ice turning into water, later â iron melting in the furnace, even later â the magnet loosing .... citations of a topological superconductor â Majorana fermions â have non- .... lowest excited state properties of lmg-based models, so semi-classical ap- ...... uniform coupling, the excitation energy becomes zero at the phase bound-.
Phase Transitions in Networks of Quantum Critical Systems
vorgelegt von MSc Phys Aleksandr Sorokin geb. in Riga von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften — Dr. rer. nat. — genehmigte Dissertation
Promotionsausschuss Vorsitzende: . Gutachter: . Gutachter:
Prof. Dr. Phys. Maria Krikunova (TU Berlin) Prof. Dr. rer. nat. Tobias Brandes (TU Berlin) Prof. Pavel Cejnar, DSc. (Univerzita Karlova)
Tag der wissenschaftlichen Aussprache:
Berlin
. November
ZUSAMMENFASSUNG
In der vorliegenden Arbeit erforschen wir Netzwerke von Vielteilchen-Quantensystemen, um neue, durch die Wechselwirkung der individuellen Knoten entstehende Quanten- und topologische Phasen zu beschreiben. Alle betrachteten Netzwerke beruhen auf dem semi-klassischen Lipkin–Meshkov– Glick-Model, und die Kopplungen zwischen den Netzwerk-Knoten werden mit entweder einer oder zwei Kopplungskonstanten beschrieben. Es wurde gezeigt, dass die uniforme anisotropische Jy –Jy Kopplung vier Quantenphasen ergibt (wenn sowie positive, als auch negative Werte für Kopplungskonstanten berücksichtigt werden), die sich in den Mean-Fields und der Entartung der Grundzustände unterscheiden. Für schwache Kopplungen befindet sich das Netzwerk in der paramagnetischen Phase. Wird es jedoch von der Selbstwechselwirkung dominiert, so wird der Grundzustand exponentiell entartet. Für starke Kopplungen ist der Grundzustand zweifach entartet und weist eine Fernordnung auf. Über die Mean-Field-Approximation hinausgehend, erhielten wir analytische Ausdrücke für die Dispersionsrelationen der Energie für ein eindimensionales Netz mit periodischen Randbedingungen. Für relativ kleine Netzwerke rechneten wir die Grundzustandsenergie durch direkte Diagonalisierung des Hamiltonians aus. Das Effekt der Einführung einer zweiten Kopplungskonstante wurde auf dem Beispiel der eindimensionale Kette (eine Erweiterung des sshModels auf Vielteilchensysteme) betrachtet, und die grundsätzliche Frage war die Untersuchung vom Wechselspiel zwischen den topologischen und Quantenphasen. Es wurde gezeigt, dass die Sprungkopplung zwischen Knoten fünf Quantenphasen ergibt, wenn sowie positive, als auch negative Werte für Kopplungskonstanten berücksichtigt werden. Die Analyse von der Quantenfluktuationen um die Mean-Field-Lösungen zeigte, dass es für dieses Model topologisch triviale und nicht-triviale Phasen gibt, die durch den Wert der symplectischen Polarisation voneinander abweichen. Die Bulk–Grenzen-Beziehung wurde auf dem Beispiel einer endlichen Kette mit offenen Randbedingungen getestet, und es wurde bestätigt, dass die Anzahl der lokalisierten Randzustände proportional zum Wert der symplectische Polarisation ist. Weitere Analyse zeigte, dass ein zusätzliches Paar von Randzuständen in der selbstwechselwirkungdominierten Quantenphase erscheint, das keinen Zusammenhang mit der Topologie hat. Außerdem, wenn eine von den Kopplungen stark wird, werden topologische Randzustände von Bulk-Bänden abgefangen und dadurch seine Lokalisation verlieren. iii
ABSTRACT
In this thesis we explore networks of many-body quantum systems, seeking to describe the new quantum and topological phases that arise from the interaction of individual sites. All the networks are based on the semi-classical Lipkin–Meshkov–Glick model, and the couplings between different sites of the network are described with either one or two coupling constants. It was shown that uniform anisotropic Jy –Jy coupling gives rise to four quantum phases (considering both positive and negative coupling constants), which are different both in the mean fields and in the degeneracy of their ground states. In the weak coupling regime the network is found in the paramagnetic phase, whereas in the strong self-interaction regime the ground state becomes exponentially degenerate. In the strong coupling regime the ground state is two-fold degenerate and has long-range ordering. Going beyond the mean-field level, we obtained analytical expressions for the energy dispersion relations in the case of a one-dimensional chain with periodic boundary conditions. For smaller networks we obtained the ground state energy using direct diagonalisation of the Hamiltonian. The effect of introducing a second coupling was studied on the example of a 1d chain – an extension of the Su–Schrieffer–Heeger model to the manybody systems –, where the interplay between topological and quantum phases was the main question. It was shown that the hopping-type coupling leads to appearance of five quantum phases if both positive and negative coupling constants are allowed. Analysis of quantum fluctuations around mean-field solutions showed the existence of topologically trivial and non-trivial phases, distinguished by the value of symplectic polarisation. The bulk–boundary correspondence was tested on the case of a finite chain with open boundary conditions, showing that the number of localised edge states is related to the value of symplectic polarisation. Further analysis showed that in the self-interactiondominated quantum phase an additional pair of topology-unrelated edge states emerge. Furthermore, in the regime when one of the couplings is strong, topological edge states get absorbed by the bulk bands and thus loose their localisation.
iv
CONTENTS
Introduction § . Phase transitions Classical phase transitions Quantum phase transitions Excited state quantum phase transitions Topological phase transitions § . Structure of the thesis Base models § . Lipkin–Meshkov–Glick model § . Su–Schrieffer–Heeger model Networks of lmg models § § § §
. . . .
Model Symmetries and limit cases Bosonisation and the mean-field energy Ground state Infinite chain Finite chain § . Energy dispersion § . Correlation functions Limit cases Equal mean fields § . Conclusions and outlook Combined lmg–ssh model § . Model Hamiltonian and symmetries Mean field and luctuations § . Bulk properties Mean field Quadratic luctuations Topological properties § . Finite chain with free open boundaries Ground state and excitation energies Magnetisation § . Conclusions and outlook v
Summary Bibliography Appendices § A Holstein–Primakoff expansion § B Localisation of edge states § C Winding number § D Conjugate gradient method § E Bogoliubov transformations
vi
INTRODUCTION 1.1
phase transitions
The phenomenon of phase transitions has been known to the mankind since the beginning of time. The human encountered it first by observing ice turning into water, later – iron melting in the furnace, even later – the magnet loosing its properties when heated and so on. Up to now the phase transitions have been thoroughly and intensively explored and classified, numerous theories were proposed and underwent rigorous tests in the experimental setups of ever increasing complexity and immense precision. 1.1.1
Classical phase transitions
Classical phase transitions occur, when the ambient conditions (temperature, pressure, etc.) cross a certain threshold, at which the substance can coexist in two distinct phases, each of which is characterised by uniform physical properties. Some of these properties – temperature, pressure and chemical potential – have to be shared between the phases to maintain the thermodynamic equilibrium, whereas others, like entropy, density or heat capacity to name but a few, can change abruptly when traversing from one of the phases into another [ ]. Phase transitions (pt) can be classified into two kinds according to the lowest order of the Helmholtz free energy derivative with respect to the temperature (pressure) that is discontinuous at the phase boundary (Ehrenfest classification). Thus, pt of the first kind are characterised by the jump in ∂F/∂T ( ∂F/∂p ) and, consequently, specific volume, enthalpy, entropy, etc; while the pt of the second kind are characterised by the jump in ∂ F/∂T ( ∂ F/∂p ) and, consequently, heat capacity, coefficient of thermal expansion, etc [ ]. The pt are related to the change of the system’s symmetry, and the phase with the higher symmetry (usually at higher temperatures) is normally called disordered or symmetric, while the phase with the lower symmetry (usually at lower temperatures) is called ordered or broken-symmetry. As a quantitative measure of the order of phases, the notion of order parameter can be introduced: in the broken-symmetry phase it can take any nonzero values, while in the symmetric phase it should be precisely zero. Thus when the system crosses a phase boundary, the order parameter changes continuously from zero to non-zero values or the other way round [ ].
1.1.2
Quantum phase transitions
It is obvious that classical phase transitions can occur only at non-zero temperatures. Nevertheless, a phenomenon with similar effects in thermodynamic parameters can occur even at the absolute zero. In contrast to classical pt, it is governed not by thermal fluctuations (as they are non-existent at K), but rather by quantum fluctuations. Thus, this phenomenon is called quantum phase transition (qpt). As the temperature is kept constant, the control parameter ξ to be varied to cross a quantum phase boundary should be one of the internal or external parameters of the Hamiltonian [ ]. When the control parameter reaches its critical value ξ = ξcr , the qpt occurs and the ground-state energy becomes a non-analytic function of ξ. This, however, is strictly the case only when the number N of particles that form the system is infinite. When N is finite, only a precursor of quantum criticality is present in the spectrum, becoming more and more apparent with the growth of N [ ]. The onset of the qpt has an effect in the spectrum of excitations as well, as the energy gap Δ between the ground state and the first excited state closes as ξ approaches ξcr . The scaling of Δ in the vicinity of ξcr is given by the power law [ ] Δ ∼ |ξ − ξcr |νz , where ν and z are critical exponents, which depend on the dimensionality of the space, the number of components of the order parameter and the general type of interactions, but usually not on the microscopic details of the Hamiltonian. 1.1.3
Excited state quantum phase transitions
The clustering of energy levels can occur not only around the ground level, as in the case of the qpt, but also in the middle of the spectrum. This phenomenon, closely related to the qpt, is called an excited state quantum phase transition (esqpt). More specifically, it is a kind of extension of the qpt on the region, where the control parameter exceeds its critical value, that is to say on the ordered phase. As the energy difference between adjacent levels Δ goes to zero, the density of states Δ− becomes singular along the path in the ξ–E space that starts with the ground state energy at the critical point [ ]. Again, the true singularity is reached only in the limit N → ∞. Quantum and excited state quantum phase transitions are strongly related to the geometry of the mean-field energy surface of the system. A qpt occurs at the point of bifurcation of the surface minima, while an esqpt and related divergence of the density of states occur at the saddle points of the surface [ ] that correspond to the minima of the ordered phase. That said, the way the energy at which an esqpt happens depends on the control
parameter is easy to find given the expressions for the mean-field stationary points of the system. 1.1.4
Topological phase transitions
It was mentioned earlier that phase transitions are associated with different symmetries of the phases [ ]. It turned out, though, that a certain class of phase transitions can happen without a change of the symmetry. The difference between phases in this case is in the underlying topological order or topology of the system, thus the name – topological phase transitions (tpt). Physically, the phases with non-trivial topological order are distinguished by the presence of edge states, which are robust under local perturbations. A wave of interest has recently arisen concerning topological order in different fields, mostly in condensed mater, where topological insulators [ – ] and topological properties of superconductors [ , ], to name but a few, were studied. There are proposals for the experimental realisations of topological states by using quantum wires with Rashba spin–orbit coupling, a combination of Zeeman splitting and proximity-induced s-wave superconductivity [ , ] and semiconductor quantum dots coupled to superconducting grains [ ]. Signatures of topologically protected states were experimentally found in nanowires coupled to superconductors [ ] and in ferromagnetic atomic chains on the surface of a superconducting lead [ ]. The manipulation of these states may have a strong impact on quantum technologies and quantum information processing. Topologically protected excitations of a topological superconductor – Majorana fermions – have nonabelian statistics, which can be used to perform braiding operations essential for the implementation of fault-tolerant quantum computing [ , ]. Condensed matter is not the only field, where topological phases can find experimental realisation. Currently topological systems can be created, for instance, using ultra-cold gases in optical lattices [ , ]. There also are proposals to realise exotic topological states in quantum optics by exploiting light–matter interactions [ ], such as fractional quantum Hall effect in arrays of coupled cavities [ , ], topological superradiance [ ] and optical realisations of the Jakiw–Rebbi model [ ]. It has been shown that metamaterials are optical analogues of topological insulators [ , ], and they can be used to perform topological pumping [ ]. Recently topological states have been observed in photonic quantum walks [ , ], phontonic quasicrystals [ ] and self-localised states in photonic topological insulators [ ]. These developments have a strong influence in material science, so that nowadays one can design phonoic topological meta-materials where elastic waves inherit topological features [ , ]. Chiral spin-wave edge modes in dipolar magnetic systems [ – ] can be mentioned as another example.
1.2
structure of the thesis
The thesis is based on two articles [ , ] published by the author in collaboration with researchers both from the TU Berlin and from abroad. The work is structured as follows. The current chapter gives a short summary on different types of phase transitions. Chapter deals with the two basic models for this work, namely, the lmg and the ssh models. The Hamiltonian of the lmg model is analysed using semi-classical approach, direct numerical diagonalisation and hp bosonisation technique. For numerical calculations the symmetries are taken into account to reduce the dimensionality of the system. Mean-field energy surface, energy spectrum and energy derivatives of the ground state are considered in the context of qpt and esqpt. The ssh Hamiltonian is analysed by calculating energy spectrum and eigenstates. The presence of edge states and their localisation length as well as topological invariants are determined in the context of tpt. Chapter deals in detail with the networks of lmg models with uniform inter-node couplings. The symmetries and limit cases are used to sketch different phases, the boundaries between which are then precisely established by hp bosonisation technique. The ground states, their degeneracies and energies are obtained for all the phases analytically for infinite 1d chains and numerically for finite chains of different geometries, and the results are checked in simple cases against the results of direct diagonalisation. The properties of the lowest excitations are then considered, followed by the analysis of ground-state correlation functions. Chapter extends the model of chapter to include alternating couplings between nodes of a 1d chain. Again, hp bosonisation is used for analysis of the ground state, its degeneracy and energy, as well as for analysis of the low-lying excitations. The Bogoliubov theory is described and applied to obtain the spectrum of excitations both in the case of a finite and infinite chains. The phase diagram is constructed for both quantum and topological phases. The topological invariants are calculated and the appearance of the edge states is observed in both the spectrum and in the calculated magnetisation of the chain. Finaly, chapter gives a short summary of the results of the work.
BASE MODELS 2.1
lipkin–meshkov–glick model
In mid-sixties, H J Lipkin, N Meshkov and A J Glick proposed a simplistic model to treat many-particle systems such as, for instance, atomic nuclei [ – ]. On the one hand, the simplicity of the model allows it to be solved exactly in some cases [ , ], while on the other hand, it is complex enough to show quantum phase transitions, subject to the change of parameters. Up to now, this model has found experimental realisations for Bose–Einstein condensates in optical lattices [ – ], whereas theoretical works suggest implementing it by means of cavity qed setups [ , ] and magnetic molecules such as manganese acetate [ ]. Further realisations are awaited in fields of Fisher information [ ] and spin squeezing [ , ]. Although the qpt is a feature of the ground state, its signatures can also be observed in the dynamics of the system [ , ]. The Lipkin–Meshkov–Glick (lmg) model is a set of N identical but distinguishable all-to-all coupled two-level systems with the level splitting ω and coupling strengths γx and γy . Its Hamiltonian is ( ) H = ω Jz − γx J x J x + γ y J y J y , ( .) N where operators J α = [ α βmomentum ] ∑ αwe have introduced the collective angular y = ⅈεαβγ J γ with σxi , σi and i σi , satisfying commutation relations J , J σzi being × Pauli matrices. Hereafter we consider only the anisotropic case when γx = γ and γy = , though the approach would remain the same for the general case, too. The Hilbert space of the lmg model is spanned by the products of states of all the two-level systems and thus has the dimension d = N . Speaking in terms of collective angular momentum operators, this corresponds to the |j, m⟩ basis, given that J |j, m⟩ = j(j + ) |j, m⟩ and Jz |j, m⟩ = m |j, m⟩. In order to simplify calculations, we can restrict the Hilbert space to the sub-space of Dicke states, i.e., states with the maximal quantum number j = N [ , ], as the collective angular momentum commutes with the Hamiltonian, i.e., [J , H ] = . The space dimension after that is reduced to d= j+ . It is possible to reduce it even further, as the Hamiltonian possesses a conserved parity [ ] z Π = ⅇⅈπ(J +j) ( . )
with eigenvalues ± . This parity acts on the basis states of the Hamiltonian as Π |j, m⟩ = (− )j+m |j, m⟩, meaning that the Hamiltonian matrix can be
{A, B} =
∂A ∂B ∂q ∂p
−
∂A ∂B ∂p ∂q
written as a direct sum of two blocks, each connecting the states with equal parities. As in the thermodynamic limit j → ∞ both of these blocks have identical spectra [ ], it suffices to take into account only the positive-parity (j + ) × (j + ) block, which contains the ground state. As was mentioned before, when the number of particles N (or, equivalently, the length of the angular momentum j) grows infinitely, we arrive at the thermodynamic limit. If we assume that the angular momentum is so long that we can neglect quantum nature thereof, that is to say, if we set commutators between different angular momentum components to zero, we restrict ourselves to the semi-classical limit. In this case dynamics of the model can be found exactly by solving Hamilton equations of motion O˙ = {O, H}, where O is an arbitrary operator, H is the Hamilton function (which is equivalent to the Hamiltonian if the quantum nature of the angular momentum is neglected and necessary parametrisation is introduced) and {·, ·} is the Poisson bracket. Time derivativos of angular momentum components can then be calculated, obtaining x J˙ = ω Jy ,
γ y J˙ = −ω Jx − Jx Jz , j
γ z J˙ = Jx Jy , j
which define at most four stationary points √ J ∗, = ( , , ±j), J ∗, = (±j γ − ω /γ, , −jω /γ),
The Bloch sphere is the set of all J vectors with the fixed length j
( . )
the first two being valid for any parameter values, and the last two only if γ ⩾ ω . As a side check here, it is worth mentioning that the dynamics is indeed restricted to the Bloch sphere, as the length of the angular momentum is conserved: ∂t J = {J , H} = . The type and the stability of stationary points change as parameters are varied. Being a minimum when γ < ω , the point J ∗ becomes a saddle point when γ > ω , as the new pair of points J ∗, takes its place as minima. In other words, a pitchfork bifurcation occurs at γ = ω , which is a mark of a quantum phase transition (qpt). Hereafter, the phase with γ < ω will be called symmetric and the other one broken-symmetry. The reason behind the latter name is that the ground state of the broken-symmetry phase is doubly degenerate, so after the transition the system will relax to either of the two minima thus breaking the symmetry. The energy of the ground state in the thermodynamic limit is given by g E = H(J ∗min ). In the symmetric phase it depends only on the level splitting: Eg = −jω , while in the broken-symmetry phase also on the coupling: Eg = −j(ω + γ )/( γ). The function Eg (γ) and its derivative are continuous around the critical value γ = ω , while the second derivative is not, thus the phase transition is of the second order (see § . . ). The semi-classical limit introduced earlier is good for describing angular momentum dynamics, but it essentially lacks any information about
quantum fluctuations, which gain great importance in the vicinity of stationary points. This work, however, concentrates mainly on ground and lowest excited state properties of lmg-based models, so semi-classical approach is not enough here. One of the possibilities to take quantum fluctuations into account is to transform the original Hamiltonian ( . ) to that of a squeezed displaced harmonic oscillator. To this end we use Holstein–Primakoff transformations (hpt) [ ] Jz = b† b − j, ( ) J + = b† j − b† b , ( ) J− = j − b† b b,
( . )
which map a long angular momentum onto bosons. Here J± = Jx ± ⅈ Jy are ladder operators and b(†) are bosonic anihilation (creation) operators, which satisfy the commutation relation [b, b† ] = . Using hpt directly in this form allows us to work with a stationary point J ∗ = ( , , −j) only. In order to be able to consider other stationary points (or to be able to make quadratic approximation around an arbitrary point of the classical energy surface) bosonic operator b has to be displaced: b = D† (α) d D(α), where α is the (complex) mean-field part of b and the displaced operator d represents quantum fluctuations. The displacement operator D is given by [ ] [ √] D(α) = exp (αd† − α∗ d) j . ( . )
The expression we get after transforming the Hamiltonian using hpt and then performing operator displacement is still not a quadratic form, but making use of the thermodynamic limit, we can expand contained square roots into a series around the point j = ∞. The Taylor expansion will have terms of O(j d ), O(j / d ), O(j d ), O(j− / d ) and so on, but, as we are interested in quadratic approximation only, we can truncate this series and keep just the first three terms. After the expansion, the Hamiltonian has the form √ ( . ) H = EMF (α)j + H ( ) (α, d) j + H ( ) (α, d), where the mean-field energy EMF depends only on α, H ( ) is linear and H ( ) quadratic in displaced operators. For extended derivation see appendix A. The normalised mean-field energy EMF (α) = −ω ( − α∗ α) −
γ
( − α∗ α)(α + α∗ )
( . )
obtained above is plotted for different values of coupling γ as a function of the mean field α in fig. . . The whole Bloch sphere is mapped onto the
EMF (a)
1 Im α
Im α
1 0
1.0
(b)
0.5 0.0
0
−0.5 −1
−1.0
−1 −1
0 Re α
1
0 Re α
−1
1
−1.5
Fig. 2.1: Normalised mean-field energy of the lmg model [see ( . )] in two different quantum phases. Level splitting ω = and coupling (a) γ = . and (b) γ = . Energy minima are marked with ◦ and saddle points with •. − .9
− −.
− .
−.
− . ∂ j ∂γ
−. −.
− .
(a)
Eg
− .8
∂ g ∂ j ∂γ E , j ∂γ
∂ g j ∂γ E
Derivatives
Energy j En
− .9
Eg
.
(b) .
. γ
.
.
−
γ
Fig. 2.2: (a) Exact energy spectrum of the lmg model. (b) Derivatives of the ground state energy. j = and ω = . The precursor of a qpt can be seen at γ = ω and that of an esqpt at γ ⩾ ω .
disk α∗ α ⩽ on the complex plane in such a way that J = ( , , −j) corresponds to α = and J = ( , , j) corresponds to the circumference α∗ α = . Positions and types of stationary points as well as the ground state energy calculated this way is identical to the expression we got in the semi-classical limit. For the sake of comparison, let us now find the ground state energy from direct diagonalisation of the Hamiltonian ( . ). As was motivated before, the Hamiltonian matrix H has dimensions (j + ) × (j + ), so it is easy to find its eigenvalues even for relatively high j values. Here, we shall set j = .
κ | , A⟩
| , B⟩
| , A⟩
κ | , B⟩
| , A⟩
| , B⟩
|L, A⟩
|L, B⟩
Fig. 2.3: Su–Schrieffer–Heeger model. Grey-shaded area marks a unit cell.
The matrix H is sparse, having non-zero elements only on the main and on the third diagonals. The spectrum of its eigenenergies depending on the coupling γ is shown in fig. . a. The ground state features a precursor of the qpt at γ = ω , as the second derivative of Eg (γ) tends to the discontinuity at this point when j → ∞ (see fig. . b). Another feature that can be seen from the spectrum is an excited state quantum phase transition (esqpt) along the line E(γ ⩾ ω ) = −jω , associated with the saddle point of the mean-field energy surface. 2.2
su–schrieffer–heeger model
Another model that will be used in this work to build upon is the Su–Schrieffer–Heeger (ssh) model [ , ]. It was proposed in the end of seventies to describe polyacetylene, a linear polymer with alternating single and double bonds. Despite being simplistic, it accounts for yet another phenomenon: a topological phase transition tpt [ , ], and its realisations, apart from the natural occurrences in unsaturated polymers, include chains of dielectric microwave resonators [ ], systems of cold atoms in an optical potential [ ], etc. The model describes a chain of L′ sites coupled with alternating strengths, along which fermionic particles may hop. Due to the staggered nature of couplings, in the limit L′ → ∞, the chain becomes translationally invariant, with the unit cell consisting of two sites: A and B. We shall keep the notion of the unit cell for finite L′ as well and label each site with the unit cell number l = , , . . . , L (= ⌈ L′ ⌉) and the species s = A, B. The Hamiltonian can then be written as
H=κ
L ∑
(|l, A⟩⟨l, B| + h.c.) + κ
L ∑ l=
l=
(|l − , B⟩⟨l, A| + h.c.),
( . )
where κ and κ are inter- and intra-cell couplins, respectively, and |l, s⟩ = |l⟩ ⊗ |s⟩. Schematically, the model is shown in fig. . . The Hamiltonian ( . ) possesses a chiral symmetry [ ]
C=
∑ l
(|l, A⟩⟨l, A| − |l, B⟩⟨l, B|),
( . )
0 Energy En
0 −0.
(d)
0
− −
.5
0
.5
⟨l′ |ψn ⟩
−0.
(c)
−
⟨l′ |ψn ⟩
0.
(b)
20
0
0
80
⟨l′ |ψn ⟩
(a)
−0.
Site l′
κ /κ
Fig. 2.4: (a) Energy spectrum of the ssh model. L = . A pair of edge states marked with red and blue appears after the tpt. (b–d) Wave function of the system at the points of corresponding colour in the panel a. Edge states show exponential decay of an amplitude.
such that CH C† = −H and C† C = C = . Operator C can also be called a particle–hole symmetry, as it implies that the spectrum of the Hamiltonian is symmetrical with respect to E = . This can be easily shown given the eigenequation H |ψn ⟩ = En |ψn ⟩: ⟩ ⟩ CH C† ψn = −En ψn , ( ⟩) ( ⟩) C† CH C† ψn = −En C† ψn , ⟩ ⟩ H ψn′ = −En ψn′ .
The main feature of the ssh model is a topological phase transition at the critical coupling κ = κ . The phase with κ > κ is called topologically trivial, whereas the other one with κ < κ is called topologically non-trivial. The physical difference between the trivial and the non-trivial phases is best seen in the finite-L case. Let us consider a chain of L = unit cells and look at the energy spectrum for different κ /κ ratios (fig. . a). Due to the presence of the chiral symmetry the upper band is an exact mirrored copy of the lower one. In both phases, the bands are separated by the gap, which tends to closing at the critical coupling ratio κ /κ = (the gap closes fully only in the limit L → ∞, though), marking the onset of the phase transition. In the non-trivial phase a pair of states close to zero energy separate from the bands and remain in the middle of the band gap. These states are localised around the edges of the chain with localisation length ν=
ln |κ /κ |
(for derivation see appendix B) as shown in figs. . b and c compared to delocalised states in the middle of the band (fig. . d) and are thus called edge states. It should be noted that the wave function of edge states is non-zero only for the sites of the A sub-lattice on the left edge and the sites of the B sub-lattice on the right edge as well as that the wave function of one edge state |−⟩ is odd, whereas that of another one |+⟩ is even with respect to the middle of the chain. Using the latter fact√it is possible to construct another that is localised on the left side pair of edge states: |L⟩ = (|+⟩ + |−⟩)/ √ on the right side. of the chain and |R⟩ = (|+⟩ − |−⟩)/ Mathematically the existence and the number of edge states can be deduced from the bulk limit, i.e., from the case when L → ∞. In this limit, due to translational invariance the states can be mapped onto the reciprocal space by using Fourier transforms (ft) |l, s⟩ = √
L
∑ k
|k, s⟩ ⅇⅈkl ,
( . )
where the wave number k = , Lπ , Lπ , . . . , (L − ) Lπ satisfies k + πZ ≡ k and Z ∈ Z. This results in the Hamiltonian ∑ ∑ (ⅇ−ⅈk |k, B⟩⟨k, A| + h.c.), ( . ) (|k, A⟩⟨k, B| + h.c.) + κ H=κ k
k
∑ which can more consicesly be written as H = k Hk , where Hk has the matrix ) ( κ + κ ⅇⅈ k ( . ) Hk = κ + κ ⅇ− ⅈ k
in the basis {|k, A⟩ , |k, B⟩}. Its eigenvalues give us two energy bands √ E± (k) = ± κ + κ + κ κ cos k
with a gap in between. The gap closes and reopens, when the coupling reaches the critical value |κ | = |κ | at k = π if κ κ > and at k = if κ κ < , marking a phase transition. If either of the couplings is zero, the bands become flat, and so the gap closes for all values of k. The matrix Hk can be expressed in terms of Pauli matrices σ , σ , σ , σ with real k-dependent coefficients: Hk = c (k)σ + c (k)σ + c (k)σ + c (k)σ .
As the ssh model does not contain on-site terms, c (k) = , and other three coefficients form a vector c(k) = (c , c , c ) = (κ + κ cos k, κ sin k, ).
The number of full revolutions that c(k) makes around the origin, while k traverses the Brillouin zone (ibz), is called a winding number, which is a function of system parameters (see appendix C for derivation): ∫ dφ(k; κ , κ ) w(κ , κ ) = dk ( . ) π IBZ dk ∫ π κ + κ κ cos k = dk . π κ + κ + κ κ cos k The winding number is a topological invariant: its value remains constant for any parameters from a certain open set (topological phase). The set boundary is marked by the bulk gap closing. For the ssh chain w can take values (for the trivial phase) and (for the non-trivial phase) and corresponds to the number of pairs of edge states in the finite chain [ ].
NETWORKS OF LMG MODELS The first complex system to be considered in this thesis is the extension of an lmg model onto a network with anisotropic couplings. Previous works have explored dynamical aspects of networks of coupled systems with global symmetries. For example, adiabatic phase transitions of networks of qubits have been investigated [ ]. In the context of quantum optics, it has been shown that arrays of coupled cavities can exhibit soliton solutions [ ], the emergence of phase transitions of light [ ] and dissipative quantum phase transitions [ ]. Recent theoretical approaches have demonstrated that cold atoms on excited bands in optical lattices could allow for the implementation of networks of non-linear multi-particle bosonic models showing collective behaviour [ , ]. Spin networks with spatial symmetries have found many experimental implementations. For instance, it has been shown that chains of trapped ions undergo a variety of quantum phase transitions when interacting with laser beams [ ]. They have also found application in detection of quantum correlations between a two-level system and environment by measuring the system only [ ]. Other experimental implementations of critical spin chains include ultra-cold polar molecules [ ] and Rydberg gases [ ], to name but a few. The major part of the material presented in this chapter is published as a regular article in Physical Review E [ ]. 3.1
model
The lmg model, on which the system considered in this chapter is based, was already discussed in § . , so now we proceed with the study of the critical behaviour of networks of such models (see fig. . ). Throughout this chapter, we shall be assuming non-directed coupling between Jy components of the nodes, so that the Hamiltonian is given by
H=
L ∑ l=
Hl −
j
L ∑
l′ >l=
y y
κll′ Jl Jl′ .
( .)
Here Hl = ω Jzl − γj Jxl Jxl is the Hamiltonian of a single lmg model, κll′ are the elements of the coupling matrix [ , ] and l, l′ = , , . . . , L are the site positions in the chain. The constants ω and γ are assumed to be positive, but κll′ can be either positive or negative. The latter case will be considered mainly for one-dimensional finite or infinite networks, restricting the number of the sites L to even values, as otherwise the effects of
ωā γ ωā
κ JĂ
Jă
JĄ
Ją
JĆ
JL−Ă
Jć
ωā
JL
Fig. 3.1: Example network of lmg models. Coupling between neighbouring sites is determined by parameter κ, self-interactions within one site by γ and level splitting of two-level systems that constitute an lmg model by ω . Each site consists of N two-level systems.
frustration or of the interplay of ferro- and anti-ferromagnetic couplings will have to be taken into account, which would considerably complicate the problem [ – ]. In most of the works on long-spin chains, the coupling is chosen to have certain continuous symmetries, most commonly by using either isotropic Heisenberg-type or anisotropic coupling [ – ]. Here, however, we shall focus on uniaxial coupling, which results in a set of global and local discrete symmetries and opens the possibility of new effects. We shall start with some general considerations and approaches to the problem and then apply these to finite networks and one-dimensional infinite chains. 3.2
symmetries and limit cases
Similarly to the basic lmg model, the combined[ Hamiltonian ( . ) com] mutes with each of the local angular momenta H , J l , thus during the time evolution all J l are individualy preserved. This justifies the use of Dicke states |j, ml ⟩ with j = N to describe each of the individual nodes and tensor products ξ
|j, m⟩ =
L ⊗ l=
|j, ml ⟩ξ
( . )
thereof to describe the system on the whole. As before, the use of Dicke states considerably reduces the dimensionality of the Hilbert space from jL to ( j + )L . In the last definition, m = (m , m , . . . , m ), −j ⩽ m ⩽ j, L l ξ ξ denotes the quantization axis and the states |j, ml ⟩ are eigenstates of the collective angular momentum operators Jlξ , such that Jlξ |j, ml ⟩ξ = ml |j, ml ⟩ξ . Another feature of the combined model is the global parity Π = exp ⅈπ [
L ∑ l=
]
(Jzl + j) ,
( . )
which is just a product of parities ( . ) of individual nodes inherited from the basic lmg model. Under the action of Π, the total angular momentum y y transforms as Π(Jxl , Jl , Jzl )Π† = (−Jxl , −Jl , Jzl ). For numerical calculations it is essential to reduce the Hilbert space of the system as much as possible, so it is natural to exploit the parity and use the basis from tensor products of eigenstates of Jzl . The operator ( . ) acts on these basis states ∑ as Π |j, m⟩z = (− ) l (ml +j) |j, m⟩z , allowing us to separate the Hilbert space into two sub-spaces – with positive and with negative parity. Similarly to the basic lmg model, in the thermodynamic limit j → ∞ the blocks of the Hamiltonian matrix corresponding to these sub-spaces have identical spectra, so for calculations we use only the positive-parity sub-space, which contains the ground state. In the result, the dimensionality of the Hilbert space is further reduced to ⌈( j + )L / ⌉. In addition to the global parity, the system is also invariant under the local reflection in the y–z plane,
Rl = ⅇⅈπ(Jl +j)Tl , x
( . ) ( y) where Tl = exp ⅈπJl Kl and Kl is the operator of charge complex conjugation with respect to the standard representation [ ] acting on the l-th site. The action of the anti-unitary local reflection operator on the angular y y momentum is Rl (Jxl , Jl , Jzl ) Rl− = (−Jxl , Jl , Jzl ). In order to get an intuition about different quantum phases of the combined model, let us begin the analysis of the system with limit cases. For convenience we shall use the notation |Gpξ ,p
,...,pL ⟩ =
L ⊗ l=
|j, (− )pl j⟩ξ
( . )
for the ground states, where pl = , and ξ = x, y, z. Such ground states are tensor products of Dicke states of individual nodes with the quantisation axis ξ. Due to the fact that we are considering limit cases, when one of the Hamiltonian parameters becomes much greater than the others, the angular momenta are forced to have components only along the quantisation axis, and so the magnetic quantum number takes values j or −j. In the limit ω ≫ γ, κll′ , the Hamiltonian reduces to
H=ω
L ∑
Jzl ,
l=
and the natural quantisation axis is ξ = z. The expectation value ⟨H ⟩ should be minimized in the ground state, thus each node should have minimal value of ml = −j. That means that there is a unique ground state |G⟩ = |Gz, ,..., ⟩, and, as the coupling between the nodes is very weak, no correlations are to be present. The state is thus paramagnetic-like (cf. ref. [ ]).
In the limit γ ≫ ω , κll′ , the Hamiltonian simplifies to L γ∑ x x H=− Jl Jl , j l=
and the natural quantisation axis is ξ = x. Due to the presence of the local symmetry ( . ), ⟨H ⟩ is minimised by states with either ml = j or ml = −j on each of the nodes independently. Thus the ground state is L -fold degenerate and is represented by the set of separable states |Gxp ,p ,...,pL ⟩ with all the possible combinations of pl . In this regime, the system consists of an ensemble of L disconnected sites with parallel spins pointing along the x direction. As a side note it can be mentioned that exponentially degenerate ground states arise naturally in the context of spin ice [ ] and spin glasses [ ]. Finally, in the strong interaction limit κll′ ≫ ω , γ, the Hamiltonian reduces to L ∑ y y κll′ Jl Jl′ , H=− j ′ l >l=
and the natural quantisation axis is ξ = y. Due to the global parity, ⟨H ⟩ is minimised by states, where all the ml have the same signs, i.e., ml = j or ml = −j collectively. The ground state is thus highly correlated and two-fold y y degenerate, including ferromagnetic states |G , ,..., ⟩ and |G , ,..., ⟩. From the analysis of limit cases we can conclude that the ground states in different limits are drastically different, so the properties between these limits should behave non-analytically at some points. This is the onset of the critical behaviour that we seek to describe in this part of the thesis. 3.3
bosonisation and the mean-field energy
Having calculated the ground state in different limit cases, it is now time to move on to the regimes, in which parameters of the Hamiltonian are of the same order. There are at least two approaches to take. One of these is to construct the Hamiltonian matrix for some (probably not very large) angular momenta J l and to diagonalise it, obtaining the energy spectrum. Then the procedure should be repeated for different points in parameter space to get the phase diagram in the end. The upside of this approach is that it is exact and gives the whole range of energies and eigenstates. The downside is the scalability, as the Hilbert space of the system, even after taking aforementioned symmetries and Dicke states into account, still grows exponentially with the number of sites in the network. We shall leave this approach for the next sections, but in the meanwhile let us focus on approximations we can make to get the lower part of the energy spectrum of the system analytically.
One possible approximation was already used in § . to treat a single lmg model, and we shall adapt it for the case of the network: Holstein– Primakoff transformations (hpt), which were introduced in ( . ), map a long angular momentum onto bosonic quasi-particles. Related works used hpt to describe low-energy magnetic excitations in time-dependent magnetic fields [ ] and the interaction of magnons in Heisenberg ferromagnets [ ]. Furthermore, in the context of spinor Bose–Einstein condensates [ , ], hpt can be used to describe the formation of periodic magnetic domains [ ]. Applying these transformations to each of the sites in the chain separately, i.e., Jzl = b†l bl − j, ( ) † † , J+ = b j − b b l l l l ( ) J− j − b†l bl bl , l =
( . )
we can transform the Hamiltonian so that it becomes dependent on the set of L creation/annihilation operators. It is then profitable to displace the operators bl so that their mean-field expectation value becomes zero. This can † be done √by applying displacement transformation bl = D (αl ) dl D(αl ) = dl + αl j, where [ √] D(αl ) = exp (αl d†l − α∗l dl ) j ( . )
and αl are complex mean-field displacements. As we are interested in the ground and lowest excited states only, we can now expand the roots in ( . ) and keep terms of up to second order in dl , following the procedure described in appendix A. In the result the system is simplified to the level of L coupled displaced and squeezed harmonic oscillators, and its Hamiltonian has the form √ ( . ) H = EMF (α)j + H ( ) (α, d) j + H ( ) (α, d),
where EMF is the mean-field energy, which depends only on complex meanfield displacements α = (α , α , . . . , αL ), whereas H ( ) and H ( ) additionally depend on bosonic fluctuation operators d = (d , d , . . . , dL ) and on their products, respectively. We shall give a more general derivation in the next part of the work. The mean-field energy is given by (see appendix A) (γ )∑ γ∑ (αl + α∗l ) − α∗l αl −ω EMF (α) = −L ω − +
γ∑ l
l
α∗l αl (αl + α∗l )
l
+
∑ l′ >l
√ √ ] [ − α∗l αl − α∗l′ αl′ . ( . ) κll′ (αl − α∗l ) (αl′ − α∗l′ )
The set of mean fields of the ground state minimises the EMF (α) function and corresponds to (one or more, depending on the degeneracy) stable fixed points of the system. The solution of a set of L equations ∂EMF (α) = , ∂α l ∂E (α) MF∗ = , ∂αl
( . )
which would give us all the critical points, cannot be obtained for the given EMF (α) analytically even for L as low as . That, in general, leaves us with the necessity of locating the critical points numerically, and we shall deal with this in the end of the next section. 3.4
ground state
3.4.1
Infinite chain
For now let us consider a special case when all the inter-site coupling forces are equal and positive and the network is a one-dimensional infinite chain with with free open boundary conditions, which mathematically is equivalent to the case of a finite chain with periodic boundary conditions (κll′ = κ if |l − l′ | = and otherwise). For such a network, the translational symmetry it possesses implies that the mean-field displacements of at least one of the ground states are equal for each and every node in the chain (cf., e.g., ref. [ ]). In this case the ground-state energy becomes a function of only one complex variable α = α = . . . = αL , thus ( . ) is simplified to (γ ) EMF (α) γ γ = −ω − (α + α∗ ) − − ω α∗ α + α∗ α (α + α∗ ) L κ + (α − α∗ ) ( − α∗ α). ( . ) The set of equations (γ ) γ γ ∂EMF (α) = − α − − ω α∗ + α∗ ( α + α∗ + α∗ α) ∂α κ + ( − α∗ α + α∗ )(α − α∗ ) = , ) ∂EMF (α) γ ∗ (γ γ = − α − − ω α + α ( α∗ + α + α∗ α) ∗ ∂α κ + ( − α∗ α + α )(α∗ − α) =
can now be solved analytically (the substitution α = x + ⅈy may be used to simplify the expressions). Taking into account the constraint α∗ α ⩽ dictated by the reality of roots in ( . ), the only possible critical points in the system are √ √ ω ω − , − . ακ± = ±ⅈ ( . ) αω = , αγ± = ± γ |κ| The meaning of the absolute value sign becomes clear if we analyse, what would happen to the ground state if κ were negative. Obviously, the Hamiltonian ( . ) is not invariant under the change of κ sign, so in order to keep H intact we have to apply a set of local gauge transformations Jzl 7→ Jzl
Jlξ 7→ (− )l Jlξ ,
κ 7→ −κ,
( . )
for ξ = x, y or, expressed in terms of bosonic operators and displacements through the hpt and ( . ), bl 7→ (− )l bl ,
dl 7→ (− )l dl ,
αl 7→ (− )l αl .
Having these gauge transformations at hand, we can easily generalise whatever result we get for positive κ onto the case κ < . The ground state, if the uniform mean-field ansatz is applicable, for instance, would transform to αl = (− )l αcr and all the energies would remain the same. That means that the minimal translationally replicable part of the chain contains two sites as opposed to the case with ferromagnetic couplings, and for the approach to work correctly, the total number of sites in the chain L should be even. As was described in § . . , quantum phase transitions occur at points of bifurcations, where energy minima of the system switch from one set of stationary points to another. To select the energy minima from critical points ( . ), let us calculate the mean-field energies EMF (αω ) = −ω ,
(γ − ω ) , γ (|κ| − ω ) . EMF (ακ± ) = −ω − |κ|
EMF (αγ± ) = −ω −
( . )
Thus, αω is the energy minimum when αγ± and ακ± do not exist, i.e., when γ < ω and |κ| < ω . Crossing the lines γ = ω and |κ| = ω , the energy minimum shifts to αγ± and ακ± , respectively, giving another two phases, so in the result we have four phases that we shall further refer to as phase I (ω dominated), phase II (κ-dominated when κ > ), phase III (γ-dominated) and phase IV (κ-dominated when κ < ) as shown in fig. . with the phase boundaries going along the lines (γ = ω , ⩽ |κ| < ω ), ( ⩽ γ < ω ,
.
D E
C
−.
κ
B .
A
−. G F (a)
.
. γ
−. −.
g
−.
Ground-state energy EMF /L
−.
−. −. −. −. −. 8
(b) DE A B C FG A Parameters
Fig. 3.3: Normalised ground-state mean-field energy of a 1d chain in the thermodynamic limits L → ∞ and j → ∞ (a) for ⩽ κ < and ⩽ γ < with solid black lines marking the phase boundaries. (b) Section along the path in panel a. ω = .
Let us take a more detailed look into the ground-state mean-field energy by going along the closed path A–B–· · · –A in the parameter space (fig. . b). The path was chosen in such a way that it crosses all the phase boundaries and goes through all the phases. The segment A–B lies in the phase I, so its ground-state energy is constant. The point B is where the pitchfork bifurcation occurs. The energy at this point is continuous and smooth, but its second derivative has a jump, meaning that the qpt here is of the second kind according to Ehrenfest classification. The part of the path from B to E lies in the phase II, where the energy depends only on the parameter κ. Thus all the points from the segment C–D are equal in energy, and the segments B–C and D–E are symmetric. At the point E there is another phase boundary. Here, in contrast to the point B, the ground-state energy is continuous, but not smooth, so the qpt here can be classified as of the first kind according to Ehrenfest. From point E to point G the path goes through the phase III, where the energy is dependent only on the parameter γ, thus the segment E–F is constant in energy, while the segment F–G is not. The last phase boundary is crossed at the point G, where the energy is again smooth, and the qpt is of the second order. The equal-mean-field ansatz has strong application limitations. While it gives appropriate results for the chains with translational invariance, its potential validity for other types of networks remains dubious. First of all, let us test this ansatz for the network that it is assumed to work well with: an infinite chain with open or periodic boundary conditions. The translational invariance of such a chain allows us to select one of the sites as a unit cell, i.e., a minimal part of the chain from which the whole chain can be constructed by translational multiplication. In this case
Ground-state energy Eg /L
−.
κ C B A
−.
j→∞ j= 8 j= j= j=
D E G F
γ
−. A
B
D E Parameters
C
F
G
A
Fig. 3.4: Normalised ground-state energy as a function of system parameters taken along the path on the inset. ω = . Thick black lines are obtained using the mean-field hpt approach; red, blue, green and purple lines show results of direct diagonalisation with finite j. The network is a selfcoupled single node, which is mathematically equivalent to an infinite 1d chain.
each node is effectively coupled to itself in Jy components with the strength κ, and after the normalisation by the length of the chain the Hamiltonian resembles the one of a single anisotropic lmg model (cf. ref [ ]): ) ∞ ( ∑ H γ x x κ y y γ κ z ω Jl − Jl Jl − Jl Jl+ = ω Jz − Jx Jx − Jy Jy = L L j j j j l=−∞
with γx = γ and γy = κ.
j
Choosing the basis set {|j, m⟩z }m=−j and holding in mind that Jx = √ (J+ + J− )/ , Jy = (J+ − J− )/ ⅈ, J± |j, m⟩ = j(j + ) − m(m ± ) |j, m ± ⟩ and Jz |j, m⟩ = m |j, m⟩, we can construct the Hamiltonian matrix with elements [ ] [ ⟨ ′ ⟩ γ+κ γ−κ m H m = δm′ ,m ω m − (j + j + m ) − j j √ √ δm′ ,m+ j(j + ) − m(m + ) j(j + ) − (m + )(m + ) ] √ √ − δm′ ,m− j(j + ) − m(m − ) j(j + ) − (m − )(m − ) .
This matrix can easily be diagonalised numerically to get the energy spectrum of the system and also the phase diagram. The result of the diagonalisation is shown in fig. . for different lengths of the angular momentum. Here, the ground-state energy was calculated at the points along the path A–B–· · · –G–A in the γ–κ space, which crosses all the phase boundaries, as shown on the inset. The convergence towards the mean-field approximation with the growth of j is obvious, so the phase diagram obtained using the equal-mean-field
ansatz should also be correct. As was already noted, it looks very similar to the diagram of an isolated anisotropic lmg, though the physical meaning of the phases is different. 3.4.2
Finite chain
Let us now somewhat generalise the problem and find the mean-field displacements corresponding to the minima of the energy for finite and not necessarily one-dimensional chains. The only assumption we shall keep is that parameters ω , γ and κ do not vary across the sites and the bonds. The method we shall use can work even if this assumption is not satisfied, but it is useful to have it for comparison with the results of the previous section. As it was noted before, in order to find minima of the ground-state mean-field energy ( . ), one has to solve simultaneous equations ( . ), which is in general not possible to do analytically even for a small number of the nodes. What we can do, though, is to locate the minima numerically, and to this end we shall use the conjugate gradient (cg) method, which is briefly sketched in appendix D. For more extensive description see, e.g., refs. [ , ]. The method is iterative by its nature, so the zeroth approximation to the vector α that minimises the ground-state mean-field energy should be given to it as an input. It seems reasonable to start with α that results from the equal-mean-field ansatz (adding some small random perturbation in case we hit a stationary point), and indeed, the iteration process converges quickly for such initial conditions. Figure . shows the results of optimisation. All in all, four different networks (in the order of growing complexity) were considered for analysis: an infinite one-dimensional chain from the previous section (mathematically equivalent to a single self-coupled lmg model), a one-dimensional chain with a finite number of sites and free open boundary conditions (fig. . d), an open chain with side chains (fig. . e) and a chain with both loops and side chains (fig. . f). First of all, let us look at the result of optimisation of ground-state mean-field displacements. We considered three representative cases with (i) γ = . , κ = . ; (ii) γ = . , κ = . and (iii) γ = . , κ = . for three different phases. It turned out that in the phases dominated by local parameters ω and γ (cases ii and iii), the equal-mean-field ansatz holds for arbitrary networks with the constraints described earlier. The ω -dominated phase has a single minimum at α = and the γ-dominated phase has a set of L minima with αl = αγ± for each l independently. The phase dominated by a non-local parameter κ is different, as the networks d–f are not uniform. Even though the general trend for the groundstate mean-field displacements to be purely imaginary holds, their absolute
(d)
−.
−. κ C B
−.
g
Ground-state mean-field energy EMF /L
A
−.
D E G F
A B C
(a)
γ
DE
FG A
−.
. ⅈ . . (e)
DE
FG A
(f)
(c) DE A B C FG A Parameters
. ⅈ . .
. ⅈ . . . ⅈ . . . ⅈ . .
. ⅈ . .
. ⅈ . .
. ⅈ . .
−.
−.
. ⅈ . .
(b) A B C
. ⅈ . . . ⅈ . .
. ⅈ . .
−.
−.
. ⅈ . .
. ⅈ . .
. ⅈ . .
Fig. 3.5: (a, b, c) Normalised ground-state mean-field energy for the networks in panels d, e, f (orange) compared to the uniform-mean-field case (black) along the path A–B–· · · –G–A on the inset. (d, e, f) Different finite-size networks of lmg models. Numbers at the sites are optimised mean-field displacements αmin of the energy minima in different phases when γ = . , κ = . (red, pt. C), γ = . , κ = . (blue, pt. F) and γ = . , κ = . (green, pt. A). The black lines on the inset are phase boundaries in the case of an infinite chain. ω = .
g
EMF /L −. −. 5
.5 κ
−. −.5
.5
(a) .5
γ
.5
(b) .5
γ
.5
(c) .5
γ
.5
−. −. 5
Fig. 3.6: Normalised ground-state mean-field energy for the networks in panels d–f of fig. . . Solid lines are phase boundaries. Dotted contours are the paths, along which plots in panels a–c of fig. . are constructed (cf. fig. . a). ω = .
values vary from site to site (see red labels in figs. . d–f), though respecting the mirror symmetry of networks’ geometry. The global parity of the Hamiltonian is still there, implying that the ground state is doubly degenerate: collective change of signs of displacements maps one minimum to the other. Figs. . a–c show the variation of the ground-state mean-field energy along the path shown on the inset of fig. . a in the κ–γ parameter space together with the respective variation in the case of an infinite chain. In the phases dominated by local parameters (approximately from A to B and g from E to A), where the equal-mean-field ansatz holds, the EMF of translag tionally non-invariant networks coincides with the EMF of an infinite chain. On the other hand, this coincidence is lost in the phases, where the nonlocal parameter is dominating (approximately from B to E). The plots of the ground-state energy also show that boundaries of the κdominated phase (as opposed to the boundary between ω - and γ-dominated phases) do not remain constant when the geometry is changed, as can be better seen in fig. . . The boundary between phases I and II can move up or down, though it always remains parallel to the γ axis, and the boundary between phases II and III always starts at the corner of phase I and can have different slopes. To finish this section, let us once again check the results we got for the mean-field case j → ∞ against the results of direct diagonalisation for finite j. Here, we have chosen rather small lengths of the angular momentum, as otherwise the matrices to be diagonalised become too large. The check was performed for two networks: the three-member ring and all-toall-coupled network of four nodes, and the resulting ground-state energies are shown in fig. . . From these figures it can be seen that even for j as low as or the exact ground-state energies qualitatively repeat the behaviour in the long-angular-momentum limit. The biggest concerns about the valid-
Ground-state energy Eg /L
j→∞ j=8 j=4
−.
−. (a)
−. −.
κ C B A
j→∞ j=4
D E G F
γ
−. 5
− .5
(b) A
B
C
D E Parameters
F
G
A
Fig. 3.7: Normalised ground-state energy as a function of system parameters taken along the path on the inset. ω = . Thick black lines are obtained using the mean-field hpt approach; red, blue, green and purple lines show results of direct diagonalisation with finite j. Networks have geometries of (a) a three-member ring and (b) four all-to-all-coupled nodes. g
ity of the check may arise for the path segment C–D, where the slope of EMF is considerable in the finite-j case, but zero when j → ∞. This concern can be lifted, if we compare fig. . with fig. . . Indeed, the same discrepancy is present in the latter case, but, following the tendency it can be seen that the slope of the segment C–D become flatter and flatter with the growth of j. Thus, we can conclude that directly diagonalised ground-state energy eventually converges to its mean-field values in the limit j → ∞, and the phase boundaries obtained in the latter case are correct. 3.5
energy dispersion
We now turn from the ground state to the lowest excited states of the system. In this section we shall concentrate on one-dimensional finite chains with periodic boundary conditions (closed loops) and without loss of generality use the ground states with equal mean fields α = α = · · · = αL = αcr , where αcr = αω , αγ± , ακ± in quantum phases I–III, respectively, and α = −α = α = · · · = −αL = αcr in the phase IV. This choice of the ground state does not impose any additional restrictions in phase III, where the ground state is highly degenerate.
g
Tab. 3.1: Normalised ground-state mean-field energy EMF /L and parameters c to c from ( . ). The network is initialised in its ground state in the respective phase with equal mean fields αl = αcr . Phase
I
±ⅈ
αcr g
EMF /L
−ω
II √
−
III
− ω /κ
±
ω +κ κ
√
− ω /γ
−
ω +γ γ
−γ
(ω +κ) (κ− γ)− ω κ κ(ω +κ)
c
ω −γ
(κ−γ)(ω +κ) + κ κ(ω +κ)
γ− ω
c
−γ
ω (ω + κ)(κ−γ)−κ (γ+ κ) κ(κ+ω )
γ− ω
c
κ
ω (ω +κ)
κ(ω +γ) γ
c
γ− ω
First, we shall calculate dispersion relations for the system. They are determined by the quadratic part of the Hamiltonian ( . ), which can be more concisely written in a form of matrix–vector products as
H
( )
= d (
†
) d H
( ) d , d†
where d = (d , d , . . . , dL ) and d† = (d† , d† , . . . , d†L ) are L-element vectorrows or vector-columns depending on the context. The equal-mean-field ansatz allows us to calculate energy dispersion relations analytically for an arbitrary large number of sites N in the chain. These relations are determined by the quadratic part of the Holstein–Primakoff Hamiltonian ( . )
H( ) = c L + c
L ∑ l=
+c
L ∑ l=
d†l dl + c
L ∑ l=
(dl + d†l )
(d†l − dl )(d†l+ − dl+ ),
( . )
where c to c are factors depending on the parameters of the system and on the critical point in use. Their expressions are summarised in tab. . . The objective now is to transform ( . ) to the form of the Hamiltonian of L uncoupled harmonic oscillators. The energy level splitting of these oscillators will give us the energy of the lowest excitations in the quadratic approximation. H ( ) contains non-local terms, which can be eliminated by using the fact that the system possesses translational invariance and can be mapped
onto the reciprocal space by Fourier transformations of bosonic operators Dk = √
L
L ∑ l=
dl ⅇ−ⅈkl ,
( . )
where new operators Dk fulfil bosonic commutation relations [Dk , D†k′ ] = δkk′ and [Dk , Dk′ ] = . Under this transformation H ( ) maps to ∑ † ∑ Dk Dk + c (Dk D−k + h.c.) H( ) = c L + c (Dk D−k ⅇ k
+c
∑ k
− ⅈk
−
k D†k Dk ⅇⅈk
+ h.c.),
( . )
which does not contain non-local terms any more. The summation index k (with the physical meaning of the wave number) runs through the sequence , Lπ , Lπ , . . . , (L − ) Lπ and is π-periodic, i.e., −k ≡ π − k. Complex exponents in ( . ) can become a nuisance for the further analysis, so it is profitable to get rid of them by restricting summation limits to positive k values, obtaining ∑ † H( ) = c L + (Dk Dk + D†−k D−k )(c − c cos k) k>
+
∑ k>
(Dk D−k + D†k D†−k )( c + c cos k).
( . )
To diagonalise this expression we can use Bogoliubov transformations (bt) [ ] D±k = uk β±k − vk β†∓k , ( . ) where uk , vk ∈ R, uk − vk = and [β±k , ↱k′ ] = δkk′ . These transformations, again, preserve commutation relations for new Bogoliubov bosonic operators βk , and applying them to ( . ), we get the final diagonalised form of the quadratic Hamiltonian ] ∑[ ε(k) − ε (k) † ( ) ε(k)βk βk + H =c L+ , ( . ) k
where ε(k) =
√
ε (k) = c − c cos k,
ε (k) − ε (k),
ε (k) = c + c cos k,
and we have returned to the summation over both positive and negative wave numbers.
As the factors c to c are independent of the number of sites L in the network, excitation energies ε(k) do not depend on it either. Here we would like to make a side note for the future reference that the products of Bogoliubov coefficients vk and uk vk can be expressed in terms of ε(k), ε (k) and ε (k) as vk =
[
ε (k) − ε(k)
]
and uk vk =
ε (k) . ε(k)
( . )
Now, if need be, we can reverse the steps to get the ground state of the Hamiltonian ( . ). From the expression of the diagonalised Hamiltonian ( . ) one can see that the ground state |G⟩ is given by the condition β†k βk |G⟩ = , that is, |G⟩ =
⊗ k>
S(χk ) D(αk ) D(α−k ) | k ,
−k ⟩ ,
( . )
√ ∑L −ⅈkl are Fourier images of mean-field displacewhere αk = L− l= αl ⅇ ments and Dk | k , −k ⟩ = . We have also used the displacement operators in the reciprocal space [ ] [ √] D(αk ) = exp (αk D†k − α∗k Dk ) j .
Similarly to ref. [ ], the ground state is a product of two-mode squeezed states with squeezing parameters χk = artanh[ε (k)/ε (k)], where ε (k) and ε (k) were defined in ( . ), and the two-mode squeezing operator [ , ] [ ] S(χk ) = exp χk (D−k Dk − D†−k D†k ) .
Analysis of ( . ) shows that the ground-state excitation energy is minimal at k = and varies with κ and γ parameters as shown in fig. . d. The energy dispersion is quadratic and gapped in the vicinity of k = , when parameters do not fall onto the phase boundaries. Traversing from phase I into phase II, the gap closes with a linear dispersion (fig. . a), marking, much like in the Ising model, the transition from an unordered paramagnetic phase to a ferromagnetic one. Here, the softening of collective excitations leads to long-range correlations, resembling the behaviour at the Ising critical point in quantum magnetism [ , ]. Crossing from phase I into phase III, on the other hand, the energy gap closes for all the wave numbers (fig. . b), so collective excitations of any possible wavelength are allowed and the two phases lack long-range ordering. At the boundary between phases II and III the form of the dispersion relation changes abruptly. Approaching it from the phase II, the gap closes with the linear dispersion in the limit γ → κ − , while approaching the boundary from the phase III, the gap closes (as between phases I and III)
.5
.5 .95 . . 5 .
ε( ; γ)
ε(k)
κ= κ= κ= κ= .5
.5
(a)
γ = .5 −π
.5 .5
.95 . . 5 .
. γ
.5
.
.5
κ
γ
c(γ)
γ= γ= γ= γ=
(d)
k= π
k
.5
ε(k)
κ = .75 κ= . κ= . 5
.5 (b)
κ = .5 −π
.5 . 5 . − . + .5
. γ
.
.5
κ
γ
c(κ)
ε(k)
γ= γ= γ= γ=
.5
π
k
.5
(e)
.5
.5 (c)
γ+κ= . −π
k
(f) π
.5
. κ
.5
.
Fig. 3.8: Excitation energy dispersion ε(k) across the boundary between phases (a) I and II, (b) I and III and (c) II and III. (d) Energy gap ε( ) as a function of γ. (e, f) Group speed c = ε′k ( ) along the phase boundaries. ω = .
for all the wave numbers in Brillouin zone (fig. . c). This jump in the behaviour of excitation energies at the boundary indicates that the qpt there is of the first kind. The linear energy dispersion around k = at the phase boundaries is phonon-like, so we can define a group speed c = ε′k ( ). The way it depends on parameters is plotted in figs. . e and . f. Following the boundary between the phases I and II, the group speed goes down and reaches its zero with a discontinuity in the derivative at the triple point, while along the boundary between the phases I and III c = . That said, the group speed at the boundary of the symmetric phase is dependent only on the value of self-interaction. Along the boundary between the phases II and III the situation is different in a sense that the group speed is discontinuous here, so we have to consider two limits: when the boundary is approached from within the phase II (fig. . e) or from within the phase III (fig. . f). In the former case c = , while in the latter one c grows with the parameters. The jump of the group speed on this boundary, thus, is the bigger, the higher are coupling values. 3.6
correlation functions
In order to further characterise phases of the system, it is useful to calculate correlations of some observables between different sites in the ground state for each of the phases. One of the obvious choices is to consider correlations between components of total angular momenta of some site l and a site l+r. Thus we are interested in functions Cξξ′ (r) =
jL
L ⟨ ∑ l=
′
ξ Jlξ Jl+r
⟩
G
,
( . )
where ξ, ξ′ = x, y, z. To simplify the notation we have defined the expectation value of an operator O in the ground state as ⟨O⟩G = ⟨G|O|G⟩. The scaling factor / j is introduced to maintain consistency with the original Hamiltonian ( . ). Now we are ready to calculate correlation functions first in limit cases, introduced in § . , and then for cases, when all the system parameters are of the same order. 3.6.1
Limit cases
Our Hamiltonian has three parameters, so, like in § . , we shall consider three limit cases, when one of the parameters is much bigger than the other two. If ω ≫ γ, κ, the ground state of the system is |Gz, ,..., ⟩ [see ( . ) for
notation]. The correlation functions Cξξ′ =
j
⟨
⟩ j Gz, ,..., Jzl Jzl+r Gz, ,..., = δξξ′ δξz
( . )
are different from zero only for z components and do not depend on the distance between the nodes r. In the limit κ ≫ ω , γ, the ground state is two-fold degenerate, so in y general we have to consider a linear combination |G⟩ = a↑ |G , ,..., ⟩ + y a↓ |G , ,..., ⟩. As before, only one correlation function is different from zero: Cyy (r) =
j
(a∗↑ ⟨G
y
, ,...,
y
y y
| + a∗↓ ⟨G , ,..., |)Jl Jl+r (a↑ |G
j j = (a∗↑ − a∗↓ )(a↑ − a↓ ) = |a↑ − a↓ | .
y
, ,...,
y
⟩ + a↓ |G , ,..., ⟩) ( . )
It can be seen that depending on the contribution of the two base ground y y states, |G , ,..., ⟩ and |G , ,..., ⟩, the correlation can vary from , when both states are equally represented, to j/ , when only one of these is present. The correlation function still does not depend on r here. Finally, in the third limit γ ≫ ω , κ, the ground state is L -fold degenerate with different sites having spin projections of either j or −j independently of one another. As in the previous case, due to the degeneracy of the ground state, in general we the ∑have tox consider a linear combination of L base ground states |G⟩ = p ap |Gp ⟩, where p runs through all the possible combinations of zeros and ones. Again, the only correlation function different from zero (∑ (∑ ) ) y y ∗ x x Cxx (r) = ap ⟨Gp | Jl Jl+r ap |Gp ⟩ j p p =
j∑
(− )pl +pl+r a∗p ap′ ′
( . )
p, p′
does not depend on r and can give values from to j/ , subject to coefficients ap . When the angular momentum j is relatively small, barriers between different ground states are low, and high probability of tunnelling between them leads to the equalisation of coefficients ap , whatever the initial state the system was prepared in. The correlation function is then equal to zero, as for half of the states (− )pl = and for another half (− )pl = − and the same is true for pl+r , so summing these terms together will give us Cxx (r) = . This argument also holds for the case κ ≫ ω , γ. Thus we can conclude that in the large-ω limit angular momenta of different sites are highly correlated (as physically they are forced into the state when all J l s are parallel). Correlations in the other two limits depend
on the initially chosen combination of ground states, but gradually disappear when j is getting smaller due to the increasing probability of tunnelling between different ground states. The only axis of correlation in these limit cases always coincides with the quantisation axis, while different components of different sites are uncorrelated. 3.6.2
Equal mean fields
We now move on to calculations of correlation functions for ground states that are fully translationally invariant. The only ground state of phase I and both ground states of phase II fulfil this requirement. In phase III, however we shall have to restrict ourselves to the subset of just two ground states from L . As we shall be using the equal-mean-field ansatz upon the Holstein–Primakoff-bosonised system, the length of an angular momentum is supposed to be big, and so the tunnelling between different degenerate ground states can be neglected. That means that once initialised in the parallel-angular-momenta state, the system remains in it, and correlations are stable in time. Because of the adopted ansatz we can use the Bogoliubov vacuum ( . ) as the ground state |G⟩. Correlation functions Cξξ′ are sums of O(j) correlations of macroscopic classical backgrounds and O(j ) correlations of microscopic fluctuations of angular momenta, much like the Hamiltonian ( . ), which is a sum of mean-field energy jEMF ∼ O(j) and microscopic fluctuations H ( ) ∼ O(j ) thereupon. [Linear terms in ( . ) vanish, as we consider only minima of EMF .] By calculating both macroscopic and microscopic parts of correlation functions in the limit r → ∞ (implying L → ∞), we can classify quantum phases according to the long-range ordering. For this we first apply hpt to the correlation functions ( . ): L L ⟩ t ∑⟨ t ∑⟨ † ⟩ dl dl + dl dl + d†l d†l Cξξ (r) = jCMF + t + L L l=
+
t L
and then ft:
L ⟨ ∑ l=
⟩ t d†l dl+r + dl d†l+r + L
l= L ∑⟨ l=
dl dl+r + d†l d†l+r
⟩
⟩ t ∑⟨ t ∑⟨ † ⟩ Dk Dk + Dk D−k + D†k D†−k L L k k ⟨ ⟩ t ∑ † † −ⅈkr ⅈkr Dk Dk ⅇ + Dk Dk ⅇ + L k ⟩ t ∑⟨ Dk D−k ⅇ−ⅈkr + D†k D†−k ⅇⅈkr . + L
Cξξ (r) = jCMF + t +
k
∑ dl = √L k Dk ⅇⅈkl , k = πn/L, n = , ,...,L −
Tab. 3.2: Classical correlations CMF and factors t to t used in the expression for microscopic correlations ( . ) between angular momenta components ξ Jlξ and Jl+r in regions with ground-state mean-field displacements αl = αcr . Phase
I
αcr ξ
x
y
z
x
±ⅈ
II √
CMF t
−
t
t
y
z
x
ω κ
γ −ω γ
(ω −κ) κ(ω +κ)
ω κ(ω +κ)
ω +κ κ ω +κ κ
−
−
±
κ −ω κ
κ − ω κ−ω κ(ω +κ)
t
t
− ω /κ
ω + ω κ− κ κ(ω +κ)
−
III
ω κ(ω +κ)
− − ωκ
√
− ω /γ y
z ω γ
(ω −γ) γ(ω +γ)
ω + ω γ− γ γ(ω +γ)
− ωγ
ω + ω γ− γ γ(ω +γ)
κ−ω κ
ω γ(ω +γ)
ω −κ κ
ω γ(ω +γ)
ω +γ γ
−ω
+γ γ
γ−ω γ γ−ω γ
As a result, restricting wave numbers to positive values [cf. ( . )], we get the following ξ–ξ correlations: ⟩ t ∑⟨ † Dk Dk + D†−k D−k + L k> ⟩ t ∑⟨ Dk D−k + D†k D†−k + L k> ⟩ t ∑⟨ † + Dk Dk ⅇⅈkr + D†−k D−k ⅇ−ⅈkr + D†k Dk ⅇ−ⅈkr + D†−k D−k ⅇⅈkr L k> ⟩( ) t ∑⟨ Dk D−k + D†k D†−k ⅇⅈkr + ⅇ−ⅈkr + L k> ⟩ ⟨ ∑ † † = jCMF + t + (t + t cos kr) Dk Dk + D−k D−k L k> ⟩ ⟨ ∑ + ( . ) ( t + t cos kr) Dk D−k + D†k D†−k L
Cξξ (r) = jCMF + t +
k>
D±k = uk β±k − vk β†∓k
where CMF and t to t are factors depending on the parameters of the system and on the critical point in use (see tab. . ), and the expectation value is taken in the ground state. Note that approaching the limit cases, the macroscopic part of correlation functions is in accordance with the result of § . .. Fourier-transformed correlation functions ( . ) can now be mapped onto Bogoliubov bosons βk , β†k obtained while diagonalising the original
Hamiltonian. The mapping gives us ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ D†k Dk + D†−k D−k = uk β†k βk + β†−k β−k + vk βk β†k + β−k β†−k ⟩ ⟨ − uk vk βk β−k + β†k β†−k , ⟩ ⟨ ⟩ ⟨ Dk D−k + D†k D†−k = (uk + vk ) βk β−k + β†k β†−k ⟩ ⟨ − uk vk βk β†k + β†k βk + β−k β†−k + β†−k β−k .
Noting that expectation values of products of Bogoliubov operators in the ground state ( . ) ⟨β†k βk ⟩ = ⟨β†−k β−k ⟩ = ⟨βk β−k ⟩ = ⟨β†k β†−k ⟩ = as well as the commutation relations [βl , β†l′ ] = δll′ , this can be simplified to ⟩ D†k Dk + D†−k D−k = vk , ⟩ ⟨ Dk D−k + D†k D†−k = − uk vk . ⟨
( . )
Now, using relations ( . ) to express Bogoliubov coefficients in terms of ε(k), ε (k) and ε (k), we can substitute ( . ) into ( . ) to obtain correlation functions in the form [ ] t ∑ ε (k) t ∑ ε (k) Cξξ (r) = jCMF + t + − − L ε(k) L ε(k) k> k> [ ] t ∑ ε (k) t ∑ ε (k) + − cos kr − cos kr. ( . ) L ε(k) L ε(k) k>
k>
In the thermodynamic limit L → ∞ the sums can be substituted with integrals, assuming dk ≡ Lπ , to get ] ∫ π[ ∫ π ε (k) t ε (k) t − dk − dk Cξξ (r) = jCMF + t + π ε(k) π ε(k) ] ∫ π[ t ε (k) t ε (k) + ( . ) − − t cos kr dk . π ε(k) ε(k) The mean-field correlations CMF = j Jlξ Jlξ of ( . ) are proportional to the square of the respective component of the angular momentum, because the ring is initialised in the ground state where all J l s are parallel and macroscopically static. Thus CMF is maximised. Microscopic correlations can be separated into two parts: a constant background [ ] } ∫ π{ ε (k) ε (k) ∞ t +t − − t dk ( . ) Cξξ = π ε(k) ε(k)
.
Cξξ − jCMF
.
ξ=x ξ=y ξ=z
ξ=x ξ=y ξ=z
ξ=x ξ=y ξ=z
(a)
(b)
(c)
. − . − . Distance r
8
Distance r
8
Distance r
8
Fig. 3.9: Microscopic parts of correlation functions Cξξ (r) − jCMF in the phases (a) I with γ = . , κ = . ; (b) II with γ = . , κ = . and (c) III with γ = . , κ = . . For all panels ω = .
and oscillations around it, which decay with r: ] ∫ π[ t ε (k) t ε (k) osc − − t cos kr dk . Cξξ (r) = π ε(k) ε(k)
( . )
In the limit r → ∞ the oscillating part vanishes, and only the background C∞ ξξ = Cξξ (∞)−jCMF is left. Plots of microscopic parts of correlation functions (fig. . ) show that the latter differ distinctly in different regions only in the background terms C∞ ξξ and the amplitude of oscillations. The fitted quasi-period of the oscillations (R = . ± . ) and their relaxation length (ρ = . ± . ) remain, where applicable, the same for all considered γ and κ parameters in the large-r limit. The necessity of taking a limit here is dictated by the fact that quasi-periods and relaxation lengths vary slightly with r (CMF is not an exponentially decaying harmonic function), but this variation becomes negligible for larger r. For the fit we used a function ( π ) Cfit (r) = A cos r + φ ⅇ−r/ρ + C∞ R
with fitting parameters A, C∞ , R, ρ and φ. The fitting domain was limited to ⩽ r ⩽ , which gave R and ρ values with adequately low standard deviations (see above). In phase I only z–z correlations persist in the long-range limit (fig. . a), repeating the behaviour of the macroscopic correlations. The same correspondence between the macroscopic and microscopic correlations is also found for phases II and III. That said, the long-range ordering persists in terms of both O(j) and O(j ) in all three phases between components Jz , Jy and Jx in phases I, II and III, respectively. Short-range correlations, on the other hand, are found between components Jξ for any ξ in any phase. The range of these correlations is around
six sites or three full oscillation wavelengths and does not depend on either the component or the phase. The last remark to make here is that plots in fig. . show only correlations between different sites, so auto-correlations at r = are not considered here as having little significance. 3.7
conclusions and outlook
In this part of the thesis we have discussed quantum phase transitions in networks of Lipkin–Meshkov–Glick systems coupled via ferromagnetic and anti-ferromagnetic interactions. Finite and translationally-invariant infinite systems were analysed both numerically – by construction and diagonalisation of Hamiltonian matrices in the basis of angular momenta eigenstates – and analytically – by limit-case analysis and by using Holstein– Primakoff bosonisation approach. In the latter case, separating the Hamiltonian into terms of different orders magnitude in j, we used the semi-classical mean-field O(j) terms to calculate the properties of the ground state and quantum O(j ) terms to get the lowest part of an excitation spectrum. Ground-state calculations showed that depending on Hamiltonian parameters κ and γ, a network can be found in one of the four quantum phases (necessarily assuming the thermodynamic limit j → ∞), which differ in mean-field displacements (equivalently, direction of a classical angular momentum vector) of individual nodes that minimise energy as well as in degeneracy (one-, two- or L -fold) of the ground state. Supporting numerical calculations allowed us to verify the results obtained from using Holstein– Primakoff approach, and iterative minimisation algorithms made it possible to follow the change in phase boundaries for finite networks when side chains or loops are added to the initially one-dimensional network. At phase boundaries ground-state energy is continuous, but has discontinuities in the first or the second derivative with respect to parameters. Excited-state levels cluster at the phase boundaries, as the excitation energy becomes zero. What we can get from analysing the O(j ) part of the Hamiltonian is the way the energy gap closes, whether the softening of the modes happens only for a single wave number or for any of them and the group speed of excitations when the energy dispersion is phonon-like. Calculations of correlation functions also emphasised the difference between phases. Being initialised in a uniform ground state, it is obvious that macroscopic correlations of order j appear in phases I–III (all J l vectors are forced to be parallel). The O(j ) correlations are of two types: short- and long-range. Long-range correlations are different from zero for the same phases and Jlξ as their macroscopic counterparts. Short-range correlations are present in all the phases and Jlξ s and have the same correlation lengths and periods of oscillations.
Possible realizations of discussed model may be based on Bose–Einstein condensates (bec) in optical lattices (cf. refs. [ – ]). Other possibilities include single-chain magnets and nanomagnets (cf. refs. [ , – ]). In a recent experiment, by superimposing an optical lattice on a harindependent transversely conmonic trap it was possible to create L = fined becs, which allows to perform quantum-enhanced magnetometry by using spin squeezing [ ]. In such an experimental set-up, our model could be realized by inducing a coupling between the condensates at different sites of an optical lattice, where each site can contain condensates consisting of up to atoms. If the dynamics is restricted to two atomic hyperfine levels, an on-site linear coupling between levels could be realized. In addition the on-site non-linear terms of the Hamiltonian ( . ) can be controlled by means of intra-well atomic interactions. In single-chain magnets the model can be realized by choosing appropriate ligands for magnetic nuclei that would create an easy axis in the chain to induce Ising-like coupling (cf. ref. [ ]). An accurate choice of ligands may also give rise to a quadratic local term in the Hamiltonian [ ]. The one-dimensional chain with periodic boundary conditions can be realized by cycling the chain as in the related work [ ]. A certain challenge for experimental realisations could be the tunnelling between degenerate ground states in phases II and III, as j will always be finite. The method we have developed in this work can be adapted to study other kinds of networks consisting of coupled mean-field-type critical systems, e.g., Dicke models [ ] and spinor Bose gases within a single-mode approximation [ ]. In the next chapter we shall extend the approaches presented here to systems that, apart from different quantum phases, can have different topological phases. As a simple example of such a system we shall be exploring chains of lmg models with alternating couplings between nodes: a hybrid of an lmg and an ssh model.
COMBINED LMG–SSH MODEL Having explored networks of quantum-critical Lipkin–Meshkov–Glick (lmg) systems with uniform couplings, a natural continuation would be to build up on this model and to extend it. At the end of the last chapter we gave some possible options for further research, and here we shall continue with one of them. The major part of the material presented here is published as a regular article in Physical Review E [ ]. 4.1
model
Instead of using uniform couplings κl′ (l′ + ) = κ between the nodes in the chain, we shall be considering the case when κl′ (l′ + ) = κ if l′ is even and κl′ (l′ + ) = κ if l′ is odd. This arrangement of couplings resembles the Su– Schrieffer–Heeger (ssh) model, so our system may be thought of as a hybrid between the latter and the lmg model. In such a system both quantumcritical and topological aspects are present, and determining how these two work together is the main goal of this part of the thesis. The lmg–ssh system is inherently semi-classical, meaning that it can be described on two levels: the mean-field level and the level of fluctuations around it. The bifurcation of the ground state on the mean-field level is accompanied by the emergence of squeezing visible on the fluctuations level, and the effect of the latter on the topological properties of the system has been recently investigated [ – ]. In our approach, we provide a microscopic and analytically tractable model to study the origin of the squeezing terms and their interplay with topological effects. 4.1.1
Hamiltonian and symmetries
As was mentioned already, we shall be working with a one-dimensional network of L sites, each of which is coupled to its nearest neighbours with two different couplings κ and κ so that a staggered structure resembling the ssh model, described in § . , is formed (see fig. . ). The unit cell of the chain consists of two sites labelled A and B (in other words, the chain itself consists of two sublattices), whose local dynamics correspond to dynamics of an lmg model, described in § . . The Hamiltonian of our system then assumes form
H=
L ∑ l=
(HlA + HlB ) −
j
L ( ∑ l=
) − + − κ J+ J + κ J J + h.c. , lA lB lB (l+ )A
( .)
ω γ κ JA
JB
J
A
ω
κ J
B
JA
JB
JLA
ω
JLB
Fig. 4.1: Chain of lmg models with alternating couplings κ and κ . Shaded elementary unit cell contains two nodes (A and B). Each node is characterised by its angular momentum Jls , level-splitting ω and strength of internal interactions γ.
where Hls = ω Jzls − γj Jxls Jxls are the Hamiltonians of individual nodes, index s = A, B labels the sublattice and l = , , . . . , L enumerates unit cells. The hopping-like coupling between neighbouring sites was preferred to the y–y coupling used previously for better consistence with the ssh model. As we shall show later, it also reduces the number of quantum phases from three to two, but, if need be, calculations can easily be modified to fit any desired coupling scheme quadratic in Jls s. For the previous model we used Dicke states with maximised j = N , and throughout the following sections we shall continue to adhere to this restriction, which is well motivated by experimental realisations of an lmg model in Bose–Einstein condensates [ ]. Having defined the Hamiltonian, it is now practical to find its symmetries, using which different quantum phases can be told apart, and related conserved quantities. In the thermodynamic limit j → ∞ some of these symmetries can be spontaneously broken, marking an onset of a quantum phase transition. Topological phase transitions, on the other hand, can occur without symmetry breaking [ ]. In that case, a change of a phase is associated with a change in system’s topology. A well-known example of this is the quantum Hall effect [ ], where the conductivity of two-dimensional electron gas in a magnetic field is given by a topological invariant. Changing the magnetic field, one comes across a phase transition between different Hall plateaus, and no symmetry is broken. Let us consider the operator
N=
L ∑
(JzlA + JzlB + j),
l=
which is the z component of the total angular momentum (up to an additive constant) or, equivalently, the total number of excitations in the system. The Hamiltonian ( . ) commutes with N if γ = , meaning that the total number of excitations is conserved. Another symmetry of the Hamiltonian is a global rotation Rθz = ⅇⅈθN around the z axis, which is continuous ( ⩽ θ < π) in the case γ = or discrete (θ = , π) otherwise.
4.1.2
Mean field and luctuations
As lmg is a semi-classical model, the number of two-level systems making up one node of the network N ≫ , and thus j ≫ . In the thermodynamic limit j → ∞ each of the nodes has a semi-classical bifurcation at the mean-field level [ ]. In addition to the limit j → ∞ we also use another thermodynamic limit L → ∞, when symmetries of the system can be broken even for finite j. Here we should note that the two limits, in general, do not commute, and taking them in different order may potentially lead to different results [ ]. Comparison of these two orderings lies beyond the scope of this thesis, though. Working in the limit j ≫ means that we can continue to use the Holstein–Primakoff transformations ( . ), mapping the Jlsξ operators onto bosonic creation and annihilation operators b†ls and bls . This approach allows us to describe semi-classical features of the system, such as bifurcations and quantum phase transitions associated with them. In order to explore quantum signatures of these bifurcations, we shall consider quantum fluctuations around semi-classical trajectories. Having that in mind, we introduce a time-dependent displacement operator ]√ } {[ Dls [αls (t)] = exp α∗ls (t)bls − αls (t)b†ls j , where [cf. ( . )] αls (t) are time-dependent mean-field displacements. With the help of Dls we define a new set of displaced bosonic operators [ ] √ ( . ) dls = Dls† (αls ) bls Dls (αls ) = bls − αls j, √ describing fluctuations around the mean fields αls j. We have dropped explicit dependence of αls on time here. To obtain dynamics of the system, we first apply the hpt ( . ) to the Hamiltonian ( . ) and then moving to displaced operators ( . ) we can obtain the Schrödinger equation
with
ⅈ
∂ |Ψ α (t)⟩ = Hα |Ψ α (t)⟩ , ∂t
|Ψ α (t)⟩ = D† (α) |Ψ(t)⟩ , ⊗ D(α) = Dls (αls ), ls
) ∂ Hα = D (α) H − ⅈ D(α), ∂t √ ∂ j ˙ † − α˙ ∗ d) j, D† (α) D(α) = ( αα ˙ ∗ − α α˙ ∗ ) + ( αd ∂t †
(
and vectors α = (α A , α B , . . . , αNA , αNB ) and d = (d A , d B , . . . , dNA , dNB ). If we are interested in fluctuations close to the semi-classical trajectories, we can make quadratic approximation to simplify the Hamiltonian Hα by expanding it into Taylor series in j− / and truncating it to √ ( . ) Hα = H ( ) (α)j + H ( ) (α, d) j + H ( ) (α, d), ˙ ∗ − α α˙ ∗ ) + EMF (α), H ( ) ∼ O(d) and H ( where H ( ) = − ⅈ ( αα The mean-field energy EMF is EMF (α) = − L ω + ω − − −
γ
κ
L ∑
∼ O(d ).
(α∗lA αlA + α∗lB αlB )
l=
[( − α∗lA αlA )(α∗lA + αlA ) + ( − α∗lB αlB )(α∗lB + αlB ) ]
l= L √ ∑ l=
L κ ∑√ l=
L ∑
)
− α∗lA αlA
√
− α∗lB αlB (α∗lA αlB + α∗lB αlA )
− α∗(l+ )A α(l+ )A
√
[ ] − α∗lB αlB α∗(l+ )A αlB + α∗lB α(l+ )A . ( . )
The displacement transformation used here can be interpreted as a transformation into a co-moving frame [ , ] with the origin given by meanfield displacements αls (t) satisfying the classical Hamilton equations ∂EMF (α) ∗ , α˙ ls = ⅈ ∂α ls
∂E (α) α˙ ls = −ⅈ MF∗ . ∂αls
( . )
Thus the vector of mean-field displacements traces the trajectory of the corresponding semi-classical system. Once we have solved this set of equations, we can calculate quantum fluctuations governed by H ( ) . In general, for a given semi-classical trajectory α(t) the quadratic part H ( ) of the Hamiltonian depends on time. But, as the objectives of this work lies with the ground states of the system, which correspond to fixed points of EMF (α), the time dependence is lost, and α˙ ∗cr = α˙ cr = . Moreover, like previously, the linear term H ( ) in ( . ) vanishes at the fixed points. As a side note, we can recall from the previous chapter that Gaussian quantum fluctuations are determined by the Hessian matrix and thus depend on the local geometry of the mean-field energy EMF (α) in the α space.
The quadratic O(j ) terms of the Hamiltonian ( . ), constituting H ( ) , are the only ones that describe quantum fluctuations around the ground state and that are responsible for the lower range of the spectrum of excitations. In order to simplify the following expressions, we shall change the notation a little. Instead of labelling the position in the chain with the unitcell and the sublattice indices, we switch to the sequential numbering of the sites. Thus, αls becomes α (l− )+p , where p = for A sublattice and p = for B sublattice. The same indexing will apply for all other variables, too. Furthermore, for open chains it is convenient to introduce the combined coupling if l ⩽ or l ⩾ L, Kl = κ if l is odd, κ if l is even.
The quadratic part of the Hamiltonian can then be written (for arbitrary α, neglecting an additive constant that does not depend on d) as
H
( )
=
L ∑ l=
+
() cl d†l dl
L ∑ l=
+
L [ ∑ l=
] ( ) cl dl + h.c.
[
] ( ) ( ) Kl cl d†l dl+ + cl dl dl+ + h.c. ,
where the α-dependent factors c to c are ()
cl
γαl (α∗l − ) + γα∗l αl (αl + α∗l ) rl ∗ ω (αl αl − ) − γα∗l (α∗l + αl ) − rl γ(αl + α∗l ) + γ(α∗l αl − ) +
=−
αl αl (Kl+ rl+ αl+ + Kl− rl− αl− ) rl ∗ ( αl αl − αl )(Kl+ rl+ α∗l+ + Kl− rl− α∗l− ) , − rl γ(α∗l − ) − γα∗l αl ( α∗l + α∗l αl − ) = rl ∗ ∗ Kl+ rl+ αl (αl αl+ − α∗l αl α∗l+ + α∗l+ ) + rl ∗ ∗ Kl− rl− αl (αl αl− − α∗l αl α∗l− + α∗l− ) , + rl −
( )
cl
( . )
( )
cl
( )
cl
(α∗l αl + α∗l+ αl+ ) − αl α∗l+ ( α∗l αl+ + αl α∗l+ ) − rl rl+ ∗ ∗ ∗ ∗ (αl + αl+ ) − αl αl+ (α∗l αl+ + αl α∗l+ ) = , rl rl+ =
,
and we used a shorthand rl =
√
− α∗l αl .
As we shall be working with both positive and negative values of κ , , it would profitable to find a way to first perform all the calculations for κ , > and then apply some kind of a transformation to obtain the results for other combinations of κ , signs. To get such a transformation, note that although the Hamiltonian ( . ) is not invariant under the interchange of signs of couplings κ 7→ (− )u κ
and
κ 7→ (− )v κ ,
( . )
where u, v ∈ { , }, there is a set of local gauge transformations that compensates this change of signs, namely, ξ ξ JlA 7→ (− )(u+v)l JlA ,
ξ ξ JlB 7→ (− )(u+v)l+u JlB ,
JzlA 7→ JzlA ,
JzlB 7→ JzlB ,
( . )
for ξ = x, y or, expressed in terms of bosonic operators, blA 7→ (− )(u+v)l blA ,
blB 7→ (− )(u+v)l+u blB .
( . )
Using these transformations we can confine all the calculations to positive couplings κ , and reach the goal of deducing the properties of the system, when κ , have different combinations of signs, from obtained results. Mean-field displacements αls and fluctuation operators dls are transformed () ( ) the same way as operators bls . The factors cl to cl from ( . ) remain invariant under the change of signs, and so does H ( ) . The arguments presented here are valid for both periodic and free open boundary conditions of a one-dimensional chain, so the approach of reconstructing the properties for positive and negative κ , from the case κ , > will be applied in the following sections for the chains with both periodic (§ . ) and open (§ . ) boundary conditions. 4.2 4.2.1
bulk properties Mean field
When an open chain becomes sufficiently long, its middle part feels almost no influence from the boundaries and thus can be treated as a translation-
ally invariant unit – a bulk. On the one hand, translational invariance makes it easy to describe its properties analytically by using transformations into the reciprocal space. On the other hand, bulk properties are strongly related to the properties of the whole finite chain, which is known as the bulk–boundary correspondence [ ]. For instance, the presence or absence of edge-localised states in the band gap can be unambiguously determined by topological invariants of the bulk. Let us begin by calculating the mean-field ground-state energy ( . ) for the bulk. Due to the translational invariance, all the mean-field displacements on the same sublattice should be equal, i.e., αlA = αA and αlB = αB . Moreover, we can assume the uniform ansatz αA = αB = α from the previous chapter in the case κ , > , and numerical minimisation of EMF (α) proves that this ansatz works well. The mean-field ground-state energy is then given by EMF (α) = L ω (α∗ α − ) −
Lγ
( − α∗ α)(α∗ + α)
− L(κ + κ )( − α∗ α)α∗ α
( . )
with critical points, given by ( . ), at αω = , ω , γ+κ +κ √ ω α κ = ±ⅈ − . κ +κ
αγκ = ±
√
−
( . )
The point αω exists for any parameter values and gives the energy EMF (αω ) = − L, the pair of points αγκ on the real axis exists only when γ + κ + κ > ω , giving the energy EMF (αγκ ) = −L
ω + (γ + κ + κ ) γ+κ +κ
and the pair of points ακ on the imaginary axis exists only when κ +κ > ω , giving the energy EMF (ακ ) = −L
ω + (κ + κ ) . κ +κ
Comparing these expressions in respective regions, one can see that the mean-field ground-state energy becomes non-analytic and its second derivative has a jump when one crosses the boundary γ + κ + κ = ω . Thus it is
obvious that in the positive-κ , regime two distinct quantum phases exist and the qpt between them is of the second order (cf. isolated lmg model). Traversing from the phase I to the phase II, the minimum at αcr = αω splits and shifts to αcr = αγκ . The points ακ never mark the minimum of EMF (α), and thus will not be analysed. After having found the bifurcation diagram for κ , > , it is straightforward to generalise it for other combinations of κ , signs by transformations ( . ). For instance, let us consider the case when κ > and κ < . The mean-field displacements are transformed according to ( . ) as αlA 7→ (− )l αlA
and
αlB 7→ (− )l αlB ,
while κ 7→ κ and κ 7→ −κ . The mean-field ground-state energy ( . ) then becomes L ∑ [
EMF (α) = − L ω + ω − − −
γ
γ
κ
L [ ∑ l= L [ ∑
(− )l α∗lA (− )l αlA + (− )l α∗lB (− )l αlB
l=
− (− )l α∗lA (− )l αlA
][
− (− )l α∗lB (− )l αlB
][
l= L √ ∑
− (− )l α∗lA (− )l αlA
l=
(− )l α∗lA + (− )l αlA
]
(− )l α∗lB + (− )l αlB
√
]
]
− (− )l α∗lB (− )l αlB
[ ] × (− )l α∗lA (− )l αlB + (− )l α∗lB (− )l αlA
+
L κ ∑√
− (− )l+ α∗(l+ )A (− )l+ α(l+ )A
l=
×
√
[
− (− )l α∗lB (− )l αlB
] × (− )l+ α∗(l+ )A (− )l αlB + (− )l α∗lB (− )l+ α(l+ )A ,
which (after simplifications) is exactly the same expression as we would get if κ were positive. Thus, all the expressions from ( . ) up to this point can be generalised to the cases κ , ≷ by simply replacing κ , with their absolute values. The resulting five quantum phases I–V are shown in fig. . b. We should note here that in the three-dimensional γ–κ , space the phase I forms an octahedron, thus in the projection on the κ –κ plane the boundary between the symmetric and the broken-symmetry phases is a square when γ < . Otherwise, when γ ⩾ , the phase I vanishes. This can be better illustrated by fig. . a.
g L EMF
−. III
.
γ =
.
II
.
.
−.
κ /ω
.8
.
− .
. κ /ω
−.
IV
(b)
− .
.
(a) − .
−.
I
−.
κ /ω
Fig. 4.2: Quantum phases and bifurcation diagram of the system. (a) Dependence of the phase boundaries on the value of γ (contour labels). For γ ⩾ only broken-symmetry phases are present. (b) Mean-field ground state energy for different parameters. Symmetric phase I has a single minimum when α = αω , while broken-symmetry phases II–V have doubly degenerate minima α = αγκ . Tab. 4.1: Mean-field displacements corresponding to the points of minima in respective phases. The shorthand a = ⅇⅈφ αγκ [see ( . )] is used. The phase φ is arbitrary if γ = and takes values or π otherwise. Phase
I
II
III
IV
V
αlA
a
a
(− )l a
(− )l a
αlB
a
−a
(− )l+ a
(− )l a
To conclude the description of the ground state and quantum phases, let us summarise mean-field displacements of the ground state (see tab. . ). In the disordered phase I the non-degenerate ground state has αA = αB = . Ground states of phases II–V are all doubly degenerate. In the phase II, like in phase I, all the displacements are equal independent of the sublattice: αA = αB . In the phase IV the displacements change signs from one node of the chain to the next one: αA = −αB . Phases III and V are a bit more complicated. Here, the unit cell has to be extended to four sites, as the displacement on an arbitrary site has the same sign as one of its neighbours and the opposite sign to the other neighbour, i.e., the signs of the displacements follow the pattern · · ·++−−++−− · · · , comparing with the pattern · · · + − + − + − + − · · · for the phase IV and · · · + + + + + + + + · · · for the phase II.
Tab. 4.2: Factors c( , is used.
, , )
from ( . ) at αls = αcr . The shorthand ζ = γ + |κ | + |κ |
Phase
I
c( ) c(
ω −
)
c( ) c(
4.2.2
II–V ζ + ζ +ζ− γ ζ(ζ+ )
γ
−γ
ζ − ζ −ζ− γ ζ(ζ+ )
−
+ ζ− − ζ ζ(ζ+ ) ζ + ζ− ζ(ζ+ )
)
Quadratic luctuations
To get the energy of the lowest excitations, we have to reconsider the quadratic part of the Hamiltonian ( . ) in the bulk of the network. We shall start again with the region κ , > and generalise the results later on. Because of the translational invariance of the bulk in that region, the factors in ( . ) do not depend on the index l, and their expressions are considerably simplified at α = αcr (see tab. . ). Bearing that in mind, we now use the Fourier transforms ( . ) to map the bosonic operators onto the reciprocal space. The newly obtained operators Dks still fulfil bosonic commutation relations ] [ [ ] (†) (†) Dks , D†k′ s′ = δkk′ δss′ and Dks , Dk′ s′ = .
The Fourier transforms allowed us to remove all non-localities in the H ( ) , so now it can be written in the form of matrix–vector products
H( ) =
∑( k
D†k
D−k
)
Hk
(
) Dk , D†−k
( . )
† where Dk = (DkA , DkB ) and D†k = (DkA , D†kB ) are two-element vector-rows or vector-columns depending on the context. The Bogoliubov matrix Hk has a block form ( ) Ak Bk , Hk = B k Ak
with self-adjoint matrices
) c( ) c( ) (κ + κ ⅇ−ⅈk ) Ak = ( ) , c (κ + κ ⅇⅈk ) c( ) ( ) c( ) c( ) (κ + κ ⅇ−ⅈk ) Bk = ( ) . c (κ + κ ⅇⅈk ) c( ) (
Next, in order to find the low-energy spectrum in the quadratic approximation, we have to diagonalise Hk . The first excitation energy for the mode k will then be given by the frequency of the k-th harmonic oscillator. We need to assert that the new operators we obtain are indeed bosonic creation/annihilation operators, but the normal diagonalisation routine does not guarantee this. So we are forced to use Bogoliubov transformations instead. This approach is sketched in appendix E, but for more detailed description we refer the reader to [ ] and to the original paper [ ]. The Bogoliubov transformation is given by a × matrix Tk such that (
Dk D†−k
)
= Tk
(
βk β†−k
)
and Tk S T†k = S. Here we introduced the expanded Pauli z matrix S = σz ⊗ as a Kronecker tensor product between the normal Pauli z matrix and the × identity matrix. Applying this transformation to the H ( ) , we obtain ∑ † βk Ek βk , H( ) = k
where Ek is a × diagonal matrix of excitation energies. Matrices Tk and Ek can be found by solving the eigenvalue problem ) ( Ek . S Hk Tk = Tk −E−k As there are two sites within a unit cell, dispersion relations ελ (k) = (Ek )λλ have two bands, labelled with λ = , such that ε (k) ⩽ ε (k) for each of the phases I–V. For the phase I Bogoliubov procedure gives us the relation √ √ (I) ε , (k) = ∓ κ + κ + κ κ cos k √ √ × − γ ∓ κ + κ + κ κ cos k ( . )
and for the other four phases II–V the relation [ (II–V) ζ(ζ − γ) + κ + κ + κ κ cos k ε, (k) = ζ ] √ ∓ (ζ + ζ − γ) κ + κ + κ κ cos k ,
( . )
where ζ = γ + |κ | + |κ |. We note here that the simultaneous change of signs of both couplings does not affect the result, whereas the change of sign in just one of the couplings shifts the dispersion relations by the phase of π, as shown in fig. . . Plots in the panels a–f there correspond to the κ
−π .8 . . . −π .8 . . .
.8 . . .
(d)
π
.
(b)
π
−π a b c
(e)
d e f
π
− . − .
(c)
−π
π
Wavenumber k
κ
.
π
−π (f)
−π
.8 . . . .8 . . .
Excitation energy ελ (k)
(a)
κ
Excitation energy ελ (k)
.8 . . .
π
Wavenumber k
Fig. 4.3: Bulk band structure as calculated from ( . ) and ( . ). Parameters ω = , γ = . , κ = . and (a–f) κ = − . , − . , − . , . , . , . .
and κ parameter values marked by respective letters in the phase diagram in the middle of the figure. As expected from the previous discussion of the lmg chains with the uniform coupling, the excitation energy becomes zero at the phase boundaries (figs. . b and . e). But this time we have two bands, and so in addition to the said behaviour there are certain points, at which the gap closes between the band and the band (figs. . c and . d). These points form the boundary between different topological phases and will be discussed in the next section. 4.2.3
Topological properties
Our system is partially based on the ssh model, which is one of the simplest configurations showing topological phase transitions (tpt). The presence of tpts in our system is thus inherent. The central question of this part of the work is the question of mutual interaction or interference between the topological and the quantum phases inherited from the base models. In particular, we shall show that the major effect of such interference is almost total delocalisation of edge states, which would normally be present in the topologically non-trivial phase, as affected by the transition into the broken-symmetry quantum phase. But – first things first. An essential physical feature that allows us to distinguish between different topological phases is the existence (and the number) of states that
are localised around the defects of the chain, such as edges, impurities, vacations and so on: edge states. These states can be calculated by direct diagonalisation of the Hamiltonian of a finite-size network, but what is more important, in the case when the only defects are the edges of the one-dimensional chain, the bulk–boundary correspondence can be used to predict their number using the bulk Hamiltonian, which in our case can easily be obtained analytically (see previous section). The key point to establishing such a correspondence is to find and to calculate some kind of a topological invariant – a number that would remain constant inside a single topological phase. Several of these, like winding number, Chern class, Euler characteristic, to name but a few, can be defined for different systems. For a one-dimensional topological insulator the winding number is an appropriate choice, which for more complex bosonic systems like ours can be generalised [ , ] to a closely related topological invariant: symplectic polarisation P=
ⅈ
π
∫
π −π
[
dT S T†k S k dk
]
dk ,
( . )
,
which is a real-valued quantity. The k-dependent Hamiltonian Hk defined by its matrix ( . ) possesses an inversion symmetry τ Hk τ = H−k , where τ = ⊗ σx , and this symmetry is sufficient for the symplectic polarisation to be quantised [ ]. More precisely, P can only take integer or half-integer values , ⁄ , , ⁄ , … and, being a topological invariant, cannot be changed by a smooth transformation that does not close the gap between the band and the band . That allows us to unambiguously define the boundary between different topological phases as the locus of points in parameter space, where the gap between the two bands closes. It can be obtained analytically from ( . ) and ( . ) if we demand ε (k) = ε (k). In the result we get that the phases are separated by lines |κ | = |κ |. The gap closes at k = π if κ κ > and at k = otherwise (cf. § . . , where these points were the other way round). By direct numerical evaluation of symplectic polarisation in different points of each phase, we can assert that phases, with |κ | > |κ | have P = and thus are topologically trivial, whereas phases, with |κ | > |κ | have P = ½ and thus are topologically non-trivial (see fig. . ). Dedicated reader can recall that earlier we have obtained the same topological phase boundary for the underlying ssh model.
theless we can still rewrite H ( ) in the form of matrix–vector products ( ) ( † ) d ( ) H = d d H † , d
where d = (d A , d B , . . . , dLA , dLB ) and d† = (d†A , d†B , . . . , d†LA , d†LB ) are Lelement vector-rows or vector-columns depending on the context. The L × L matrix H has a block form ) ( A B . H= B A As before, we cannot simply diagonalise this matrix in an ordinary way, as the resulting operators will not necessary obey bosonic commutation relations, so once again we have to apply Bogoliubov theory [ ]. In short, to get the eigenstates and eigenenergies in the new Bogoliubov bosonic operators β, β† , we have to solve the eigenvalue problem ) ( E , ( . ) SHT = T −E where S = σz ⊗ L , columns of T are the eigenstates and E is the diagonal matrix of energies. Due to the inversion symmetry, which the system possesses, the spectrum is symmetric with respect to ε = , but we will only be interested in the case ε > as negative excitation energies are unphysical. Let us take a look at how the spectrum changes when one of the couplings (κ , for instance) is varied, while other parameters are kept constant (see figs. . a and . b). We shall consider only κ , > as other combinations of signs will give similar results. Furthermore, we shall set local parameters γ = . and ω = . In order to cover all the first quadrant phases (I′ , I′′ , II′ and II′′ ) we shall choose two different κ couplings: κ = . covers the phases I′ , II′ and II′′ (fig. . a), whereas κ = . covers the phases I′ , I′′ and II′′ (fig. . b). In the first case the tpt occurs in the broken-symmetry quantum phase, but in the second one in the symmetric phase. As we shall show later these two cases are rather different. The energy spectrum itself consists, much like in the case of a pure ssh model, of two bands. In the phase I′ the lower-lying states of each band decrease, but the high-lying states increase in energy with κ . This eventually leads to either the touching of the bands or the lower band touching the zero-excitation-energy line. The first case marks the tpt, while the second one marks the qpt. Let us first concentrate on the transition within the symmetric quantum phase, i.e., between phases I′ and I′′ (fig. . b). As we pass into the nontrivial tp I′′ , the gap reopens, but a pair of states – one from the upper and another from the lower band – remains in the middle of the gap. These are
the edge states, and their localisation length is correlated with the distance in energy scale between them and the closest states in the bands. If in the phase I′ the upper states of the lower band and the lower states of the upper band were energetically approaching each other, in the phase I′′ they are getting farther away from each other again. The energy of the edge states remains constant, so up to now the spectrum repeats the one of an ssh model. The qpt at κ = . in fig. . b changes the behaviour of the lower band. The energy of all the levels of that band starts to increase with κ , but the energy of the edge states is scarcely affected. This eventually leads to the situation, when the edge states get absorbed by the lower band, and although the topological invariant has not changed, their localisation at the chain boundaries is lost completely. If the qpt occurs before the tpt, the situation is a bit different (fig. . a). After the qpt at κ = . all the levels of both bands increase in energy, and at the tpt at κ = . the edge states that would normally have appeared get absorbed by the lower band immediately. So the effect of the tpt is only that the lowest level from the upper band switches to the lower one when the gap closes. ‘Edge states’ in the sense of strong localisation at the edges do not form in this regime. Another interesting new feature that appears in the spectrum is the separation of a couple of the lowest-lying states from the lower band in the broken-symmetry qp. This effect can not be predicted from the analysis of the bulk Hamiltonian, as it has no connection to topological invariants. It appears only in the broken-symmetry phases irrelevant to the topological phase (cf. figs. . a and . b). One of the possible explanations for appearance of these sub-band states may be connected with the boundary effects, i.e., deviations of the mean-field ground-state displacements α from the uniform bulk case close to the edges of the network. The edges in this case can be treated as effective impurities resulting in localised eigenstates [ ]. 4.3.2
Magnetisation
Up to now, the obtained results have been somewhat abstract, so now we would like to turn to some experimentally measurable quantities closely related to the edge states of our system. As the latter consists of coupled angular ⟨ z ⟩ momenta, one of the natural choices for such an observable would be Jls . But, because the translational unit of the system contains two ⟨ sites, ⟩it is more promising to choose the unit-cell magnetisation Ml = JzlA + JzlB . To calculate it, we can use the approaches described in the earlier sections of this thesis. First we use hpt to map the angular momenta onto bosonic operators Jzls = b†ls bls − j,
then we split bls up into their mean-field expectation values αls and quantum fluctuations dls operators, so ⟨ √ ⟩ √ Ml = (dlA + αlA j)(d†lA + α∗lA j) ⟨ √ √ ⟩ + (dlB + αlB j)(d†lB + α∗lB j) − j = (α∗lA αlA + α∗lB αlB − )j
+ (α∗lA ⟨dlA ⟩ + α∗lB ⟨dlB ⟩ + h.c.)
+ ⟨d†lA dlA ⟩ + ⟨d†lB dlB ⟩ ( ) ( )√ ( ) = Ml j + Ml j + Ml .
√ j ( . )
As we can see, the magnetisation consists of contributions of three different orders in j. The O(j / ) terms actually do not contribute to the mag(†) netisation, as the expectation values ⟨dls ⟩ = due to the displacement performed earlier. The O(j) terms form the macroscopic mean-field magnetisation, which depends on α only, so it remains constant for any reasonably low number of normal-mode excitations in the system. Its dependence on the position in the chain is shown in fig. . for two sets of parameters: one from the topologically trivial and another from the topologically non-trivial regions of broken-symmetry quantum phase II. The whole chain clearly splits into two parts: edges (the first two unit cells counting from the ends) and the bulk. The transition from the edge to the bulk is more abrupt in the non-trivial tp, while in the trivial tp it is more gradual. Speaking of the symmetric qp, the mean-field magnetisation in it is constant and can not be split up into the bulk and the edges, because α = , and so ( )
Ml
Bogoliubov transformation between the old bosons d and the is ( new ) bosons ( β) d β =T † β d†
= (α∗lA αlA + α∗lB αlB − ) = − .
Arguably the most interesting part of ( . ) are the O(j ) terms, which form microscopic fluctuations on top of the mean-field magnetisation. It is in the terms of this order in which one can see the presence and the localisation of the edge states. In order to calculate M( ) , we first have to select a suitable state to take the expectation values in. Here again, the Bogoliubov transformations come in handy, as a reasonable choice of such a state would be |λ⟩ = | · · · λ · · · L ⟩ with a single excitation in the λ-th Bogoliubov mode. With that we can analyse the differences in the magnetisation of different edge and bulk states. If earlier we were mostly interested in the eigenenergies E of the eigenvalue problem ( . ), now we shall need the eigenvectors as well. Because of the block form of the Hamiltonian, the matrix of eigenvectors T also has a block form ) ( U V∗ , T= V U∗
)
Mean-field magnetisation M(
−.
−.
− .8
− Unit cell index l
Fig. 4.6: Mean-field magnetisation in the broken-symmetry phases as a function of position in the network. Parameters ω = , γ = . , κ = . and (•) κ = . for the non-trivial tp and (•) κ = . for the trivial tp.
and thus the expectation value of d†p dp is ⟨λ|d†p dp |λ⟩
= ⟨λ| = ⟨λ| = ⟨λ| = ⟨λ|
L ( ∑
n,m=
L ( ∑
n,m=
L ( ∑
n= L [ ∑ n=
Vpn βn +
U∗pn β†n
)(
Upm βm +
∗ † Vpm βm
)
|λ⟩
) Vpn V∗pm βn β†m + U∗pn Upm β†n βm |λ⟩
) |Vpn | βn β†n + |Upn | β†n βn |λ⟩
(
|Vpn | + |Upn |
= |Vpλ | + |Upλ | +
L ∑ n=
)
] β†n βn + |Vpn | |λ⟩
|Vpn | ,
where < p ⩽ L is the sequence number of the site in the network. It should be noted here that M( ) still implicitly depends on the mean-field displacements α through the Bogoliubov transformation matrix T or, equivalently, on the Bogoliubov eigenvectors. Besides, it is always non-negative. Let us now look at the microscopic magnetisation in the non-trivial broken-symmetry phase II′′ , as the most feature-rich case. For analysis we have chosen two different κ values, . and . , while keeping the other parameters ω = , γ = . , κ = . constant. As M( ) is dependent not only on the position in the chain l, but also on the chosen Bogoliubov state λ, for each parameter set we have considered three different λ sets: the subband edge states λ = ( , ), the in-band bulk states λ = ( , ) and interband edge states λ = ( , ). The results are shown in figs. . c–h. Both
pairs of edge states are almost degenerate for considered parameter values, and the individual states within these pairs, due to the symmetry reasons, can be chosen in such a way that the one is localised on the one end of the chain, whereas the other one on the other. To make plots more clear, we plot the sum of M( ) for these pairs of states and apply this pairwise approach to the bulk states as well. First of all, the microscopic magnetisation within the bands (figs. . d and g) is delocalised: it is rather low and almost independent of the position in the chain. The M( ) in the topology-unrelated sub-band edge states is substantially localised on the edges with the localisation length about three unit cells, and no magnetisation in the middle of the chain. The topology-related inter-band states, as was discussed earlier, behave differently depending on the chosen κ . When these states are energetically well-separated from the bands (fig. . f), they show a sharply localised intensive magnetisation at the edges with the localisation length of about two unit cells. On the other hand, when these edge states get absorbed in the lower band (fig. . c), the intensity and localisation is almost totally lost. 4.4
conclusions and outlook
In this part of the thesis we have discussed the interplay of quantum and topological phase transitions in chains of Lipkin–Meshkov–Glick models, coupled with alternating interaction strengths. These chains possess the features of both underlying models: semiclassical bifurcations of the ground state, inherited from the lmg model, coexist with the different topological phases, inherited from the ssh model. We analysed finite and translationally invariant infinite networks in quadratic approximation using hp bosonisation techniques. The latter allowed us to split the Hamiltonian into terms of different orders of magnitude in the length of an angular momentum j, which were later dealt with separately. We then used the semiclassical mean-field terms of O(j) to calculate ground-state properties, starting from the energy and concluding with the macroscopic magnetisation, whereas quantum terms of O(j ) were used in the calculations of the excitation spectra and microscopic fluctuations of magnetisation. In comparison with the previous part of the thesis, we somewhat generalised the derivation to include the possibility of further extension of the research to systems far from the ground state. Mean-field ground-state calculations showed that depending on the values of Hamiltonian parameters γ, κ and κ , the system can be found in one of the five quantum phases: one symmetric and four broken-symmetry phases with non-degenerate and doubly-degenerate ground states, respectively. Devised approach allowed us to restrict the further calculations to just the regions of positive κ , or two quantum phases and easily obtain the
properties for other phases from that. The calculations on the bulk of the chain, or, in other words, on the infinite translationally invariant chains, have resulted in the two-band structure, in which the two gaps – one between the bands and the other between the lower band and the zero of energy – can close for different parameters marking either a qpt or a tpt. From expressions for the band energies, we obtained the analytical condition for the topological phase boundary. We also used the bulk Hamiltonian to establish a topological invariant – symplectic polarisation in this case – which keeps its value for any Hamiltonian parameters from a single topological phase and relates to the number of edge states in the corresponding finite chain. Based on this topological invariant, we constructed the final phase diagram, containing all the combinations of topological and quantum phases. All the calculations in this section were heavily based on the Fourier and Bogoliubov transformations. For the calculations on finite chains we worked in the quadratic approximation only, as direct diagonalisation of the Hamilton proved to be too resource-consuming. The ground state was optimised by iterative optimisation algorithms starting from the bulk ground state in order to account for the edges of the chain. We then obtained excitation spectra by Bogoliubov transformations of the Hamiltonian for different sets of κ and κ . The two types of edge states were observed in these spectra: ‘regular’, topology-related states were predicted by the symplectic polarisation of the bulk, while ‘irregular’, topology-unrelated states appeared only in the brokensymmetry quantum phase. An interplay between topological and quantum properties mainly resulted in the absorption of the topological edge states by the lower band of excitations in the broken-symmetry phases. As a result, these absorbed states were shown to lose their inherent localisation. In the last section of this part, we dealt with the observable that can potentially be measured experimentally to prove or disprove the existence of the edge states. For the role of such an observable, we have chosen the unit-cell magnetisation. Like the Hamiltonian, the magnetisation also consists of terms of different orders. The O(j) terms constitute the semiclassical mean-field magnetisation, which is closely related to the mean-field displacements of the ground state and do not depend on the excitations. The terms of O(j ) form the quantum fluctuations on top of the meanfield magnetisation and are closely related to the eigenstates of the system. The magnetisation of the bulk states from within the bands proved to have low intensity and no localisation, whereas the magnetisation of energetically isolated edge states showed strong localisation at the ends of the chain. The magnetisation in topology-related edge states that got absorbed by the lower band is qualitatively the same as the bulk magnetisation, that is to say, these states totally loose their localisation for corresponding parameter regimes.
The outlooks for potential experimental realisation are the same as for the previous model, as the Hamiltonians are different only in the coupling strengths. The work can be further extended to more complex topological networks, such as two-dimensional graphene lattice, and to the dynamical cases when the system is initialised far away from the ground state. The framework described here could be used for these purposes without any substantial alterations.
SUMMARY In this work we have explored the properties of the ground state and the low-lying excited states of networks of quantum-critical systems. A central role was allotted to the Lipkin–Meshkov–Glick (lmg) model: a set of N twolevel systems that mutually interact with some controlled strength γ in the external field of intensity ω . This model is semi-classical in a sense that N ≫ is usually considered, and has a second order continuous quantum phase transition at the critical γ value. The main objective of this thesis was to determine, what happens to the phase diagram and properties of low-energy states when several (possibly, infinitely many) lmg models are coupled together to form a network. Two main cases were considered, namely, when the couplings between the nodes are all equal and when two different couplings are used alternatively for bonds that share a node. We started with a more particular former case and extended it afterwards to a more general latter one. The description of networks started with the analysis of the Hamiltonian symmetries and of limit cases, when one of the control parameters considerably dominates the Hamiltonian of the system. This allowed us to determine three distinct phases – one disordered and two ordered ones (for positive parameters), – which have qualitatively different ground states, mainly in the sense of degeneracy and correlations. The analysis continued for parameters of the same order by mapping angular momenta onto bosonic operators and making a quadratic approximation, thus restricting the problem to the lower part of the energy spectrum. The two parts of transformed Hamiltonian of different orders in j were then considered separately starting from the mean-field O(j) contributions. g The ground-state mean-field energy EMF for periodic and infinite 1d networks was determined analytically using the ansatz of uniform meanfield displacements, whereas for more complex finite-size networks this ansatz served as an initial input to the iterative minimisation algorithm. g Having the expressions for EMF , we have determined the phase boundaries and obtained the phase diagram for the system, which depends on the geometry of the network. For smaller networks analytical and iterative numerical calculations were backed-up by direct diagonalisation of respective Hamiltonians. The approach was then extended to the negative coupling strengths to account for anti-ferromagnetic systems. This was done using the gauge transformation of bosonic operators and led to description of one more
broken-symmetry phase with alternating states of neighbouring nodes. The O(j ) terms of the Hamiltonian govern the microscopic fluctuations around the mean-field states, and in this work they were used for analysis of the lower part of the excitation spectrum in the 1d chains with translational invariance. We found analytic expressions for the energy gap Δ between the ground and the first excited states and showed that it closes differently at different phase boundaries. Another point of interest was the question of correlation functions in the infinite chains, which, much like the Hamiltonian, were shown to consist of terms of different orders in j. Mean-field-level correlations are closely related to the mutual orientation of semi-classical angular momenta of the ground state, thus two phases had positive correlations, one negative and one showed no correlations on this level. Microscopic correlations were shown to have oscillating form, around the background, which is zero or non-zero depending on the phase and components. In the next part of the work we extended the model to account for alternating couplings between the sites, focusing on finite and infinite 1d chains. The bosonisation approach was used again, but this time the derivation was generalised to account for the dynamics of the system, not only its stationary points, although non-stationary trajectories lye beyond the scope of this thesis. After applying quadratic approximation, the Hamiltonian split again into O(j) and O(j ) terms, for which the coefficients were found in the closed form. As before, the gauge transformations were introduced to account for the negative coupling values. The O(j) terms form the mean-field energy, based on which we found the critical points of the system, constructed the phase diagram and identified one symmetric and four brokensymmetry phases. For the analysis of the O(j ) terms we used Fourier and Bogoliubov transformations so that the Hamiltonian matrix could be diagonalised correctly. In the result we got the two-band excitation energy dependence on the wavenumber, which showed that the excitation energy is zero at the quantum phase boundaries, whereas the closing of the gap between the two bands marks another type of phase transition – a topological pt. With that we performed the calculations of topological invariants and classified the topological phases, thus obtaining the total phase diagram for the system. Diagonalisation of the O(j ) Hamiltonian in the direct space allowed us to obtain the spectrum of excitations. The two types of edge states were located and described, and the absorption of the in-gap topological edge states by the lower band was observed. In order to make a bridge to the possible experimental realisations, the mean-field and microscopic magnetisation in the chain was calculated, and the effects of the edge and bulk states were observed therein.
acknowledgements Every extensive work, even if written by a single author, builds up on inputs of many people, and this thesis is no exception. First of all, I would like to thank my advisor, Prof. Tobias Brandes for his valuable directions, sugestions and ideas, as well as his tact and understanding. Next, I would like to thank Dr. Victor Manuel Bastidas Valencia for his efforts of introducing me to the field, for his patience and for always being open for discussions. Many thanks go to my immediate collaborators on the projects, Martin Aparicio Alcalde and Georg Engelhardt (and Victor, too, of course), for fruitful work and many hours of discussions. A special thank you goes to Wassilij Kopylov for spending a certain time of his vacation reading the thesis through and making valuable comments on the comprehensibility, as well as for proofreading the German abstract (if some problems are still present, only the author is to blame, of course). I would also like to thank Nadežda and Pëtr Žgun for their moral support and comments on the thesis from an outside perspective, as well as all the members of the group for interesting and educational seminars and for a friendly atmosphere during the last three years. Last but not least, I am grateful to German Academic Exchange Service (DAAD) for financial support and high degree of genuine interest in the course and results of this doctoral project.
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APPENDICES a
holstein–primakoff expansion
Holstein–Primakoff transformations (hpt) can be used to describe a long angular momentum or a system thereof in terms of bosonic creation and annihilation operators. Although these transformations are exact only if j = ∞, they give a good approximation when J is sufficiently long. Moreover, only a couple of terms in the series expansion is needed if we are interested in the small neighbourhood around the fixed points of the classical energy surface, which substantially simplifies analysis. √ First, introducing displaced bosonic operator d = b − α j with α ∈ C in hpt
we get expressions
Jz = b† b − j, ( ) J+ = b† j − b† b , ( ) J− = j − b† b b,
√ Jz = d† d + (α∗ d + αd† ) j + (α∗ α − )j, ] [ √ J+ = b† j( − α∗ α)j − (α∗ d + αd† ) j − d† d [ √] √ √ d† d + (α∗ d + αd† ) j † ∗ = jΩ(d + α j) − jΩ √ √ = jΩ(d† + α∗ j)( − x) , √ √ J− = jΩ( − x) (d + α j),
x≡
d† d+(α∗ d+αd† ) jΩ
√
where Ω = − α∗ α. As we are working in the long angular momentum limit, x is a small quantity, so we can perform Taylor series expansion of square roots around x = ( − x) = − x − x + O(x )
(A. )
and substitute the expanded expressions into the Hamiltonian. Being interested in the lowest excitations of the system only, we can keep things simple by making the harmonic approximation, i.e. keeping only the terms of O(d ), O(d ) and O(d ) in the Hamiltonian. As an example, let us consider a single lmg model. Its Hamiltonian is
H = ω Jz − γj Jx Jx
j
at most quadratic in Jα operators, so three terms of the expansion (A. ) are sufficient to account for all the terms in the Hamiltonian, up to O(d )in other words to make a quadratic approximation. Substituting, expanding and neglecting the terms of O(d ) and higher, we get [ ] γ H = ω (α∗ α − ) − ( − α∗ α)(α + α∗ ) j ) ] ( [ √ γ ∗ ∗ ∗ ∗ ∗ d + h.c. α+α +α α− α α − α + ω α d− j ] [ α∗ α γ ∗ − (α + α ) + + − α∗ α [ ] γ + ( α∗ + α∗ α − )d + h.c. ] [ γ γ ∗ ∗ (α + α ) − ( − α α) + ω d† d. +
Thus the mean-field energy, linear and quadratic parts of the Hamiltonian ( . ) are obtained. Let us consider another example: a network of lmg models given by ( . ). The O(j ) and O(j / ) terms are a little too long to write them out, but let us focus on the O(j) terms. Keeping only these, allows us to neglect everything containing bosonic operators in the hpt, simplifying the latter to Jzl = j(α∗l αl − ), √ ∗ J+ = jα − α∗l αl , l l √ − α∗l αl . J− = jα l l
y
The Jxl and Jl operators can be expressed through the ladder operators J± l as Jxl = y
Jl =
− J+ l + Jl − J+ l − Jl
ⅈ
√ j = (α∗l + αl ) − α∗l αl , =
j
ⅈ
(α∗l − αl )
√
− α∗l αl ,
and substituting into the Hamiltonian, we obtain
H j
=ω
L ∑ l=
+
L
(α∗l αl − ) −
L ∑
l′ >l=
= EMF
γ∑ l=
(α∗l + αl ) ( − α∗l αl )
κll′ (α∗l − αl )(α∗l′ − αl′ )
√
− α∗l αl
√
− α∗l′ αl′
b
localisation of edge states
In the topologically non-trivial phase of ssh model a pair of states appears in the band gap (see § . ). Their energy is close to zero if the chain is finite and open and becomes exactly zero if the system is translationally invariant, i.e. either in the thermodynamic limit L → ∞ or when periodic boundary conditions apply. In order to estimate the characteristic localisation length of an edge state L ∑ |ψ⟩ = (al |l, A⟩ + bl |l, B⟩), l=
let us assume that the chain consists only of the bulk. The action of the Hamiltonian
H=κ
L ∑
(|l, A⟩⟨l, B| + h.c.) + κ
L ∑ l=
l=
on this state H |ψ⟩ =
(|l − , B⟩⟨l, A| + h.c.)
results in recurrent equations for coefficients κ al + κ al+ = , κ bl + κ bl− = ,
if we neglect the boundary. Therefore, al = aL
(
κ − κ
)L−l
and bl = b
(
κ − κ
)l−
.
After simple algebraic transformations, these expressions result in ] [ κ |al | = |a | exp (L − l) ln , κ ] [ κ |bl | = |b | exp (l − ) ln . κ
Thus the absolute values of coefficients in the basis expansion of an edge state decay exponentially moving into the bulk of the chain, and the localisation length is ν=
c
ln |κ /κ |
.
winding number
By definition the winding number is the number of full revolutions of vector c(k) around the origin as k traverses the Brillouin zone (see § . ). This can
be written as w =
π
∫
k∈IBZ dφ(k)
or alternatively ∫ dφ w= dk , π IBZ dk
where φ is the polar angle of the vector c. Let us consider two vectors c(k) and c(k + dk) = c(k) + dc(k) of infinitesimally close arguments. Then . . [c, c + dc]z = c sin dφ = c dφ , [c, c + dc]z = [c, dc]z , which allows us to express the derivative [ ] dφ dc c, = dk c dk z and the winding number w=
π
∫
π
c
[
dc c, dk
]
dk z
As was shown in § . , for the ssh model c = (κ + κ cos k, κ sin k, ), dc = (−κ sin k, κ cos k, ) dk and combining everything together we arrive at the winding number ∫ π κ + κ κ cos k w= dk . π κ + κ + κ κ cos k d
conjugate gradient method
The conjugate gradient (cg) method was originally developed to iteratively solve large systems of linear equations, but in was adapted for nonlinear optimisation problems [ ]. Faster convergence and robustness are its main advantages over more straightforward methods like steepest descent. The idea of the non-linear cg method is as follows. Suppose we have to minimise a scalar function f (x) of a vector argument x = (x( ) , x( ) , . . . , x(N) ). The algorithm starts with some initial guess x , at which the function has the value f = f (x ). The first iteration is made in the direction of negative gradient p = −∇f (x ), so the next approximation x = x + α p . The scalar α may be found using the linear search to minimise f (x ). Up to this point the cg method coincides with the steepest descent method.
Next, instead of choosing p in the direction of −∇f (x ), we correct it by mixing in the previous direction p so that p = −∇f (x ) + β p . This allows us to evade a possible scenario of going back and forth along the same directions for a long time due to the mutual orthogonality of the gradients in two consecutive iterations. The value of β may be chosen in different ways, but in this work we stick with the modified Polak–Ribière approach when
βk = max
{
∇f (xk ) [∇f (xk ) − ∇f (xk− )] , ∥∇f (xk− )∥
}
.
Having x and p we can continue with further iterations until the algorithm converges. For each step k only the gradients ∇f (xk ) and ∇f (xk− ) are needed, so we do not have to keep the whole history in the memory. The pseudo-code for the algorithm looks like this [ ]: Input: x p ← −∇f (x ) k← while |∇f (xk )| > ε: αk ← arg minα f (xk + αpk ) xk+ ← xk + αk pk βk ← max {∇f (xk ) [∇f (xk ) − ∇f (xk− )] /∥∇f (xk− )∥ , } pk+ ← −∇f (xk+ ) + βk+ pk k←k+ end e
bogoliubov transformations
Suppose we have a system described by the Hamiltonian H (d) in the second quantisation. The operators d , d , . . . dL constituting the vector d obey bosonic commutation relations [di , d†j ] = δij and [di , dj ] = . Also suppose that H (d) is quadratic in d, that is to say
H (d) =
L ( ∑ i,j=
()
( )
cij d†i dj + cij di dj + h.c.
)
and possibly contains a free term c( ) that will have no impact whatsoever on the further derivations. This expression can be written more concisely
using matrix–vector products
(
H (d) = d† · · · d†L d
d .. . ( ) ) dL ( † ) d · · · dL H † = d d H † . d d .. . d†L
()
The hermiticity of the Hamiltonian implies that cii are real and without the loss of generality we can choose cijα = cjiα . Then the matrix H has a block structure () ( )∗ ( ) ( ) c ··· c L c ··· c L . .. .. .. .. .. .. . . . . . ( ) () () ( ) ( ) ··· cLL cL · · · cLL A B cL H = ( )∗ ( )∗ () ( )∗ = B∗ A , c ··· c L c ··· c L .. .. .. .. .. .. . . . . . . ( )∗ ( )∗ () () cL · · · cLL c L · · · cLL
where the matrix A is self-conjugate and the matrix B is symmetric. The main objective of Bogoliubov approach is to find a transformation T that would diagonalise the matrix H in a way that would preserve canonical commutation relations for the new set of operators. That said, the transformation matrix should have a block form [ ] )( ) ( ) ( ( ) d β β U V∗ = T = V U∗ β† β† d† Substituting this expression into commutation relations, we get that U U† − V∗ VT = UT V − VT U =
and for the matrix H to be diagonalised, ) )( ) ( ( UE U A B = −V E V B∗ A
with diagonal matrix E has to be satisfied. This leads to the eigenvalue problem ) ( E , SHT = T −E
where S = σz ⊗ L is an expanded Pauli z matrix, solving which we can obtain both the energies of normal modes and the expressions for the new Bogoliubov operators in terms of original ones.
