj,na†j ′ ,n′ i
(E.51)
APPENDIX
F
Perturbative continuous unitary transformations This appendix describes the method of perturbative continuous unitary transformations (PCUT) used in chapter 5 to derive the effective Hamiltonian for the Shastry-Sutherland model. The idea of the method is to use a unitary transformation to map the initial system onto an effective system hopefully easier to solve. This appendix only presents an overview of the method and more details can be found in [81, 82, 98–101]. Let us consider a system described by an Hamiltonian H. It is always possible to define an effective Hamiltonian Heff obtained by a unitary transformation U (U † U = UU † = 1) of H (F.1) Heff = U † HU. Supposing that |ψn i is an eigenstate of H with eigenvalue En H|ψn i = En |ψn i
(F.2)
|ψeff,n i = U † |ψn i
(F.3)
then the state will be an eigenstate of Heff with the same eigenvalue: Heff |ψeff,n i = = = =
U † HUU † |ψn i U † H|ψn i En U † |ψn i En |ψeff,n i.
(F.4)
The expectation value hψ|O|ψi of an operator O on a state |ψi can also be obtained as the expectation value of the transformed operator Oeff = U † OU on the transformed state |ψeff i = U † |ψi: hψn |O|ψn i = hψn |UU † OUU † |ψn i = hψeff,n |Oeff |ψeff,n i.
(F.5)
166
Perturbative continuous unitary transformations
Instead of performing one unitary transformation U, Wegner [102] as well as Glazek and Wilson [103, 104] proposed to perform infinitely many infinitesimal unitary transformations e−η(l)dl such that Z l ′ ′ (F.6) U = U(l = ∞) with U(l) = L exp − η(l )dl 0
where the generator η(l) is given by η(l) = −U † (l)∂l U(l) and L denotes the l-ordering operator which sorts from left to right according to ascending values of l. Although it is in principle possible to evaluate the effective Hamiltonian Heff = U † HU by using the equation (F.6), it is extremely difficult. It is easier to solve the flow equation ∂l H(l) = [η(l), H(l)]
(F.7)
where H(l) = U † (l)H(0)U(l) is the effective Hamiltonian obtained with the transformation U(l) and H(0) = H is the initial Hamiltonian. The choice of the generator η(l) is crucial in order to obtain an effective Hamiltonian which is easier to solve than the initial Hamiltonian. In this work, we used the generator proposed by Mielke [105] and by Knetter and Uhrig [81]. This generator produce an effective Hamiltonian which conserves the total number of quasi-particles and can be evaluated in a perturbative approach. To be more specific, we follow the description by Knetter and Uhrig [81] and consider a system whose Hamiltonian can be written as H = H0 + λ
N X
Tn
(F.8)
n=−N
where H0 is an Hamiltonian which has an equidistant spectrum bounded from below {E0 , E1 , E2 , · · · } with E0 < E1 < E2 < · · · . We denote by Ui the eigenspace having energy Ei . The second part of the Hamiltonian is a small perturbation (λ ≪ 1) and the operators Tn are such that if |ψi ∈ Ui then Tn |ψi ∈ Ui+n , i.e. it increments (n > 0) or decrements (n < 0) the number of elementary excitations. For the Shastry-Sutherland model, H0 contains the intra-dimer couplings and Tn increases (or decreases) the total number of triplets by n. In the following we will only consider the case N = 1 which is sufficient for the Shastry-Sutherland model. The general form of the effective Hamiltonian H(l) is given by ∞ X X H(l) = H0 + λk F (l, m)Tm1 Tm2 · · · Tmk (F.9) k=1
m
where m = (m1 , m2 , · · · , mk ) with mi ∈ {0, ±1} and F (l, m) are real-valued functions which have to be determined. The generator is then defined as η(l) =
∞ X k=1
λk
X m
sgn(M(m))F (l, m)Tm1 Tm2 · · · Tmk
where M(m) =
k X i=1
mk
(F.10)
(F.11)
167 One can now derive the flow equations for the coefficients F (l, m) by using equation (F.7) with the Hamiltonian (F.9) and the generator (F.10) with the initial condition H(l = 0) = H. The flow equation can then be solved numerically and the resulting coefficients F (l = ∞, m) can be obtained exactly up to high order k as rational numbers. After the transformation, the effective Hamiltonian can be written as Heff = H(∞) = H0 +
∞ X k=1
λ
k
′ X m
C(m)Tm1 Tm2 · · · Tmk
(F.12)
where C(m) = F (∞, m) and the prime indicates that the sum is restricted to the terms satisfying M(m) = 0. For the Shastry-Sutherland model, this restriction reflects the conservation of the total number of particles. For small λ, one can easily extract a perturbative effective Hamiltonian by keeping only the terms corresponding to an order k smaller than a given maximum order and neglecting the other terms. Finally, the Hamiltonian (F.12) has to be normal ordered, which is not a trivial task in higher order. One way to do this is to evaluate matrix elements of the effective Hamiltonian (F.12) on finite clusters which are chosen such that one obtains results in the thermodynamic limit using the linked cluster theorem [82]. In order to obtain the relevant amplitudes for an n-particle operator, one has to substract all m = 0, 1, · · · , n − 1-particle contributions [106] which arise on a finite cluster. Clearly, the size of the cluster needed to obtain a correct result in the thermodynamic limit increases with the desired perturbative order and the memory needs increases. This effect is larger in higher dimensions and gives one restriction of the PCUT. Additionally, the number of operators in Hamiltonian (F.12) which have to be applied on a given cluster increases strongly from order to order. This sets another constraint on the PCUT because at one point the time for the calculation is too large. For the specific case of the Shastry-Sutherland model, we always used a N = 64 square cluster (actually there are two different ones depending whether the first dimer is chosen to be a horizontal or a vertical one) and we obtained a maximum order 15 in our calculation. The relatively high order in perturbation reached for a two-dimensional problem is a direct consequence of the very special property of the Shastry-Sutherland model that the ground state is exactly the product state of singlets, i.e. there are no operators in the Hamiltonian which create triplets out of the vacuum. This reduces drastically the number of states in intermediate steps during the calculation which have to be stored in the computer. Note that this property will change when additional couplings are added to the Shastry-Sutherland model (assuming these coupling destroy the property of the system to have the product state of singlets as an exact ground state).
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[96] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007. [97] A. G. del Maestro and M. J. Gingras, J. Phys.: Condens. Matter 16, 3339 (2004). [98] S. Dusuel and G. S. Uhrig, J. Phys. A 37, 9275 (2004). [99] F. Mila and K. P. Schmidt, Strong coupling expansion and effective hamiltonians, Springer Series in Solid State Sciences, in press, 2008. [100] C. Knetter, Perturbative continuous unitary transformations: spectral properties of low dimensional spin systems, PhD thesis, University of Köln, 2003. [101] K. P. Schmidt, Spectral properties of quasi one-dimensional quantum antiferromagnets., PhD thesis, University of Köln, 2004. [102] F. J. Wegner, Ann. Phys 3, 77 (1994). [103] S. D. Glazek and K. G. Wilson, Phys. Rev. D 48, 5863 (1993). [104] S. D. Glazek and K. G. Wilson, Phys. Rev. D 49, 4214 (1994). [105] A. Mielke, Eur. Phys. J. B 5, 605 (1998). [106] C. Knetter, K. Schmidt, and G. Uhrig, Eur. Phys. J. B 36, 525 (2004).
Curriculum Vitae Julien Dorier
EPFL - SB - ITP - CTMC BSP 718 CH-1015 Lausanne +41 21 693 05 14 julien.dorier@epfl.ch
Date of birth: 7th dec 1977 Nationality: Swiss
EDUCATION 2004 - 2008 EPFL: Doctoral student. 1999 - 2004 University of Lausanne and EPFL: Physics diploma. 1996 - 1998 Ecole des Arches (Lausanne): Maturité fédérale scientifique. SCHOOLS AND CONFERENCES 2008 - Lecture Introduction à la programmation parallèle, EPFL. 2007 - Summer school Highly frustrated magnets and strongly correlated systems: from non-perturbativ approaches to experiments, Trieste (Italy). - Annual meeting of the SPS, Zurich. - Swiss Workshop MaNEP, Les Diablerets. 2006 - MaNEP summer school, Saas Fee. - Lecture Field theory approaches to quasi-one-dimensional magnetism, Ian Affleck (3ème cycle de la physique en Suisse romande). - Lecture The quantum Hall effects, Pascal Lederer (3ème cycle de la physique en Suisse romande). - SPS/MaNEP meeting, EPFL. - CECAM workshop Novel theoretical aspects of frustrated spin systems, Lyon (France). 2005 - Lecture Méthodes à N corps en matière condensée, Nicolas Macris and Dionis Baeriswyl (3ème cycle de la physique en Suisse romande). - Lecture Interferences and interactions in mesoscopic physics, Laurent Levy (3ème cycle de la physique en Suisse romande). - Lecture Theory and phenomena of unconventional superconductivity, Manfred Sigrist (3ème cycle de la physique en Suisse romande). - Summer school X training course in the physics of correlated electron systems and High-Tc superconductors, Vietri sul Mare (Italy). - MaNEP workshop, Les Diablerets. 2004 - Systèmes vitreux, dynamique lente et vieillissement, Jean-Philippe Bouchaud (3ème cycle de la physique en Suisse romande).
PRESENTATIONS AND POSTERS 2007 2006 -
Poster at the Swiss Workshop MaNEP, Les Diablerets. Presentation at the Annual meeting of the SPS, Zurich. Presentation at Réunion Systèmes fortement corrélés, EPFL. Poster at the SPS/MaNEP meeting, EPFL. Poster at the CECAM workshop Novel theoretical aspects of frustrated spin systems, Lyon (France). 2004 - Poster at Journée des doctorants, EPFL. PUBLICATIONS 1. K. P. Schmidt, J. Dorier and F. Mila, Magnetization plateaux in an extended ShastrySutherland model, Proceedings of HFM 2008, to appear in Journal of Physics: Conference Series (2008). 2. F. Mila, J. Dorier and K. P. Schmidt, Supersolid phases of hardcore bosons on the square lattice: Correlated hopping, next-nearest neighbor hopping and frustration, Proceedings of YKIS2007, to appear in Progress of Theoretical Physics (2008). 3. J. Dorier, K.P. Schmidt and F. Mila, Theory of magnetization plateaux in the Shastry-Sutherland model, Phys. Rev. Lett, in press (2008). 4. K.P. Schmidt, J. Dorier and A. Läuchli, Solids and supersolids of three-body interacting polar molecules in an optical lattice, Phys. Rev. Lett, in press (2008). 5. K.P. Schmidt, J. Dorier, A. Läuchli and F. Mila, Supersolid phase induced by correlated hopping in spin-1/2 frustrated quantum magnets, Phys. Rev. Lett. 100, 090401 (2008). 6. K.P. Schmidt, J. Dorier, A. Läuchli and F. Mila, Single-particle versus pair condensation of hard-core bosons with correlated hopping, Phys. Rev. B 74, 174508 (2006). 7. J. Dorier, F. Becca and F. Mila, The Quantum Compass Model on the Square Lattice, Phys. Rev. B 72, 024448 (2005).
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