Saha Institute of Nuclear Physics, India between August 2009 and June 2013, ... also like to thank Dr. Jonathan Keeling and Justyna Cwik, with whom I have ... Finally we consider how the strong local exciton-phonon (ex-ph) coupling modifies ...
Strong electron-phonon interactions in some strongly correlated systems
Sahinur Reja Department of Physics University of Cambridge
A thesis submitted for the degree of Doctor of Philosophy August 2013
To my loving family.
Preface This thesis describes work undertaken at the Cavendish Laboratory and at Saha Institute of Nuclear Physics, India between August 2009 and June 2013, under the supervision of Prof. Peter Littlewood and Prof. Sudhakar Yarlagadda. It does not exceed 60,000 words. Chapter 1 and 2 summarises previous developments in the field, including the relevant background for the work in this thesis. Subsequent chapters present original research. The research presented is my own work and includes nothing which is the outcome of work done in collaboration or which has been submitted for a previous degree except where specified in the text and Acknowledgements. Parts of this thesis have been published or submitted for publication as follows: Chapter 3. Sahinur Reja, Sudhakar Yarlagadda, and Peter B. Littlewood. Phase diagram of the one-dimensional Hubbard-Holstein model at quarter filling. Phys. Rev. B 84, 085127 (2011). Chapter 4. Sahinur Reja, Sudhakar Yarlagadda, and Peter B. Littlewood. Correlated singlet phase in the one-dimensional Hubbard-Holstein model. Phys. Rev. B 86, 045110 (2012). Chapter 5. Justyna A. Cwik, Sahinur Reja, Peter B. Littlewood, and Jonathan Keeling. Organic Polaritons —strong Exciton-Phonon-Photon coupling. Submitted for publication; preprint: arXiv:1303.3720, 2013
Acknowledgements Being a Cavendish-Saha Research Fellow, the first joint collaborative PhD student of Cavendish Laboratory and Saha Institute of Nuclear Physics (SINP), India, my PhD was supported financially by Cambridge Commonwealth Trust and Centre for Applied Mathematics and Computational Science (CAMCS), SINP. My travel expenses to conferences and schools were provided by TCM group, Cavendish Lab and CAMCS, SINP. I am very grateful to have had the opportunity to be a part of this collaboration. I feel very lucky to have been supervised by Professor Peter Littlewood at Cavendish Lab and Professor Sudhakar Yarlagadda at SINP. I thank them for their patient support and guidance, for offering a great many physical insights, and for introducing me to a great many interesting research problems. I would also like to thank Dr. Jonathan Keeling and Justyna Cwik, with whom I have collaborated more recently. My time in TCM, Cavendish Lab has been a very happy one, largely thanks to the pleasant people I have met. Particular mention is due to Richard Brierley, Ahsan Zeb, Gareth Conduit, Cele Creatore, Sitikantha Das. I learnt a lot from lectures and private discussions with David Khmelnitskii and Ben Simons. Like everyone in TCM, I am indebted to Helen Verrechia, Michael Rutter and, latterly, David Taylor for keeping things together. I also appreciate the stimulating research environment at SINP, Kolkata. There I have enjoyed the company of and discussions with Muzaffar, Debasis Samanta, Ravindra Pankaj, Amit Dey, Sourish, Najmul Haque and many others. I would like to thank Mr. Kausik Das who has helped me in computational facilities. Finally, none of this would have been possible without the support of my family, especially my father and my younger brothers, Sohel and Hasanur.
Summary The interplay of electron-electron (e-e) and electron-phonon (e-ph) interactions in correlated electron systems leads to coexistence of or competition between various phases such as superconductivity and charge order, and also modifies the dynamical response of excitations such as polarons and excitons. In this thesis we consider two types of systems that have a common ingredient of strong coupling to localized vibrational modes. One is the Hubbard-Holstein model as a minimal model to discuss competing ground states in condensed matter systems such as transition metal oxides, and organic conductors. The second is a model for localized (Frenkel) excitons which naturally have both strong exciton-phonon coupling and strong coupling to photons, where we are interested in the potential effects of e-ph coupling on the Bose-Einstein condensation of exciton-polaritons. This model is relevant for some organic materials (such as cyanine J-aggregates) introduced into optical microcavities. We first address the one-dimensional Hubbard-Holstein (HH) model in both the strong e-e and the strong e-ph interactions limit. We utilise a controlled analytic approach (that takes into account dynamical quantum phonons) to derive an effective electronic hamiltonian after averaging out the vibrational degrees of freedom using second order perturbation theory. We then solve this model numerically for finite chains and obtain a ground state phase diagram at various fillings. We find that the strong e-ph coupling stabilises a correlated nearestneighbour (NN) singlet phase in these fillings. This phase is quite distinct from a Peierls-like ( bond-order) wave and would be absent in the pure Hubbard model. We then analyse this correlated NN singlet phase by mapping the effective electronic hamiltonian onto the well-understood one-dimensional t-V model with large repulsion. This also gives a distinct advantage to access bigger system sizes which would not be possible with effective electronic hamiltonian; furthermore, it also helps identify the numerically elusive KT transition. Because the physics is dictated by the t-V model, we find that CDW and superfluidity occur mutually exclusively with CDW resulting only at one-third filling while superfluidity man-
ifests itself at all other fillings. So CDW and superconductivity are incompatible in one-dimensional HH model. In a technical advance, we present a modification of the world-line quantum monte carlo (WQMC) method, which helps us to perform the calculations for larger sized systems than modified Lanczos algorithm, in particular to accurately determine the BEC fraction. Finally we consider how the strong local exciton-phonon (ex-ph) coupling modifies the collective behaviour of exciton-polaritons and the phase transition to superradiant phase at finite densities in Dicke-Holstein model, using a meanfield theory within path-integral approach which treats the photon and phonon fields as coherent states. We find a re-entrant behavior producing a Mott lobe, even if the detuning is less than the critical detuning. We show that the strong ex-ph coupling can drive the normal state to superradiant state transition to be first order as we vary exciton-photon coupling. We show that our results match onsite exact diagonalisation at large ex-ph coupling as expected. Our theory predicts first order transitions with decreasing magnitude of first order character as the exciton-phonon coupling is reduced. Only at zero ex-ph coupling is the phase boundary everywhere second order.
Contents Contents
vi
List of Figures
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1 Introduction 1.1 Motivation and Overview . . . . . . . . . . . . . . . 1.2 Strongly correlated systems . . . . . . . . . . . . . 1.3 Electron-phonon coupling . . . . . . . . . . . . . . 1.3.1 Charge-density wave: Peierls transition . . . 1.3.2 BCS superconductivity . . . . . . . . . . . . 1.3.3 High Tc superconductors: Phase diagram . . 1.4 Exciton-photon-phonon coupling and condensation 1.4.1 Quantum coherences and condensation . . . 1.4.2 Excitons . . . . . . . . . . . . . . . . . . . . 1.4.3 Microcavity polariton . . . . . . . . . . . . . 1.4.4 Polariton condensation . . . . . . . . . . . . 1.4.5 Organic polariton . . . . . . . . . . . . . . .
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1 1 2 5 6 8 9 12 12 15 16 18 21
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22 22 23 24 27 27 29
2 The Models 2.1 Hubbard model . . . . . . . . . . . . . . . . . . . 2.1.1 Mott Transition . . . . . . . . . . . . . . . 2.1.2 t-J Transformation . . . . . . . . . . . . . 2.2 Holstein model . . . . . . . . . . . . . . . . . . . 2.2.1 The interaction hamiltonian and the model 2.2.2 Polarons by self-trapping . . . . . . . . . .
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CONTENTS
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Hubbard-Holstein model . . . . . . . . . . . . . . . . . . . . . . . Dicke model and phonon . . . . . . . . . . . . . . . . . . . . . . .
31 33
3 Phase diagram of one-dimensional strong coupling Hubbard-Holstein model 37 3.1 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . 40 3.1.2 Effective Electronic Hamiltonian . . . . . . . . . . . . . . . 41 3.2 Region of validity of our theory . . . . . . . . . . . . . . . . . . . 46 3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.1 AF Cluster to Correlated Singlets Transition . . . . . . . . 48 3.3.2 Transition from correlated singlets to spins in one sublattice 52 3.3.2.1 Transitions at smaller g . . . . . . . . . . . . . . 53 3.3.2.2 Transition for larger g values . . . . . . . . . . . 54 3.4 The Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4 Analysis of the correlated singlet phase of the Hubbard-Holstein model 4.1 t-V1 -V2 hard-core-boson (HCB) model . . . . . . 4.2 CDW correlations . . . . . . . . . . . . . . . . . 4.3 Superfluid density . . . . . . . . . . . . . . . . . 4.4 BEC occupation number . . . . . . . . . . . . . 4.5 WQMC for BEC fraction . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . .
one-dimensional . . . . . .
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5 Organic polaritons: First order normal to superradiant tion 5.1 Part I: Analysis with Direct Hamiltonian . . . . . . . . . . 5.1.1 Coherent state Path integral formulation . . . . . . 5.1.2 Saddle point analysis . . . . . . . . . . . . . . . . . 5.1.3 Results and Discussions . . . . . . . . . . . . . . . 5.1.4 Zero temperature case . . . . . . . . . . . . . . . . 5.1.4.1 Condensate state λ > 0 . . . . . . . . . .
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57 58 60 63 66 68 71
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82 84 84 86 86 87 87 88 94
6 Conclusions 6.1 Hubbard-Holstein model in one-dimension . . . . . . . . . . . . . 6.2 Dicke-Holstein model . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95 95 97 98
5.2
5.3
5.1.5 Finite temperature case . . . . . . . . . . . . . . . . Part II: Analysis with Lang-Firsov transformed hamiltonian 5.2.1 Variational Lang-Firsov transformation . . . . . . . . 5.2.2 Zero temperature case . . . . . . . . . . . . . . . . . 5.2.2.1 Condensate state, λ > 0 . . . . . . . . . . . 5.2.3 Finite temperature case . . . . . . . . . . . . . . . . 5.2.3.1 Phase diagram at λ → 0 . . . . . . . . . . . 5.2.3.2 General solutions with λ 6= 0 . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Appendix A The formulation of perturbative correction in electronic operator form
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Appendix B Projection onto singly occupied subspace
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Appendix C Definition of n-particle normalized clustering probability [NCP(n)]
References
105 107
viii
List of Figures 1.1 1.2 1.3 1.4 1.5
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
4.1 4.2 4.3
Schematic of the behaviour of the dispersion curve due to Peierls transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The electron-electron attractive interaction mediated by phonon − the BCS mechanism. . . . . . . . . . . . . . . . . . . . . . . . The schematic of temperature-doping phase diagram of superconducting cuprates. . . . . . . . . . . . . . . . . . . . . . . . . . . . The structure of a microcavity and the polariton dispersion . . . . Sketch of optical parametric amplification (OPA) process and experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different hopping processes contributing to second order perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic representation of the hopping processes . . . . . . . . . n-particle normalised clustering probability NCP(n) . . . . . . . . Plot of the correlation functions . . . . . . . . . . . . . . . . . . . Plot of correlation functions in the correlated singlet phase . . . . Correlation functions depicting AF order and paramagnetic phase in one sublattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . More correlation functions . . . . . . . . . . . . . . . . . . . . . . The phase diagram of the Hubbard-Holstein model at quarterfilling for a twelve-site system in the t/ω0 = 1.0 plane . . . . . . .
7 8 10 16 20
44 45 49 50 51 52 53 55
The phase diagram at various fillings and the structure factor plots 58 WQMC plot of the structure factor . . . . . . . . . . . . . . . . . 60 The correlation function and structure factor plot using WQMC . 61
ix
LIST OF FIGURES
4.4 4.5 4.6 4.7
The transformations of the model to equivalent representations used in our analysis . . . . . . . . . . . . . . . . . . . . . . . . . . Superfluid density for an infinite system . . . . . . . . . . . . . . Superfluid density decaying exponentially with system size for the CDW state at one-third filling and large NNN repulsion V2 . . . . . Plots of BEC occupation number n0 , obtained from modified Lanczos (open circles) and WQMC (crosses) . . . . . . . . . . . . . . .
This shows the variation of λ with ρ for different detuning ∆ with S = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 (a) This shows the variation of λ with excitation density ρ for ∆ = 1. (b) This shows the corresponding chemical potential µ − ωc 5.3 The normal state and superradiant energy solutions vs excitation density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 (a) This shows the variation of λ with excitation density ρ for ∆ = −2. (b) This shows the corresponding chemical potential µ − ωc 5.5 (a) This shows critical temperature Tc − ρ second order phase boundary for ∆ = 1. (b) This shows the corresponding Tc − µ phase boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The critical exciton-photon coupling separating normal and superradiant (SR) states at different temperatures by second order phase boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 The free energy plot at very low temperature . . . . . . . . . . . . 5.8 The typical free energy extrema solutions . . . . . . . . . . . . . . √ 5.9 Comparison between critical g N from exact on-site diagonalisation (left) and phonon mean-field path-integral approach . . . . . 5.10 The g − µ phase diagram for smaller exciton-phonon coupling S . 5.11 This shows a colour map for the photon field λ on T − µ plane for the parameters S = 2, g = 2, ∆ = 4 and Ω = 0.1. . . . . . . . . . .
62 64 65 66
5.1
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83
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Chapter 1 Introduction 1.1
Motivation and Overview
Strongly correlated systems are often accompanied by strong electron-phonon(eph) coupling along with the prominent electron-electron (e-e) interactions. The interplay of e-e and e-ph interactions in these correlated systems leads to coexistence of or competition between various phases such as superconductivity, chargedensity wave, spin-density wave. In another example, exciton-phonon coupling can modify the collective behavior of polaritons in an optical microcavity. In this thesis, we discuss several models regarding these physical systems that show such interplay, in particular systems of correlated electrons and exciton-polaritons. In the rest of this chapter, we present background interests on the physical systems, with a qualitative discussion of different interesting phenomena such as superconductivity and the Peierls transition in correlated electron systems and the Bose-Einstein condensation (BEC) of exciton-polaritons in optical microcavity. In chapter 2 we introduce the basic theoretical models which we will study in later chapters. In chapter 3 we consider the one-dimensional Hubbard-Holstein (HH) model in the limit of both strong e-e and strong e-ph coupling and present a phase diagram at quarter-filling where an interesting phase − the correlated nearest-neighbour (NN) singlet phase occurs. In chapter 4 we present a detailed analysis of this phase which occurs at other fillings too, and show that chargedensity wave and superconductivity are mutually exclusive in the one-dimensional
1
HH model. In chapter 5 we explain how the strong local exciton-phonon coupling modifies the phase transition to the superradiant phase at finite densities in the Dicke-Holstein model, using a mean-field theory within a path-integral approach. In chapter 6 we accumulate all the results obtained in our calculations presented in chapter 3, 4 and 5 along with the suggestions for possible future works. There are three appendices, discussing technical details. The first discusses our approach to carrying out perturbation theory and obtaining the ground state energy. The second describes the projection onto singly-occupied subspaces to calculate the effective electronic hamiltonian for one-dimensional HH model. And the third gives the definition of n-particle normalized clustering probability [N CP (n)] which has been used to study the transitions between the phases in the phase diagram of one-dimensional HH model.
1.2
Strongly correlated systems
One of the most challenging aspects in theoretical condensed matter and materials physics is to produce a systematic way of describing strongly correlated quantum many-body systems. These theories indeed would have very wide range of consequences in various fields like condensed matter physics (e.g. Hubbard model vs. high Tc superconductivity, light-matter systems like semiconductor quantum dots or more recently the organic materials in optical microcavity), and quantum chemistry. The understanding and the classifications of metals and insulators in terms of occupation of electronic bands was the first striking success of the quantum theory of condensed matter systems. The band theory (e.g. as embodied by density functional theory) is basically a single particle theory that assumes each electron moves in a periodic potential created by the lattice ions and other electrons, i.e. each electron responds to the average properties of the surrounding environment, neglecting any further correlations. The single particle eigenstates with their corresponding energies taking continuous values on some distinct energy intervals, constitute the energy bands. Though band theory has been successful in many respects, soon after its appearance de Boer and Verwey[1] reported many properties of transition-metal
2
oxides (the most famous example being NiO) directly in disagreement, even in a qualitative way, with the band structure calculations. These seemingly simple materials (e.g. NiO) with a partially filled 3d-band, would be expected to be a good conductor in band theory. But in fact these materials are insulators with large conductivity gap. The reason behind this insulating behavior was first conceptualised by Mott and Peierls[2]. They pointed out that the failure of band theory might be due to the neglect of strong Coulomb repulsion (a correlation effect) between d-electrons. Later on, Mott[3; 4; 5] and Hubbard[6] showed that the system would be indeed an insulator if the repulsion is larger than the bare electron bandwidth. In the last decades, an impressive number of novel materials with electronic structures that are neither simply free-electron-like nor completely ionic, but a mixture of both, were discovered. Examples of these materials that have rich variety of properties, include: Cuprate superconductors and iron based superconductors, intrinsically an-
tiferromagnetic Mott insulators due to interactions amongst electrons in localized 3d-shells but develop high temperature superconductivity when doped[7]. Quantum Dots, which are tiny pools of electrons in semiconductors that
act as artificial atoms and have discrete energy levels due to quantum confinement. As the gate voltage is changed, the Coulomb repulsion between electrons in the dot leads to the Coulomb Blockade, whereby electrons can be added one by one to the quantum dot. Also these quantum dots couple strongly to photon modes in an optical microcavity, forming new quasiparticles called polaritons which can form a condensed ground state like a Bose-Einstein condensate[8; 9]. Cold atomic gases, in which the interactions between the neutral atoms are
governed by a two-body resonance (Feshbach resonance), can be tuned by external magnetic fields to create a whole new world of strongly correlated quantum fluids[10]. Heavy Fermion compounds, in which localized magnetic moments immersed
3
within the metal give rise to electron quasi-particles with effective masses a thousand times bigger than that of a bare electron[11]. Fractional Quantum Hall systems, where the interactions between elec-
trons in the lowest Landau level of a two-dimensional electron fluid give rise to a incompressible state with quasi-particles of fractional charge and statistics[12]. The conventional treatment of the systems with interacting electrons was developed by Landau and is called Fermi-liquid theory which uses the notion of adiabaticity and the Pauli exclusion principle. If we turn on the interaction slowly in a non-interacting Fermi gas, according to Landau, its ground state would adiabatically transform into the ground state of the interacting system. Thus the theory assumes a one-to-one correspondence between the low energy excitations of a Fermi gas and a Fermi liquid system. These elementary excitations in Fermi liquid are called quasi-particles and form a well defined Fermi surface due to the Pauli exclusion principle which blocks the scattering inside the Fermi sea. This implies that the Fermi liquid properties can be described by the same expressions as in the Fermi gas, but with some normalised parameters such as effective mass, magnetic moment etc. But the experimental studies of many strongly correlated materials can not be understood within the framework of Fermi liquid theory. Beside the strong electronic correlation, many of these materials have the characteristics of low dimensionality (quasi-one and quasi-two dimensional) which is subjected to strong quantum fluctuations making the theoretical treatment even more difficult. In recent years, a large number of analytical approaches accompanied with more or less uncontrollable approximations have been applied to condensed matter systems. Beside those, different numerical algorithms have been developed to solve model Hamiltonians of various strongly correlated systems. These numerical techniques however can handle only small sized systems, making the calculated properties in the thermodynamic limit unreliable. So the complete theoretical understanding of different properties of strongly correlated electron systems remains challenging till today. And out of the variety of condensed matter systems, our interests include the systems like cuprates, manganites, and exciton-photon
4
systems in organic microcavity where lattice vibrations play a crucial role. These systems are indeed important to understand not only the fundamental physics but also for technological importance.
1.3
Electron-phonon coupling
The topic of electron-phonon (e-ph) coupling is very important in many-body theory along with the prominent electron-electron (e-e) interactions. In many materials the lattice vibrations can indeed mediate an attractive interaction between electrons to form Cooper pairs which then move with zero electrical resistance at low temperature, leading to the fascinating phenomenon called superconductivity. But the same e-ph interaction scatters the electrons and is responsible for electrical resistance at high temperature in metals. Also due to e-ph coupling, a one-dimensional metal is always unstable to the formation of a charge-density wave, accompanied by Peierls distortions. A wealth of strongly correlated materials show the evidences of strong e-ph interactions besides the ubiquitous e-e interactions. For instance: angle-resolved photoemission spectroscopy (ARPES) experiments in high temperature superconductors (cuprates and pnictides), manganites [13; 14; 15], molecular solids such as fullerides [16] and DFT calculations in the recently discovered material LaO0.5 F0.5 BiS2 [17] all indicate strong e-ph coupling. The interplay between superconductivity and charge-density wave is a subject of immense ongoing focus and is observed in various materials such as layered dichalcogenides (e.g., 2H-TaSe2 , 2H-TaS2 , and NbSe2 ) [18], recently in pristine 1T -TiSe2 under high-pressure[19], bismuthates (e.g., BaBiO3 doped with K) [20; 21], doped spin ladder cuprate Sr14 Cu24 O41 [22; 23], quarter-filled organic materials [24; 25], non-iron based pnictides (e.g., SrPt2 As2 ) [26] etc. In this section, we present a qualitative discussion about some important phenomena where e-ph interactions play a crucial role. These include topics such as the charge-density wave or Peierls transition, conventional superconductivity with a qualitative description of phonon mediated Cooper pair formation, and finally high temperature superconductors with their exotic properties summarised in a temperature-doping phase diagram.
5
1.3.1
Charge-density wave: Peierls transition
In many materials with a highly anisotropic band structure, the interaction between electrons and lattice distortions can lead to a novel type of ground state called the charge-density wave (CDW) characterised by a gap in single particle excitation spectrum and an electronic charge modulation accompanying the periodic lattice distortion (PLD)[27; 28]. Rudolf Peierls (1930)[29] was the first to point out that a one-dimensional metal is always unstable to the formation of a CDW, even if the e-ph coupling is weak. Though the occurrence of charge-density waves (CDWs) in materials with twoor three-dimensional band structure are not rare, they are predominantly onedimensional phenomena. As an concrete example, we consider a one-dimensional chain of atoms, with lattice constant a, and an electronic density which makes sure that the Fermi wave-vector kF falls within the band (see the schematic in Fig.1.1). In the absence of electron-electron or e-ph interactions, the ground state at zero temperature corresponds to a band with filled momentum states upto kF . It is a metal. Now if we introduce a periodic lattice distortion (PLD) with the periodicity Q = 2kF ; namely move the nth atom in the chain to a new position, Rn = na + u0 Cos(Qna)
(1.1)
Here we assume the amplitude of PLD u0 ≪ a. If the atoms have a PLD with period 2π/Q, they will produce a new potential V (r) seen by the electrons with same period which causes the formation of a band gap at the Fermi level. It is also evident that the amplitude of the Fourier component VQ ∝ u0 and we can write VQ ∝ gQ u0 where gQ is e-ph coupling constant. As the energy gap at kF is given by |VQ |, this means that an energy levels just below Fermi level are lowered by an energy proportional to the atomic displacement |u0 |, (and the unoccupied states just above the Fermi level is raised by the same amount). So, due to the PLD, there is an overall energy lowering calculated by adding the energy changes of all occupied states, Egain = A(u0 /a)2 ln(u0 /a)
6
(1.2)
Figure 1.1: Left panel: The lattice with one electron per site i.e. at half-filling, lattice constant a and uniform charge density ρ(r), has a dispersion with filled momentum states upto Fermi wave-vector kF = π/2a. Right panel: The periodic lattice distortion with periodicity Q = 2kF produces a gap at Fermi energy, hence a CDW transition. in the limit of u0 ≪ a and constant A depending on gQ . The PLD also introduces an elastic energy proportional to (u0 /a)2 (Rice and Str¨assler, 1973). So, the total energy adds up to the form, E(x) = Ax2 ln(x) + Bx2
(1.3)
which always has a minimum at a non-zero displacement. The system lowers its energy by lattice distortion, accompanying a CDW with periodicity determined by Fermi wave-vector, viz. 2π/2kF . So for an arbitrary band filling, the period of CDW may be incommensurate with the underlying lattice. The CDW ground state at low temperature, becomes a usual metallic state as temperature increases because the gain in electronic energy is reduced due to thermal excitation of electrons across the gap. The second order phase transition between CDW and metallic states is known as the Peierls transition. Generally, the materials consisting of weakly coupled molecular chain along which the electrons can move freely but restricted to perpendicular direction, are the good candidates to show the Peierls transition. Organic conductors, of which tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) is the prototype of one group of solids, have been widely studied[30]. The inorganic linear-chain
7
materials like NbSe3 and K0.3 MoO3 show the CDW transitions at relatively high temperature of 145K and 180K, respectively[31].
1.3.2
BCS superconductivity
The most striking feature of a superconductor is that its electrical resistivity drops abruptly to zero when the material is cooled below the transition temperature. This phenomenon was discovered by Kammerlingh Onnes in 1911, three years after he first liquefied helium. A superconductor also has the property, called the Meissner effect, to expel all magnetic flux from its interior below the transition temperature. Superconductivity involves an ordered state of conduction electrons in metal, caused by the residual attractive interaction at the Fermi surface. The nature and origin of the ordering in conventional metals, were explained in a seminal work by Bardeen, Cooper and Schrieffer − the BCS theory[32; 33] − some 50 years after its discovery! The essential ingredient of their theory is the interaction between freely moving electrons and the underlying structural basis of the solid i.e. the lattice of ions. A passing electron attracts the lattice ions, creating a slight ripple (quanta of lattice vibration i.e. phonon) towards its path. Because of the time scale of electron set by Fermi energy is much smaller than the lattice relaxation time set by Debye freSuper conducting lattice quency, another electron moving in opposite direction is attracted to that displacement. This leads to an attrac- Figure 1.2: The electron-electron attractive interaction between the two elec- tive interaction mediated by phonon − trons mediated by phonon and the for- the BCS mechanism. e
e
e
e
mation of a bound pair of electrons, called Cooper pair (schematic for the mech-
8
anism shown in Fig.1.2). BCS superconductivity can be viewed as a condensed phase of the Cooper pairs analogous to the Bose-Einstein condensation(BEC), but with important difference that the electrons in Cooper pairs have only weak attraction, and so the size of a Cooper pair can be many lattice spacings. BCS theory represents the special case where the temperatures for the formation of cooper pairs and their condensation coincide. To explore fermionic condensation, we consider the BCS hamiltonian[34]: H=
X k,σ
εk c†k,σ ck,σ −
g X † c c† c−k′ +q,↓ ck′ ,↑ 2 k,k′ ,q k+q,↑ −k,↓
(1.4)
which describes the electrons represented by fermionic operator ck with band dispersion εk and g is the attractive interaction strength. In the weak coupling limit, where no bound pair exists above the transition temperature, the mean-field theory with coherent fermionic wave function involving pairs of fermions: |ψBCS i =
Y uk + vk c†k,↑ c†−k,↓ |0i
(1.5)
k
produces the transition temperature accurately[35]. For three dimensional systems, it is given by,
1 Tc = ωD exp − gν
(1.6)
where ωD is the Debye frequency and ν is the electronic density of states at the Fermi energy. Also BCS theory of lattice coupling to electrons, was supported by Isotope effect where the transition temperature is inversely proportional to the square root of nuclear mass.
1.3.3
High Tc superconductors: Phase diagram
In 1986 Bednorz and M¨ uller[7] reported the discovery of the high temperature superconductivity (HTSC) in copper oxide material, with a surprisingly then high transition temperature at 35 K. Just few months later, the finding was dramatically confirmed in the same class of material, that became supercon-
9
ducting at 93 K[36]. Since then, a large number of similar materials (such as YBa2 Cu3 O7 , Bi2 Sr2 CaCu2 O8+x etc), called cuprates, were synthesised, having transition temperature above the boiling temperature of liquid nitrogen (77 K) and hence opening up the potential technological applications. These are twodimensional layered structure (perovskite) with superconductivity taking place in a copper oxide plane. Despite the intensive research, the physical mechanism for underlying pairing of electrons in superconducting cuprates is still far from being fully understood. Initially the very nature of d-wave pairing suggested the mechanism through spin-excitations [37; 38; 39], but nevertheless the recent experimental results provide evidences that the e-ph interaction plays an important role in these materials. Though some evidences have come from Raman spectroscopy and neutron scattering also, but many results from angle-resolved photoemission spectroscopy (ARPES) experiments [40; 41; 42; 43; 44; 45; 46] have suggested that several specific phonon modes may be of relevance in cuprate superconductors. Also these strongly correlated materials exhibit many anomalous properties besides the superconductivity. The common physical properties of cuprates can be summarised in the generic temperature-doping phase diagram shown in Fig.1.3 All electron properties of high temperature superconductors(HTSCs) depend on the doping and show a variety of exotic phases in temperaturedoping phase diagram. HTSCs are dielectrics and antiferromagnetic(AF) insulators without doping. This is due to onsite e-e coulomb repulsion. Superconductivity arises at large doping behind the magnetically ordered phase. In yttrium containing systems, the AF and superconducting phases adjoin to- Figure 1.3: The schematic of gether. The experiments showed that temperature-doping phase diagram the charge carriers have the ’hole’ char- of superconducting cuprates. acter for all class of HTSCs.
10
The region far beyond the optimal doping (where the superconducting transition temperature Tc is maximum) is metallic and can be explained by Fermi liquid theory. A signature of the Fermi liquid behavior is quadratic temperature dependence of in-plane resistivity, coming from electron-electron interaction. But it is still debated to what extent Fermi liquid theory applies to the normal state of HTSCs in general. The normal state of BCS superconductors is a Fermi liquid and the superconductivity appears as an instability of the Fermi sea. The normal state of optimally doped HTSCs is, however, not a standard Fermi liquid since it exhibits anomalous metallic behavior. For example the in-plane resistivity scales linearly with temperature. The normal state around optimal doping is therefore often referred as a strange metal. The microscopic mechanism for this strange metal is still controversial. The normal state of HTSCs has very peculiar behavior. Inelastic magnetic scattering of neutrons indicates the existence of strong magnetic fluctuations in the doped regions, even beyond the limits of AF phase. This suggests a important role of AF fluctuations in HTSCs. In HTSCs, there exists an excitation gap, called pseudogap (PG) which is present in the absence of the phase coherence, i.e in the non-superconducting specimens. It appears at the temperature below a characteristic temperature T ∗ which depends on doping. Its nature is still not completely understood. But one viewpoint tells that PG arises in connection with the formation of magnetic states which compete with superconducting states. The resolution of this dilemma is further complicated by the strong anisotropy of the superconducting gap. Some recent discoveries[47; 48] of charge-density wave in underdoped HTSCs may provide a great insight to understand this phase. Recently in 2008, a second class of high temperature superconducting materials, based on iron and arsenic, were discovered[49]. Again ARPES experiment[50] shows the evidence of the strong e-ph interaction in LiFeAs. Though these materials, called pnictides, superconduct at lower temperatures than most cuprates − often only below 40 K, they have provided the theorists a new arena for testing their ideas and hope to discover the other classes of high temperature superconductors.
11
1.4
Exciton-photon-phonon coupling and condensation
Microcavity polaritons (superpositions of the photons and excitons confined in optical microcavity) are bosonic quasi-particles with low effective mass and arise in the strong coupling regime[51; 52] where the exciton-photon interaction strength exceeds the excitonic optical transition and cavity-mode damping rate. These exciton-polaritons have been observed to form a Bose-Einstein condensate, and a non-thermalized version − a polariton laser[8; 9]. However, with exception of the materials such as GaN and ZnO with large exciton dipole moments[53; 54; 55], the strong coupling regime has been only reported at cryogenic temperatures in microcavities with inorganic materials characterised by the small binding energy (∼ 10 meV) extended Wannier-Mott excitons. In contrast, organic molecules, characterised by localized (Frenkel) excitons with high binding energy (∼ 1 eV), naturally have strong optical coupling which leads to BEC and polariton lasing at room temperature. In addition, excitons in organic microcavities have strong coupling to localized molecular vibrational modes (i.e. phonons) which introduces a new set of behaviours and possibilities not present in inorganic systems.
1.4.1
Quantum coherences and condensation
One of the surprising features of quantum many-body systems is the appearance of macroscopic phase coherence, leading to Bose-Einstein condensation (BEC) where a fraction of particles go to the lowest quantum state. Actually BoseEinstein statistics allows bosons − the particles with integer spin − to undergo a condensate state when the associated de Broglie wavelength becomes comparable to their average separation. In quantum statistical physics, the phenomenon of BEC is represented in terms of occupancy of single-particle states which are occupied according to the Bose distribution function nk , indexed by the momentum k. nk =
1 eβ(Ek −µ)
12
−1
(1.7)
Here we have considered a grand canonical ensemble of non-interacting bosons of mass m each and Ek = k 2 /2m, the kinetic energy of the boson and β = 1/kB T (~ = 1 throughout the thesis). The total number of particles is then fixed by N=
X k
nk =
Z
dE
D(E) −1
eβ(E−µ)
(1.8)
where the density of states D(E) ∝ E d/2−1 in dimension d and this equation basically determines the chemical potential µ as a function of temperature T . As we cool the system, the Bose factor nk becomes sharply peaked in the vicinity of the chemical potential µ. So for a fixed number of particles N , lowering the temperature requires an increase in µ. Remarkably, for dimension d > 2, the integral remains finite at non-zero temperature even if the chemical potential reaches the bottom of the particle energy band i.e. at µ = 0. This gives the critical temperature as, kB Tc ∼
n2/d m
(1.9)
where n is the particle density. Bellow this temperature µ remains fixed at the bottom of the band and the zero-momentum state gets populated with a macroscopic number of particles. However the non-interacting picture does not answer the central question: why do the condensate particles accumulate in a single lowest energy state? It is due to interactions between the particles which lead to an extensive energy difference between populating a single state and two almost degenerate states[56]. So we consider a weakly interacting dilute Bose gas model[34]: H=
X k
(ωk − µ)b†k bk +
U X † b b† b−k′ +q bk′ 2Ld k,k′ ,q k+q −k
(1.10)
where Bose particles represented by the operators bk with dispersion ωk , have the contact interaction U . This model can be analysed using mean-field approach, taking the variational wave function as coherent state, |βi = exp(βb† ) where β = hbi. For a chemical potential µ < 0, the minimisation of energy corresponds
13
to a trivial solution β = 0. But for µ ≥ 0, there is a phase transition to state p with |β| = µLd /U [34]. The phase coherent BEC state breaks the global gauge symmetry of the hamiltonian (i.e. invariant under the global phase rotation b → beiφ ) in Eq.(1.10) and leads to a low energy Goldstone mode with linear dispersion and hence superfluidity[34]. Other than the excitons and microcavity polaritons with inorganic and more recently organic materials, which will be discussed below, there exists a wide variety of systems that show macroscopic phase coherence and condensation. The dilute atomic gases are the prominent example systems where Bose
condensations are seen unambiguously[57; 58]. Due to the large mass and low densities of the particle in atomic gases, the transition temperature in such systems, is extremely low; typically 100 nK. Such low temperatures are achieved by employing the laser cooling[59; 60; 61] and magnetic trapping[62]. Liquid helium, as discovered by Kapitza[63] and Allen and Meisner [64],
has a superfluid (zero viscosity) transition at temperatures around 2.17 K which is a much higher temperature than the atomic condensates as it is at much higher density. Superconductivity represents, a rather different example of condensation of
Cooper pairs, mentioned in section[1.3.2]. Recently there has been huge interest in dilute Fermionic gases [10; 65;
66; 67] near Feshbach resonance which allows the inter-atomic interaction strength to be tuned by changing the magnetic field adiabatically and unbound atoms can be bound into molecules[10]. This makes it possible to investigate the crossover between a BCS state for weak attractive Fermions to BEC state of bound molecules. Finally here comes the laser where a macroscopic large number of photons
have almost same energy and momentum − a coherent state. An optical cavity with a gain medium, sandwitched by two partially reflecting mirrors, is strongly pumped, either electrically or by a high energy radiation.
14
Pumping leads to population inversion: a non-thermal distribution with more particles in higher energy levels than lower levels. Then stimulated emission from the inverted oscillators gives rise to coherent photons, even with entirely incoherent pumping. Unlike the above examples, it is far from thermal equilibrium and strongly pumped.
1.4.2
Excitons
After superfluid helium, excitons in semiconductor have long been considered a promising candidate for BEC at temperature of few kelvin, accessible by cryogenic techniques[68; 69; 70]. An exciton is an electron-hole pair, bound by Coulomb interaction and is formed when an electron is excited to empty conduction band leaving behind a hole in filled valence band. Excitons are analogous to hydrogen atoms but due to the strong dielectric screening in solids, the binding energy of the exciton is of the order of 1−100 meV and Bohr radius ∼ 10−100˚ A , extending over tens of lattice spacings. Because of the bosonic nature and small effective mass−typically of the order of electron mass, excitons are an obvious candidate for BEC at temperature of few kelvin. There have been several studies and early claims of exciton BEC in three-dimensional semiconductors (see for recent review [71; 72]) and for theoretical models for exciton condensation[73]. However, the finite lifetime of excitons, usually of the order of nanoseconds, poses a problem for Bose condensation of excitons. The illumination by light with energy in resonance with the band gap or with higher energies, both create the hot excitons. So to observe the condensation, the excitons have to be thermalized and cooled before the electron-hole pairs recombine. So there has been search for exciton condensation in variety of materials in which the recombination process is suppressed. Cu2 O [74] is one of such materials in which recombination is dipole forbidden, giving exciton a long lifetime. But again at the high density required for condensation, the recombination by an Auger process becomes important. More recently, there has been a lot of experimental efforts on coupled quantum well systems under an external applied electric field, leading electrons and holes to be physically separated by a tunnel barrier [75; 76; 77; 78; 79; 80; 81]. Despite intensive effort, exciton condensation has not been conclusively observed[72].
15
1.4.3
Microcavity polariton
The confinement of photons emitted by excitons in a semiconductor optical microcavity, leads to a quantum mixture of photonic and excitonic states, usually called microcavity exciton-polaritons. Though the idea of exciton-polariton was first conceptualised by Peker[82] and Hopfield[83] in bulk semiconductors, it is necessary to consider microcavity polaritons to observe the condensation.
Figure 1.4: (a) The simplified schematic of an optical microcavity containing a quantum well placed at the antinodes of cavity mode and sandwitched between two distributed Bragg reflectors. The conserved in-plane momentum of the photon k|| can be written in terms of emission angle (θ) as c|k|| | = ELP k sin(θ). This implies that there is a one to one correspondence between the polariton momentum and emission angle. (b) The respective dispersions of exciton and cavity mode produce the upper (UP) and lower (LP) polariton branch in the strong coupling limit. A microcavity is constructed by a planar Fabry-Pe´rot resonator containing a set of narrow quantum wells, sandwitched between two distributed Bragg reflectors (DBRs). The quantum wells are placed at the antinodes of the cavity photon mode to ensure maximum coupling between photon and exciton. The schematic diagram of an optical microcavity is shown in Fig.1.4(a). Here the Bragg mirrors are engineered by quarter wavelength layers of alternating refractive indexes, and so they can provide a high quality mirror − though still not perfect and a photon can escape the system. However ’strong coupling’ occurs when the rate at which
16
the photons are converted to excitons and vice verse is greater than the rate at which the photons escape from the cavity[84]. The photon momentum is quantized in the transverse direction due to confinement. So the photon dispersion in terms of in-plane momentum k|| , ωk||
r c 4l2 π 2 = + k2|| 2 n L 2clπ cL 2 ≈ + k nL 4nlπ || k2|| 2πnl ; mp = = ω0 + 2mp cL
(1.11)
where l is the quantization index for transverse momentum, n is the refractive index and L is the effective length of the cavity [85]. Here ω0 defines the bottom of the photon band and should be close to a resonance to the energy needed to create a bound exciton. This in turn sets the photon mass mp ∼ 10−4 me (me being the electron mass) for typical materials[85]. A simple model describing the interaction between the exciton and photon is given by[83; 86], X
ΩR † + + H = ψk bk + h.c. 2 k i Xh = EL L†k Lk + EU Uk† Uk ωk ψk† ψk
Eb†k bk
(1.12)
k
Here ψk and bk represent the photon and exciton operators for the mode k with energy ωk and E respectively. The exciton mass is large as compared to photon mass, so it is reasonable to assume the exciton energy to be dispersionless. The Rabi frequency ΩR describes the rate of interconversion between excitons and photons, so depends on the dipole matrix element for a single photon to excite a single exciton[87; 88]. The eigenstates of this hamiltonian is the linear superpositions of exciton and photon, with eigenenergies of upper and lower polaritons, EkU,P
q 1 2 = E + ωk ± (E − ωk )2 + ΩR 2
17
(1.13)
The dispersion shown in Fig.1.4(b) for upper and lower polaritons has a typical anticrossing form. The resonance between exciton and photon (i.e. E = ω0 ) gives the k = 0 gap to be the Rabi splitting ΩR which determines the excitonphoton coupling strength. The anticrossing between exciton and photon modes i.e. strong coupling, also has been observed for a single quantum dot in a semiconductor microcavity[89] and in a photonic band crystal nanocavity[90].
1.4.4
Polariton condensation
Polaritons are half-matter and half-light quasi-particles in an optical microcavity. The microcavity is an attractive system to study BEC and lasing at relatively high temperatures. Due to their strong coupling nature, microcavity polaritons allow the proposal of non-equilibrium condensation[91] as an intermediate state between laser and BEC, identifying the condensates as a laser without the usual population inversion. The polariton mass is much less than the exciton mass due to its photon component. Since the BEC transition temperature is inversely proportional to the mass, polariton lasing is possible even at room temperature[53]. However, a steady-state polariton must be pumped continuously as the lifetime of polariton is of the order of picoseconds[92]. So polariton condensation is inherently a nonequilibrium phenomenon. The basic principles of the experiments on polariton condensate are similar. The semiconductor microcavity is pumped by radiation to create the excitations which then relax and scatter to form a polariton condensate if the strong pumping crosses a threshold. The macroscopic gathering of polaritons in the lowest energy state depends on how the system is pumped. Incoherent pumping creates a random polariton population which then re-
lax towards the lower momentum states which in tern stimulate the final relaxation to the bottom of the lower polariton branch and get condensed in the strong-coupling regime. As the polariton has a finite lifetime (∼ picoseconds), the photon escapes the cavity and is detected. The escaping light conserves the energy and in-plane momentum of the polaritons, so the coherent emission of light from the microcavity can be used to image both the real-space and momentum-space shape of polariton BEC and the
18
dispersion. However, at high excitation density where the bosonic character is lost, a microcavity can be characterised as electron-hole plasma in weak-coupling regime and there can be lasing with coherent light emission. The onset of macroscopic occupation of the ground state by stimulated scattering of lower polaritons has been reported[93], where two thresholds in photoluminescence were identified, separating weak-coupling lasing from strong-coupling polariton stimulation. If polaritons are injected coherently close to point of inflection on lower
polariton branch at momentum kp , a weak probe at zero momentum can stimulate a pair of polaritons to scatter to k = 0 and k = 2kp states while conserving the energy and momentum, and gets amplified strongly [94; 95]. This process is called optical parametric scattering, illustrated in Fig.1.5. However without the probe beam, this stimulated scattering, called optical parametric oscillation (OPO), can be observed[96]. In this case, above a threshold pump power, the state with polaritons only at pump wave vector becomes unstable, and scatters to lower and higher energy states. The experiments on polariton show very large nonlinearities including bistability, and so provide a great opportunities to develop polariton-based devices[97; 98; 99] To present the polariton BEC as a quasi-equilibrium transition, we need the polariton population to be in thermal equilibrium − an essential property of BEC. But the formation of a BEC in the context of polaritons, corresponds to a transition as the density is increased at a fixed temperature, from the quasi-thermal distribution to a distribution with a macroscopic occupation of the ground state. The experiments show that polariton condensates can be formed without being in thermal equilibrium between the condensate and non-condensed polaritons, above a threshold density by developing a sharply peaked distribution around a single ground state [92; 101]. Polariton BEC provides the possibility of direct observation of coherence properties of a condensate by using the coherence of the emitted light. By retro-reflecting the emission, the light from a position r can be interfered with the light from r′ . Varying the path difference between the two images, the different intensity fringes determine the strength of the correla-
19
Figure 1.5: Left panel: Sketch of optical parametric amplification (OPA) process where a pump beam (filled circles) can scatter (conserving the energy and momentum) two polaritons into an signal (lower momentum) and idler (higher momentum) mode, and hence the probe in the signal mode would be amplified. Right panel: Experimental data showing OPA behaviour[100] . tion function g (1) (r, r′ ) and the presence of a coherent condensate to be directly confirmed [92]. There has been debate about how the polariton condensate is similar to the equilibrium condensates[102]. For a polariton condensate, the coherence time can be upto 150ps which is significantly larger than the individual polariton of which lifetime is few picoseconds[92]. As a result, the behavior of polariton could be different from equilibrium condensate. As an example, in equilibrium condensate, we get a linear Bogoliubov dispersion for the elementary excitations due to the broken phase symmetry. But in polariton condensate, this behavior gets modified due to continuous pumping and decay, and there is instead a diffusive mode[103; 104; 105] not preventing the superfluid behavior[106]. Elementary excitations have been observed[107; 108], and appear to be consistent with these predictions [8; 108]. Also the two-dimensional nature of these polariton condensate should show a power law decay in spatial coherence correlations, according to the theory of the Berezinskii-Kosterlitz- Thouless (BKT) transition[109]. A recent experiment on large polariton condensates has demonstrated power-law decay with
20
a different exponent from equilibrium transitions, which is attributed to the presence of continuous pumping[110]. The similarities between polariton condensates and equilibrium condensates still remain a matter of active debate[111; 112].
1.4.5
Organic polariton
Till now we have discussed the non-equilibrium polariton condensation, parametric process in optical microcavities with inorganic quantum wells. Along with the inorganic MCs, there has been a development of strongly coupled organic MCs[113], by using different kinds of optically active organic materials, among which the cyanine dye J-aggregates films are probably the most typical one. A J-aggregate can be thought as a chain of dye molecules linked by electrostatic interactions. Unlike the inorganic materials, organic molecules have localized (Frenkel) excitons with large binding energy and oscillator strength. This allows the organic MCs to be in the strong-coupling regime with Rabi splitting upto 300meV[114], and the possibility to observe easily the polariton condensations even at room temperature [115]. Recently coherent emission as in inorganic MCs (i.e. polariton lasing) has been observed in MCs composed of organic semiconducting crystal of anthracene at room temperature[116]. A detailed review about organic MCs can be found in Refs.[117; 118]. In addition to the strong exciton-photon coupling, these systems have strong coupling to localized molecular vibrational modes which introduce a new set of behaviours such as strongly coupled vibronic replicas[119; 120] and possibilities that are not present in inorganic systems. Organic polariton condensation provides a natural venue to explore the quantum many body effects arising from simultaneous strong exciton-photon and exciton-phonon coupling. This vibronic coupling indeed can play a complex and crucial role in energy transfer in light harvesting complexes[121; 122; 123]. Along with the fundamental interests in room temperature BEC, organic polaritons provide a novel route to electrically pumped ultra-low thresholds organic lasers[124].
21
Chapter 2 The Models 2.1
Hubbard model
The celebrated Hubbard model is a minimal description of electron correlations in condensed matter theory, consisting of two competing characteristics of electrons: the kinetic energy (hopping of localized electrons from site to site) tries to
delocalise electrons, leading to metallic behavior. On site Coulomb interactions tend to localise the electron onto sites, driving
the system to a Mott insulator which is magnetic. The Hubbard model in its modern form was systematically introduced independently by Gutzwiller[125; 126], Hubbard[6] and Kanamori[127], to study magnetism in transition metals. The hamiltonian in second quantised form is given by, X † X H = −t cjσ cj+1σ + h.c + U nj↑ nj↓ jσ
j
= Ht + HU
(2.1)
where t is the nearest-neighbour (NN) hopping amplitude of localized electrons with spin σ = (↑, ↓) represented by creation(annihilation) operators c†jσ (cjσ ) where njσ = c†jσ cjσ being the number operator, and U measures the on site Coulomb interaction. Although this model appears to be oversimplified, it is
22
believed to exhibit various interesting phenomena including metal-insulator transition, different magnetic orders, Tomonaga-Luttinger liquid, and superconductivity. A comprehensive review[128; 129] of single- and multi-band Hubbard model in context of non-conventional superconductors can be found in Ref.[130].
2.1.1
Mott Transition
This is a metal-insulator transition due to large onsite Coulomb interaction which forbids the electrons to hop when the system has one electron per site, i.e. at half-filling. In this strongly correlated state, the mutual Coulomb interaction between the electrons can drive the system from a metallic to an insulating phase with properties very different from those of a conventional band insulator. For a qualitative understanding of this transition, let us consider a lattice system of size L at half-filling. Strictly speaking, at t = 0 the ground state spectrum contains L copies of two energy levels separated by U and all electrons are at the lower levels on each site. Now if we put one extra electron, it can be in any one of the upper energy levels. For further simplification, lets assume the L lower-level electrons have ↓ -spins, and the extra one has ↑-spin. Then the hopping of this extra electron broadens the upper energy level into a tight binding band of width ∼ 2zt where z is the coordination number. Similarly, in a system with L − 1 spinpolarised electrons, considering the motion of a single hole, a similar argument gives a broadening to form a band of the same width at lower energy level. These are referred as Hubbard subbands[6] which are essentially occupation-number dependent band structure. The very existence of the upper subband depends on the presence of electrons in the lower subband. The splitting of two subbands is a correlation effect. At large U , an energy gap exists between the subbands, which corresponds to the insulating behavior, called Mott insulator. As U is decreased, the gap becomes narrower and vanishes at the critical value Uc ∼ zt, which gives a insulator-metal transition driven by electronic correlation. Indeed, this type of metal-insulator transition has been found for some Mott insulators[131], where the bandwidth t, and thereby U/t, is controlled by pressure or doping. More recently, the Bose-Hubbard model has
23
been used to describe the superfluid to Mott insulator transition in the system of ultracold atoms trapped in optical lattices[132].
2.1.2
t-J Transformation
In the strong coupling limit U/t ≫ 1, we have a Mott insulating phase at halffilling. Experimentally it is often found that the low-temperature phase of the Mott insulator is accompanied by the anti-ferromagnetic ordering of the local moments. Indeed the low energy physics of the Hubbard model in Eq.(2.1) for the Mott insulating phase can be described by anti-ferromagnetic Heisenberg model which can be derived by taking the hopping term Ht as a weak perturbation to the ground state of Hubbard interaction HU . To implement the perturbation theory, we seek a canonical transformation of the hamiltonian in Eq.(2.1): ˜ = e−tS HetS H = H − t[S, H] +
t2 [S, [S, H]] + · · · 2
(2.2)
Here the hopping integral t has been used to control the order of perturbation. To eliminate the hopping contributions linear in t we put Ht = t[S, HU ] which defines the operator S. The transformed hamiltonian then becomes, ˜ ≈ HU − t [S, Ht ] + O(t3 ) H 2
(2.3)
By applying an ansatz[34], tS = (Ps Ht Pd − Pd Ht Ps )/U , with the projection operators Ps and Pd which project the states onto singly- and doubly-occupied subspaces respectively, we assure the cancellation of first order term in t. This can be easily verified, using the properties of the projection operators: Ps + Pd = 1, Ps Pd = 0, HU Ps = 0
(2.4)
Now using the ansatz defining the operator S along with the operator properties Ps Ht Ps = Pd Ht Pd = 0 and projecting the transformed hamiltonian in
24
Eq.(2.3) onto singly occupied subspace, we get, ˜ s = − 1 Ps Ht Pd Ht Ps Ps HP U
(2.5)
Here, the operator Ht Pd Ht sandwiched between the projection operator Ps , can have two distinct contributions: A sum of two-site terms where starting with a singly occupied state an
electron jumps to NN sites making it doubly occupied and jumps back to the original site. This corresponds to virtual hopping process and any such term will be weighted (apart from signs and overall factor U −1 ) by a factor of t2 . Another sum of three-site terms corresponding to the processes where an
electron jumps to NN site and then hops to next-nearest neighbouring site instead of coming back to original site, the weight factor being t2 again. To this end the explicit calculations[133] of Eq.(2.5) involves the operator decomposition of hopping term as, Ht = Ps Ht Ps + Pd Ht Pd + Ps Ht Pd + Pd Ht Ps
(2.6)
and in terms of trivial identity ciσ = ciσ [(1 − niσ ) + niσ ], we can write Ht as, Ht = −t +
n
Xh
hijiσ
(1 − ni¯σ )c†iσ cjσ (1 − nj σ¯ ) + ni¯σ c†iσ cjσ nj σ¯
ni¯σ c†iσ cjσ (1 − nj σ¯ ) + h.c
oi
; (σ, σ ¯ ) = (↑, ↓)
(2.7)
Here the first and second terms in Eq.(2.7) conserve the number of doubly occupied sites whereas third term mixes the singly- and doubly-occupied subspaces. Simple comparison of Eq.(2.6) and (2.7) gives, Ps Ht Pd = −t
X
hijiσ
(1 − ni¯σ )c†iσ cjσ nj σ¯
25
(2.8)
Finally putting this expression in Eq.(2.5), we obtain: 2 ˜ s = − t Ps {H (1) + H (2) }Ps Ps HP U
(2.9)
with the expressions: H (1) =
XX (1 − ni¯σ )c†iσ cjσ nj σ¯ nj τ¯ c†jτ ciτ (1 − ni¯τ )
(2.10)
hiji στ
H (2) =
XX
hijki στ
(1 − ni¯σ )c†iσ cjσ nj σ¯ nj τ¯ c†jτ ckτ (1 − nk¯τ )
(2.11)
Here, the symbol hiji denotes a summation in which i and j are NN sites and hijki represents a summation in which i and k are NN sites to j and i 6= j. The rearrangement of the terms in Eq.(2.10) indeed gives the anti-ferromagnetic Heisenberg interaction as[134]: X 1 t2 ˜ =J Si · Sj − ni nj − H (2) H 4 U
(2.12)
hiji
P where ni = σ niσ and the action of Ps on both sides is implied. The spin operator Si and the exchange constant J are given by: Si =
1X † 4t2 ciσ [σ]στ ciτ ; J = 2 στ U
(2.13)
where [σ] represent Pauli spin matrices. So the perturbation theory above shows that for large U the electrons at half-filling have the tendency to adopt an antiferromagnetic ordering of their spins. Physically this is easy to understand. Antiparallel spin ordering helps the electrons to reduce energy by hopping which would be forbidden by Pauli exclusion principle if the spins are arranged in parallel. This mechanism involving two step processes was first formulated by Anderson[135], and is known as Superexchange. The behavior of the Hubbard model is very difficult to solve when we dope away from half-filling. the hole doping into the half-filled system introduces vacancies into the lower Hubbard subband which can propagate through the lattice.
26
For a low density of holes, the strong coupling Hubbard system may be described by the effective t-J Hamiltonian (neglecting three-site terms): Ht−J ≈ −t
X
c†iσ ciσ
iσ
X 1 +J Si · Sj − ni nj 4
(2.14)
hiji
again the projection onto singly occupied subspace is implied.
2.2
Holstein model
The Holstein model[136; 137] has been used for many years to study the effect of lattice vibrations on the motion of electrons, such as the formation of polarons and bipolarons by self-trapping of charge carries, or the Peierls transition accompanied by a CDW. This model describes the tight-binding electron on a lattice with eph interactions. Before the discussions of the model and the self-trapping of electrons, we will derive the e-ph interaction hamiltonian − the main ingredient of the Holstein model.
2.2.1
The interaction hamiltonian and the model
The basic hamiltonian of electron-phonon system is given by H = He + Hp + Hep ;
Hp =
X
ωqλ (φ†qλ φqλ + 1/2)
(2.15)
qλ
where He consists of free electron and Coulomb interaction parts, Hp describes the normal modes of vibration (phonon) of the solid, ωqλ being the frequency of vibrational mode λ. The interaction between electron and phonon is given by the sum of the contributions of individual ions: Hep =
X
V (ri );
V (ri ) =
i
X j
Vep (ri − Rj )
(2.16)
Following the Born-Oppenheimer approximation, electrons are assumed to be very fast moving objects while the ions are considered to be vibrating around
27
their equilibrium positions. Assuming the lattice displacement xj to be small, we can Taylor expand the potential Vep around the equilibrium ion position R0j as,
Vep (ri − Rj ) ≈ Vep (ri − R0j ) + xj ·
∂Vep (ri − R0j ) + O(xj 2 ) + ... ∂Rj
(2.17)
The first term in Eq.(2.17) represents a periodic potential due to equilibrium positions of ions. This periodic potential leads to Bloch states for the motion of electrons and are assumed to be known. Neglecting the higher order terms in the displacement xj , our interest is on e-ph interaction term: V (r) =
X j
xj ·
∂Vep (r − R0j ) ∂Rj
Assuming that the electron-ion potential possesses a Fourier transform, we have: 1 X Vep (k)eik·r N k X i X ik·r e Vep (k)k · xj e−ik·R0j V (r) = N k j
Vep (r) =
(2.18)
Using the definition of the displacement xj as an expansion in terms of phonon creation and destruction operators as, 1/2 X i X ~ −ik·R0j (φk+G,λ + φ†−k−G,λ ) xj e = ξk+G,λ N j 2ρνωk+G,λ G,λ
(2.19)
where ρ and ν are the density and the volume of the system. Here G represents reciprocal lattice vectors and the phonon states k+G are defined within first Brillouin zone. The phonon polarization vector ξk+G is considered to be real but changes sign with momentum direction.
28
So we can write the interaction potential from Eq.(2.18) as, Vep (r) =
X
e
i(k+G)·r
k,G,λ
Vep (k + G)(k + G) · ξk,λ
~ 2ρνωk,λ
1/2
(φk,λ + φ†−k,λ )(2.20)
Here we have restricted the sum over k to be within first Brillouin zone, but the sum over G allows the potential Vep (k + G) to interact with higher Fourier components. The potential Vep (r) acts upon the electrons and the e-ph interaction hamiltonian can be obtained by integrating this potential over the electronic charge density ̺(r),
Hep = =
Z
d3 r̺(r)Vep (r) 1
ν 1/2
X
k,G,λ
Mk+G ̺(k + G)(φk,λ + φ†−k,λ )
Mk+G = Vep (k + G, λ)(k + G) · ξk,λ
~ 2̺ωk,λ
1/2
(2.21) (2.22)
Finally, the Holstein model, we will be considering, includes the tight-binding electrons interacting with dispersionless phonon localized on lattice sites. The model in one-dimensional lattice reads: Hhol = −t
X jσ
(c†j+1σ cjσ + h.c) + ω0
X
φ†j φj + gω0
j
X
njσ (φ†j + φj )
(2.23)
jσ
where first term represents electron tunnelling from site to site, φ†j and φj are the creation and destruction operators for dispersionless optical phonon at site j, with frequency ω0 . The last term includes the the density of electrons at site interacting with local lattice displacements, with coupling constant g.
2.2.2
Polarons by self-trapping
Non-interacting itinerant electrons in a solid occupy one-electron Bloch states. The mutual interaction between the these charge carriers and the collective lattice deformations i.e. phonons, may lead to the formation of new quasiparti-
29
cles. The composite entity − an electron dressed by a phonon cloud is called a polaron[138; 139]. Since the lattice deformations follow the electron while moving from site to site, one of the most important properties in the ground state of polaron is the increase in electron’s effective mass. The fundamental theoretical question in the context of polaron physics asked by Landau[138] is the possibility of a local lattice instability making the quasiparticle effective mass large, that traps the charge carrier, upon increasing the e-ph coupling. Such trapping is energetically favourable over the itinerant electron band if the carrier’s binding energy exceeds the deformation energy to produce the potential trap. Depending upon the extension of phonon cloud in these polaronic trapped states, the polarons are referred as large-polaron with the extension of phonon cloud exceed few lattice constants and small-polaron where phonon cloud is localized to a single site due to extremely large quasiparticle effective mass. Selftrapping does not break the translational invariance, but the itinerant electrons move with extremely narrow bandwidth. These self-trapped states of electrons dressed with phonon cloud can be found in alkaline earth halides, group-IV semiconductors, organic molecular crystals[140], high-Tc cuprates[141], and colossal magneto-resistance manganites[142; 143]. The Holstein model in Eq.(2.23) has three parameters: the electron hopping integral t, the phonon frequency ω0 and the e-ph coupling strength gω0 . As all three parameters represent some energy scales, we can characterise this model by two independent dimensionless parameters: The first one is the adiabaticity parameter α = ω0 /t which determines
the fastness and slowness of the component subsystems of the model i.e., electrons and phonons. In the adiabatic limit α ≪ 1, the motion of electron gets affected by the lattice distortions. In contrast the lattice deformation is presumed to adjust instantaneously to the position of the electron, in the anti-adiabatic limit α ≫ 1. The second one, the coupling constant g, plays a crucial role in small-
polaron (strong coupling) theory. Equivalently, we can define a parameter λ which is the ratio of polaronic energy ǫp = g 2 ω0 when confined to a single site, and the electron bandwidth 2dt, d being the dimension of the system.
30
The cross-over from essentially free carriers to heavy quasi-particles i.e. the self-trapped states takes place at λ ⋍ 1 and much sharper in higher dimensions[144; 145]. A comprehensive review of the Holstein model can be found in Ref.[146].
2.3
Hubbard-Holstein model
With the brief qualitative discussions of onsite electron-electron (e-e) and e-ph interactions above through Hubbard and Holstein model respectively, in this section we present a qualitative discussion of the Hubbard-Holstein (HH) model which includes both e-e and e-ph interactions. The analysis of the HH model using a controlled analytic approach along with the numerical simulations, will be the subject matter of Chapter.3 and Chapter.4. Our focus will be specifically on the materials where the electrons are coupled to localized optical phonons, which may be of relatively high frequency. The interplay of e-e and e-ph interactions in these strongly correlated systems leads to coexistence of or competition between various phases such as superconductivity, CDW, spin-density-wave (SDW) phases, or formation of novel non-Fermi liquid phases, polarons, bipolarons[141; 147], etc. It is of particular interest to consider both the strong coupling regime, and the crossover to the non-adiabatic limit where the lattice response is not slower in comparison to the heavy effective mass of the correlated electron system. The simplest framework to analyse the effects of e-e and the concomitant e-ph interactions is offered by the Hubbard-Holstein model whose Hamiltonian is given by: Hhh
X † X † = −t cj+1σ cjσ + h.c. + ω0 aj aj jσ
+gω0
j
X
njσ (aj + a†j ) + U
X
nj↑ nj↓ .
(2.24)
j
jσ
Here, njσ ≡ c†jσ cjσ with c†jσ and a†j being the creation operators at site j for an electron with spin σ and a phonon respectively. The Hamiltonian describes a tight-binding model with hopping amplitude t, a set of independent oscillators characterised by a dispersionless phonon frequency ω0 , along with an onsite Coulomb repulsion of strength U and an e-ph interaction of strength g where elec-
31
P tron density nj = σ njσ couples to the local lattice displacement xj ∝ (a†j + aj ). The Holstein and the Hubbard models are recovered in the limits U = 0 and g = 0 respectively. In this thesis, we will consider the HH model in both strong e-e and e-ph coupling limit. The e-ph interaction term can be eliminated by a Canonical transformation, called the Lang-Firsov (LF) transformation[148]. But this makes the electrons hopping in a very narrow band because of the phonon cloud surrounding the electrons. The essence of LF transformation is to shift the equilibrium positions of lattice vibrations with an energy gain corresponding to polaron-formation. Since we will be using this transformation extensively, it will be helpful to have some mathematical expression to support the above qualitative statements. We employ the LF transformation in the context of HH model in Eq.(2.24) ˜ hh = eS He−S , where the value of S should be taken in such a way that after as H transformation the e-ph interaction term gets cancelled, and hence S = −g
X jσ
njσ (aj − a†j )
(2.25)
The electronic operators will be transformed as: †
eS cjσ e−S = cjσ Xj ; Xj = eg(aj −aj )
(2.26)
where it can be seen that each electronic operator is now combined with exponential of phonon operators which we are calling phonon clouds. Also the phonon operators get transformed as: eS aj e−S = aj − gnjσ
(2.27)
by which the independent Einstein oscillator term and the e-ph interaction term in Eq.(2.24) become: eS
X
ω0 a†j aj e−S = ω0
j
X j
S
e gω0
X
njσ (aj +
a†j aj − gω0
a†j )e−S
X
c†jσ cjσ (aj + a†j ) + g 2 ω0
jσ
= gω0
jσ
X jσ
32
X
n2jσ (2.28)
jσ
njσ (aj +
a†j )
2
− 2g ω0
X jσ
n2jσ (2.29)
Now it is easy to see that the e-ph interaction term gets cancelled. Also P now the additional term g 2 ω0 jσ n2jσ will account for the polaronic energy gain P g 2 ω0 jσ njσ and the reduction of the Hubbard interaction to effective one: Ueff = U − 2g 2 ω0 . To gain insight into the rich physics of the Hubbard-Holstein model, several studies have been conducted (in one-, two-, and infinite-dimensions and at various fillings) by employing various approaches such as quantum Monte Carlo (QMC) [149; 150; 151; 152; 153; 154], exact diagonalisation [155; 156; 157], density matrix renormalisation group (DMRG)[158; 159], dynamical mean field theory (DMFT) [160; 161; 162; 163; 164; 165; 166; 167], semi-analytical slave boson approximations [168; 169; 170; 171; 172], large-N expansion [173], variational methods based on Lang-Firsov transformation [174; 175], Gutzwiller approximation[176; 177], and variational cluster approximation[178].
2.4
Dicke model and phonon
The simplest model, the Jaynes-Cummings model in quantum optics is for a single two-level system interacting with light. On the other hand the Dicke model[179], originally introduced to study the radiative decay of a gas, treats a group of atoms as two-level oscillators interacting with a single mode of the radiation field through electric-dipole transition between these two states. Due to the effect of disorder in 2D system (e.g. quantum wells made of organic semiconductors in optical microcavity), the exciton states get localised spatially which interact strongly with light[87; 180]. We can generalise the Dicke model to describe the cavity polaritons in disordered electric systems. If the electronic excitations are smaller as compared to the wavelength of light, we can describe them as pointlike objects, just like molecular excitations as in Dicke model. The localised excitations are basically considered as the excitons whose energy can be labelled by site index n, associated with a particular eigenstate of the disordered potential. In low density limit, the interaction between the excitons can be neglected. The Pauli exclusion principle restricts the excitons to have no more than a single excitation. The generalised Dicke model for localised, physically separable and saturable
33
excitons can be described by the hamiltonian with rotating wave approximation (RWA) [181]: †
Hd = ωc ψ ψ +
X h ǫn n
2
(b†n bn
−
a†n an )
+
g(b†n an ψ
+
ψ † a†n bn )
i
(2.30)
where ψ † is the creation operator for the photon field with frequency ωc , an exciton at site n is represented by a two-level system with b†n and a†n being the fermionic creation operators corresponding to upper and lower energy levels with the hard-core constraint: b†n bn + a†n an = 1
(2.31)
and g being the interaction strength between exciton and photon, is taken to be site independent as the amplitude of photon field in long wavelength limit remains almost constant over the sample (2D quantum well). The exciton after absorbing a photon gets excited and emits the photon when comes to ground state. Path integral approach: Mean-field results Now, we will give a brief account of the equilibrium properties of the model given in Eq.(2.30) using Mean-Field (MF) theory[181; 182] within a path-integral approach which we will be using in the thesis. Consider a grand-canonical ensem˜ d = Hd − µL, where the chemical potential µ controls the total excitation ble, H number L of photons and excitons, given by: L = ψ†ψ +
1X † (bn bn − a†n an ) 2 n
(2.32)
The coherent-state path-integral representation for the grand canonical par-
34
tition function Q can be given as constrained functional integral, Q = Σ =
Z Z
D[ψ]
Y n
β
¯ n Φn − 1)e−Σ D[Φn ]δ(Φ
"
¯ τ +ω dτ ψ(∂ ˜ c )ψ +
0
X
¯ n Mn Φn Φ
n
(2.33) #
(2.34)
We have introduced the Nambu spinors on each site to simplify the notation as, bn an
Φn =
!
The matrix M is given by
∂τ + ǫ˜n
Mn =
g ψ¯
gψ ∂τ − ǫ˜n
where we have defined ω ˜ c = ωc − µ, ǫ˜ = (ǫ − µ)/2. Functional integration over the fermionic degrees of freedom, gives the effective action in the photon field ψ. By saddle-point analysis, the partition function is dominated by the stationary photon field ψ(τ ) = λ assuming to be real, which produces the minimum action. The fluctuation spectrum can be obtained by integrating over Gaussian fluctuation around this stationary value λ. The static, uniform value of photon field λ satisfies the condition: ω ˜cλ = g2λ
tanh(βE) ; 2E
E=
p
ǫ˜2 + (gλ)2
(2.35)
which describes a mean-field condensate of coupled exciton polarisation and coherent photon. The mean-field density of the total number of excitations is given by: ρ = λ2 −
1 ǫ˜ tanh (βE) 2E
(2.36)
Note that the photon field acquires an extensive occupation and λ2 can be defined as the photon density in the condensate. The above results are for a homogeneous
35
system where excitons at all sites have the same energy. But for an inhomogeneous broadened band of exciton energies, the Eq.(2.35) for λ and Eq.(2.36) for density should be averaged over exciton energies. Phonon in Dicke model Finally the theory of polaritons in optical cavity made of organic materials in which molecular vibrations are important, will be the subject matter in Chapter.(5). The theory of excitons in molecular crystals has been extensively developed by Davydov[183] and Agranovich[184]. However, the model describing organic molecules interacting strongly with light is less developed which includes the works, focused on rate equations to calculate the luminescence spectrum[185; 186] and relaxation processes on J-aggregate microcavities[187], as well as the effects of disorder on the spectrum[188; 189]. Focusing on low densities, these works have generally used a theory of a weakly interacting gas of polaritons, in some cases explicitly derived[190] from models of molecules as saturable absorbers and are not so accurate at higher densities.
36
Chapter 3 Phase diagram of one-dimensional strong coupling Hubbard-Holstein model In this chapter, we consider the Hubbard-Holstein (HH) model in one-dimension given in Eq.(2.24) to present a phase diagram at quarter-filling. In contrast to earlier approaches, we utilise a controlled analytic approach (that takes into account dynamical quantum phonons). Our method uses both the strong eph coupling limit g > 1 and the strong Coulomb coupling limit U/t > 1 to generate an effective t − J model with displaced oscillators that can then be treated perturbatively, and generates longer-range interactions in an effective Hamiltonian. This model we then solve numerically for finite chains. We then obtain a phase diagram of the one-dimensional HH model at quarter-filling, which is strikingly similar at other fillings also discussed in the next chapter. Our effective Hamiltonian comprises of two dominant competing interactions – spin-spin AF interaction and nearest-neighbour (NN) electron repulsion. In addition, three types of hopping also result – the NN hopping with reduced band width, next-nearest-neighbour (NNN) hopping, and NN spin-pair σ¯ σ hopping. As the e-e interaction U/t is increased, the system sequentially transforms from an AF cluster phase to a correlated singlet phase followed by CDW phase(s) with (e-ph coupling g dependent) accompanying spin order. The most interesting
37
feature is that, at intermediate values of U/t and for all strong e-ph couplings (2 ≤ g ≤ 3) considered, a phase comprising of correlated NN singlets appears which suggests the possibility for superconductivity occurrence. In this paper, we are not considering the situation U < 2g 2 ω0 where onsite bipolarons form. Thus the controversies related to onsite bipolarons, such as those mentioned in Ref. [191; 192], are not pertinent to our treatment involving nearest-neighbour singlets which occur when U > 2g 2 ω0 .
3.1
Effective Hamiltonian
We briefly outline below the procedure to get an effective electronic hamiltonian for the HH model given in Eq.(2.24) Although we obtain the effective Hamiltonian here in one-dimension only, our approach is easily extendable to higher dimensions LF = as well. We first carry out the Lang-Firsov (LF) transformation [148], Hhh P † eT Hhh e−T where T = −g jσ njσ (aj − aj ) and get the following LF transformed Hamiltonian: LF Hhh = −t
X
† (Xj+1 c†j+1σ cjσ Xj + h.c.) + ω0
jσ
2
−g ω0
X
a†j aj
j
X j
2
nj + (U − 2g ω0 )
X
nj↑ nj↓ ,
(3.1)
j
†
where Xj = eg(aj −aj ) and nj = nj↑ + nj↓ . Next, we express as follows our LF transformed Hamiltonian in terms of the composite fermionic operator d†jσ ≡ c†jσ Xj† : LF Hhh = −t
X † X † aj aj dj+1σ djσ + h.c. + ω0 j
jσ
+ (U − 2g 2 ω0 )
X j
ndj↑ ndj↓ − g 2 ω0
X j
ndj↑ + ndj↓ ,
(3.2)
where ndjσ = d†jσ djσ . On dropping the last term, which is a constant polaronic energy, we recognise that Eq.(3.2) essentially represents the Hubbard Model for composite fermions with Hubbard interaction Uef f = (U − 2g 2 ω0 ). In the limit
38
of large Uef f /t, using standard treatment involving a canonical transformation, we get the following effective Hamiltonian written to second order in the small parameter t/Uef f [133]: "
X † X † dj+1σ djσ + h.c. + ω0 aj aj
Ht−J−t3 = Ps −t + J
jσ
~j · S ~j+1 − S
X j
Xh
+ t3
j
ndj ndj+1 4
!
d†j σ¯ dj+1σ d†j−1σ dj σ¯ + h.c.
jσ
− t3
Xh
i
d†jσ dj+1σ d†j−1¯σ dj σ¯ + h.c.
jσ
i
#
Ps ,
(3.3)
4t2 ~ where ndj = ndj↑ + ndj↓ , J = U −2g 2 ω , t3 = J/4, Si is the spin operator for a 0 spin 1/2 fermion at site i, and Ps is the single-occupancy-subspace projection operator. Furthermore, the last two terms with coefficient t3 (= J/4) are the three site terms which when omitted from the above Hamiltonian Ht−J−t3 yield the well-known t − J Hamiltonian (for the composite fermionic operators djσ ). The effective t − J − t3 Hamiltonian, given in Eq.(3.3), can be re-written in terms of the original fermionic operators cjσ as
Ht−J−t3 = H0 + H1 ,
(3.4)
where H0 = −te−g +J
2
X jσ
X
Ps
j
+ −
−g 2
Je 4
−g 2
Je 4
X † Ps c†j+1σ cjσ + h.c. Ps + ω0 aj aj
j
~j · S ~j+1 − nj nj+1 Ps S 4
X jσ
X jσ
h i Ps c†j σ¯ cj+1σ c†j−1σ cj σ¯ + h.c. Ps
h i Ps c†jσ cj+1σ c†j−1¯σ cj σ¯ + h.c. Ps ,
39
(3.5)
and H1 = −te−g
2
X jσ
h i Ps c†j+1σ cjσ (Y+j† Y−j − 1) + h.c. Ps .
(3.6)
In the above equation, we have separated the Ht−J−t3 Hamiltonian into (i) an electronic part H0 which is essentially a modified t−J −t3 Hamiltonian containing 2 a NN hopping with a reduced amplitude (te−g ), electronic interaction terms with the same interaction strength J, three site terms with reduced amplitude 2 Je−g /4, and no e-ph interaction; and (ii) the remaining perturbative part H1 which corresponds to the composite fermion terms containing the e-ph interaction with Y±j ≡ e±g(aj+1 −aj ) . Furthermore, since J/4 n ≥ 1/5) in our t-V1 -V2 model, we find that the values of n0 seem to increase more slowly with system size [see Figs. 4.7(a), √ 4.7(c), and 4.7(e)] – this being due to smaller coefficients of N resulting from interaction effects. Moreover, we also note [from Figs. 4.7(b) and 4.7(e)] that the √ value of n0 [i.e., the coefficient of N in the expression for n0 ] decreases due to repulsion.
67
4.5
WQMC for BEC fraction
In this section we present our trick added to world-line quantum Monte Carlo (WQMC) to calculate the BEC fraction for our t-V1 -V2 model. We will discuss, in brief, the usual WQMC approach [202; 208] adapted for calculating correlations for the model Hamiltonian given below: Hb =
X
Hj =
j
X
[−T (b†j bj+1 + H.c.) + V1 nj nj+1 + V2 nj nj+2 ].
(4.8)
j
Since this is quite similar to the t-V model, we can employ the checkerboard P j P H j . It is decomposition Hb = H1 + H2 where H1 = H and H2 = j odd
j even
important to note that both H1 and H2 consist of independent two-site pieces. Because of the decomposition, it becomes easier to evaluate the expectation value of an operator A given by hAi =
T r[Ae−βHb ] , T r[e−βHb ]
(4.9)
with A involving only number operators (such as ni nj ) or NN hopping operators (such as b†j bj+1 + H.c.). Now we calculate the partition function: Z = T r[e−βHb ] X = hi1 |U1 |i2L ihi2L |U2 |i2L−1 i...hi3 |U1 |i2 ihi2 |U2 |i1 i. i1 ,...,i2L
Here Ui = e−∆τ Hi , β = L∆τ , and each of |i1 i, ...,|i2L i form a complete basis set in the occupation number representation. Here the world lines are the locus of the particles in the imaginary time (τ ) direction. For the density-density correlation function hni ni+l i (which is the expectation value of a diagonal operator), the above procedure of inserting 2L time slices yields the simple form hni ni+l i =
1 h[hiL |ni ni+l |iL i + hiL+1 |ni ni+l |iL+1 i]iQMC , 2
where h iQM C represents average over many QMC passes. Notice that we have
68
concentrated only on L and L + 1 time slice indexes although expectation value can be taken over all the 2L time slice indexes for better statistics. As for hb†j bj+1 + H.c.i (which corresponds to a non-diagonal operator), WQMC procedure yields hb†j bj+1
hiM |(b†j bj+1 + H.c.)Uk |iM +1 i iQMC , + H.c.i = h hiM |Uk |iM +1 i
where, for odd (even) values of j, we take k = 1 (2) and even (odd) M . However, as regards obtaining expectation value of (b†j bj+m + H.c.) for m > 1, the simple procedure (involving checkerboard decomposition) given above is not applicable; moreover, other suggested procedures in the literature are complicated [202]. Here, we propose an alternate simple method for evaluating hb†j bj+m + H.c.i for m > 1 and thus obtaining the BEC occupation number n0 =
1 X hΨ0 |b†i bj |Ψ0 i, N i,j
(4.10)
with |Ψ0 i being the ground state. To the WQMC method mentioned above, we add our trick to construct |Ψ0 i as a linear combination of the basis states |φi i in P P ai |φi i with a2i = 1. Once the occupation number representation, i.e., |Ψ0 i = i
i
we get a good estimate of the ground state |Ψ0 i, we can calculate the expectation values of any operator. After equilibrium (which is attained after several QMC passes), we run the simulation for a sufficient number of QMC passes and store the basis states corresponding to time slices L and L + 1 in each pass. It is obvious that some of the basis states will occur more frequently. The frequency of occurrence of a basis state |φi i is proportional to the probability (a2i ) of its occurrence in the expansion of the ground state |Ψ0 i. Now, the coefficients ai can be taken as real because the Hamiltonian is real and consequently |Ψ0 i can also be taken as real. Furthermore, all ai can be taken to be positive for the following reason. Firstly, the expectation values of NN and NNN interaction terms remain unaffected by
69
the sign of ai . Next, the expectation value of the hopping term is given by − T hΨ0 |b†l bl+1 |Ψ0 i = −T = −T = −T
X i,j
X i,k
X
hφi |ai (b†l bl+1 )aj |φj i] hφi |ai ck |φk i] ai c i .
(4.11)
i
This value is minimised when ai and ci have the same sign. Then, if we take ai P ai |φi i, we can take all to be positive for all i, ci > 0 for all i. Thus in |Ψ0 i = i
ai to be positive and real. Let |Ψi i and Ei be the eigenstates and the eigenenergies of the Hamiltonian with E0 being the ground state energy. For sufficiently large β, we approximate the ground state by |Ψi =
X i
r
hφi | exp[−βH]|φi i |φi i, Z
(4.12)
because then |Ψi =
X
≈
X
i
i
s
hφi |
r
hφi |Ψ0 i exp[−βE0 ]hΨ0 |φi i |φi i Z
P
j
|Ψj ihΨj | exp[−βH] Z
X ≈ hφi |Ψ0 i|φi i = |Ψ0 i,
P
k
|Ψk ihΨk ||φi i
|φi i
(4.13)
i
P since the partition function Z = i hΨi | exp[−βH]|Ψi i ≈ exp[−βE0 ]. Our WQMC approach to n0 has been benchmarked against the modified Lanczos method for small system sizes (see Fig. 4.7). The number of passes needed to estimate |Ψi turns out to be an order of magnitude larger than that needed for obtaining correlation functions by WQMC. We take |Ψi to be the state that produces an estimate of the kinetic energy hΨ|K|Ψi (with K being the kinetic energy operator) that is closest to the usual WQMC estimate
70
hhφi | exp[−βH]K|φi i/hφi | exp[−βH]|φi iiQMC where hiQMC denotes a quantum Monte Carlo average over various states |φi i.
4.6
Conclusions
In this chapter, we have analysed the correlated NN singlet phase predicted by the effective Hamiltonian of the Hubbard-Holstein model by essentially mapping the Hamiltonian onto the well-understood one-dimensional t-V model with large repulsion. Because the physics is dictated by the t-V model, we find that CDW and superfluidity occur mutually exclusively with CDW resulting only at n = 1/3 while superfluidity manifests itself at all other fillings. We also show that the √ the BEC occupation number n0 for our model scales as N similar to the n0 for a HCB tight binding model; additionally, we demonstrate numerically (using a new WQMC method and a modified Lanczos algorithm), at n 6= 1/3, that the n0 for our model is smaller than the n0 for a HCB tight binding model. We close by observing that, while CDW and superconductivity seem to be incompatible in the one-dimensional HH model, experimental results (such as those reported in Refs. [18; 20; 209]) suggest that they can coexist in higher dimensions. Furthermore, the vanishing of BEC fraction for the HH model is again an artifact of the one-dimensionality and should make way to non-zero fractions for higher dimensions just as in the case of the xxz model [205].
71
Chapter 5 Organic polaritons: First order normal to superradiant transition As discussed earlier in Sec.1.4.5 and 2.4, this chapter provides a theoretical model of organic polaritons in an optical microcavity with active organic materials like anthracene or cyanine dye J-aggregates where molecular vibrations are important. This model treats the organic molecules as saturable absorbers (two-level systems) of which electronic excitations (excitons) interact simultaneously with a single mode photon and the local molecular vibrations (phonon). This is basically an extension to the Dicke model[179], allowing us to work with higher excitation densities. The hamiltonian for the model which we term the Dicke-Holstein (DH) model is, Hdh = ωc ψ † ψ +
X
g(b†n an ψ + ψ † a†n bn ) + Ωφ†n φn
n
+
(
En + 2
√
) # SΩ † φn + φn (b†n bn − a†n an ) 2
(5.1)
Here b†n and a†n are the creation operators for excited and ground states respectively for nth molecule (considered as two-level system) with the hard-core condition: b†n bn + a†n an = 1 and N is the number of sites. Each molecule is coupled to a single photon mode with frequency ωc and represented by creation oper-
72
ator ψ † . The molecule-photon coupling strength is denoted by g at each site. The molecules after absorbing a photon get excited and emit the photon when come to ground state. To this we include the coupling expressed in terms of the Huang-Rhys parameter S, between the two-level systems and the local vibrations ∝ (φ†n +φn ) of the molecules along with the free Einstein oscillator with frequency Ω. The exciton has energy En . We consider the effects of strong molecular exciton-phonon coupling on collective behavior. We investigate how the phase transition to a superradiant state[210; 211; 212] at finite density[181; 182] is modified by the coupling to phonons, employing two approaches.While the experiments are currently far from equilibrium, we focus in this chapter on the equilibrium behaviour, in order to establish a basis from which the out-of-equilibrium physics can be compared. Firstly, we analyse the model given in Eq.(5.1) directly using a mean-field theory within path-integral approach. Secondly, we use the same formalism but with the hamiltonian after a canonical transformation called the Lang-Firsov transformation[148] which does not change the system but changes the previous mean-field theory giving an extra variational parameter to minimise the free energy.
5.1
Part I: Analysis with Direct Hamiltonian
In this section we investigate the model given in Eq.(5.1) using a coherent state path-integral approach within saddle point analysis i.e. a mean-field theory because in the strong coupling limit the photon and phonon states can be considered √ as coherent states. Note that the shift of the oscillator φ → (φ − S/2) is equiv′ alent to a relabelling of the exciton energy En = En − SΩ. This is useful in describing the results for different values of exciton-phonon coupling constant S, shown below.
5.1.1
Coherent state Path integral formulation
We consider the grand canonical partition function Q for the hamiltonian defined by Eq.(5.1) using the chemical potential µ to constrain the total number of
73
excitations L as, Q = T r[e−β(Hdh −µL) ] 1X † L = ψ†ψ + (b bn − a†n an ) 2 n n
(5.2)
The coherent-state path-integral representation for the partition function Q can be given as constrained functional integral, Q = S =
Z Z
D[ψ]
Z Y n
β 0
D[φn ]D[ηn ]δ(¯ ηn ηn − 1)e−S
"
(5.3)
# X ¯ τ +ω dτ ψ(∂ ˜ c )ψ + {φ¯n (∂τ + Ω)φn + η¯n Mn ηn } n
We have introduced the Nambu spinors on each site to simplify the notation as, bn an
ηn =
!
The matrix Mn is given by
Mn =
∂ τ + δn
gψ
g ψ¯ ∂τ + δn √
¯
˜ c = ωc − µ and ǫ˜n = (En − µ). Here we have defined δn = ǫ˜n + SΩ(2 φn +φn ) , ω Now transferring the fermionic and phononic integrals into the action, we can write the partition function Q in terms of an effective action S[ψ] as, Q = S[ψ] =
Z
Z
′
D[ψ]e−S[ψ] β 0
¯ τ +ω dτ ψ(∂ ˜ c )ψ −
(5.4) X
Kn
(5.5)
n
In the above expression for Kn , we perform functional integrations over fermionic
74
operators ηn , Kn = ln = ln
Z
Z
Dφn e
−
Dφn e
−
Rβ
dτ φ¯n (∂τ +Ω)φn
Rβ
dτ φ¯n (∂τ +Ω)φn +tr
0
0
det(Mn )
ln(Mn )
(5.6)
We present the results for the homogeneous system where each exciton has the same energy E, but shifted due to phonon coupling to E ′ and the phonon displacements are of same amplitude, independent of site index n.
5.1.2
Saddle point analysis
The mean field approximation for the photon field is quite good. Also for strong phonon coupling S, we can treat the phonon field to have a stationary mean field value φ0 , neglecting the phonon fluctuations. So the electron and phonon contribution to the mean-field action is given by, − K[ψ, φ0 ] = β φ¯0 Ωφ0 − ln det(M0 )
(5.7)
where the matrix is given by,
and δ =
√ ¯ ǫ˜+ SΩ(φ+φ) . 2
M0 =
∂τ + δ g ψ¯
gψ ∂τ − δ
Now from Eq. (5.5) we get,
S[ψ, φ0 ] =
Z
β 0
¯ τ +ω dτ ψ(∂ ˜ c )ψ − N K[ψ, φ0 ]
(5.8)
Here first to make the single mode photon field consistent with the summation √ over exciton sites, we can rescale ψ → ψ N . Treating the photon field to have a stationary mean-field value, we calculate the asymptotic form of the partition √ function by taking the stationary values of the fields (i.e. ψ = λ N , φ0 = α which are taken to be real). So the partition function in terms of saddle point
75
action S[λ, α] for the homogeneous system is given by, Q ≈ e−N S[λ,α] ;
S[λ, α] = β ω ˜ c λ2 + Ωα2 − ln detM0
(5.9)
Here the last term of the above expression of S[λ, α] can be evaluated by summing over the fermionic Matsubara frequencies ωm . Remember that the calculation of the partition function Q should be restricted to only physical states by inserting a phase factor in the expression for Q which causes the cancellation of the contributions of zero occupied and doubly occupied sites and the partition sum only includes the physical states of singly occupied sites. Due to Popov and Fedotov[213], this is just equivalent to shift the Matsubara fermionic frequencies as ωm = (2m + 3/2)π/β. In Matsubara frequency representation, ln detM0 =
X ωm
ln (iωm )2 − δ 2 − (˜ g λ)2
= ln[2 cosh(βξ)] where ξ =
p δ 2 + (˜ g λ)2 ; δ =
√ ǫ˜+2 SΩα 2
(5.10) (5.11)
√ and g˜ = g N
Free energy and minimisation conditions: Finally the mean-field free energy is given by, F N
1 = − lnQ0 β = ω ˜ c λ2 + Ωα2 −
1 ln[2 cosh(βξ)] β
(5.12)
We now minimise the free energy given in Eq. (5.12) with respect to the variational parameters λ and α and the minimisation conditions respectively are given by: tanh(βξ) ω ˜ c λ = g˜2 λ 2ξ √ Sδ tanh(βξ) α = 2ξ
76
(5.13) (5.14)
Here we notice that λ = 0 is always a solution to these equations. The density equation: Now we find the expression for the total excitation density ρ = hLi corresponding to the chemical potential µ. From the partition function Q0 at the saddle point, we have the usual formula, 1 ∂ lnQ0 βN ∂µ 1δ = λ2 − tanh (βξ) 2ξ = λ 2 + ρx
ρ =
(5.15) (5.16)
where ρx is the exciton density.
5.1.3
Results and Discussions
Now we extract the mean-field results by solving the self-consistent equations for λ and α given in Eq.(5.13) and (5.14) respectively, allowing us to express the phonon field α in terms of the chemical potential µ and other parameters as, α=
√
Sω ˜ c (˜ ωc − ∆s ) 2(1 − SΩ˜ ωc )
(5.17)
where the bare detuning ∆ = ωc − E, the energy difference between the cavity and exciton mode, has been used to define ∆s = (∆ + SΩ). Also the phonon displacement in Eq.(5.14) can be expressed in terms of exciton density ρx as, √ Sρx α=− 2
where
ρx = ρ − λ 2
(5.18)
The sole effect of the exciton-phonon coupling is to modify the detuning which depends on the exciton-phonon coupling strength S and the exciton density ρx as, ∆0 = ∆s + 2SΩρx
77
(5.19)
We notice that the effective detuning ∆0 has the same value ∆ for all S when there are no excitations of any kind i.e. at λ = 0 and ρx = −0.5. That was the motivation to subtract the S-dependent term from the exciton level E, so that we can compare the results for different S. Here ∆0 grows linearly with the exciton density ρx for a given value of S, and this is of course a physical effect the Holstein polaron. Now taking the value of α in Eq.(5.17), we can solve the Eq.(5.13) for λ 6= 0 as, √ 1 2 2 tanh β δ + λ ω ˜c = √ 2 δ 2 + λ2 √ ω ˜ c − ∆s + 2 SΩα δ= 2
(5.20) (5.21)
Here we have scaled all the energy parameters by the exciton-photon coupling g˜. For a given set of parameters (i.e. S, ω ˜ c , Ω, β, ∆), we can solve this transcendental Eq.(5.20) for ω ˜ c as a function of photon field λ and we use these values to calculate the the total density of excitations.
5.1.4
Zero temperature case
At zero temperature, we have tanh(βξ) = 1 and free energy in Eq.(5.12) becomes, F0 =ω ˜ c λ2 + Ωα2 − ξ N
(5.22)
and the Eq.(5.17) for phonon field is also valid for zero temperature. We should mention that these results at zero temperature can be deduced by variational mean-field theory by taking the wave function ansatz as the coherent state for photon and phonon field and BCS like wave function for the fermions: |λ, α, u, vi = eλψ
† +αφ†
(vb† + ua† )|0i
(5.23)
where |0i is the vacuum state with no particles in any level of the excitons and λ = hψi, α = hφi, u and v all are taken to be real with u2 + v 2 = 1. For the normal state (NS) when the photon condensate λ = 0, we have the √ phonon field α = ± S/2 of which the positive solution gives the minimum energy
78
which is from Eq.(5.22) [omitting the exciton energy shift i.e. E ≡ E ′ ], ΩS |˜ ωc − ∆| Fn =− − N 4 2 5.1.4.1
(5.24)
Condensate state λ > 0
For superradiant (SR) state λ > 0, using Eq.(5.17) and (5.21), we solve the photon field λ in Eq.(5.20) for a given ω ˜ c = ωc − µ: 1 ω ˜c = √ 2 δ 2 + λ2
(5.25)
λ
We present the results by considering two different detuning ∆ = 1 and −2. Fig.5.2(a) shows the variation of photon field λ and the corresponding variation in chemical potential µ (measured with respect to photon mode ωc ) with total excitation density ρ in Fig.5.2(b). 1.0 The S = 0 curves are identical to S=0, ∆=0 previous results, and let’s recapitulate 1 2 3 the qualitative behavior because it will 0.5 guide the discussion of the further results. If the detuning is small enough, the growth of the order parameter λ is 0.0 -0.5 0.0 0.5 1.0 monotonous with excitation density ρ, ρ but when ∆ ≥ 2 the photon compoFigure 5.1: This shows the variation of nent grows and then shrinks to zero λ with ρ for different detuning ∆ with at ρ = 0.5 due to saturable nature S = 0. of excitons, growing beyond that (see Fig.5.1). This is called in different contexts a Mott lobe, because for low excitation number, the polaritons are essentially excitonic in character − one has exciton condensation mediated by photon fluctuations, and so the condensate is suppressed when the exciton level is filled. At ρ > 0.5, the chemical potential jumps abruptly to just below the photon frequency ωc , and the condensate is now photon-like. Now due to the phonon coupling, we have an effective detuning ∆0 defined in Eq.(5.19) which is increased with exciton density ρx and S. So starting with
79
1.0
-0.4 (a)
(b)
Ω=0.5, ∆=1, T=0
-0.8
λ
0.6
S=0.0 S=0.9 S=1.0 S=1.1 S=1.2 S=1.3
µ−ωc
0.8
-0.6
-1.0 -1.2
0.4
-1.4 0.2 -1.6 0.0
-1.8 -0.4 -0.2 0.0
0.2
0.4
0.6
0.8
1.0
-0.4 -0.2 0.0
ρ
0.2
0.4
0.6
0.8
1.0
ρ
Figure 5.2: (a) This shows the variation of λ with excitation density ρ for ∆ = 1. (b) This shows the corresponding chemical potential µ − ωc a detuning less than the critical detuning ∆c = 2 for re-entrant to occur, at sufficiently large S and appropriate ρx , the effective detuning ∆0 can cross ∆c to give a re-entrant behavior. To make the case, we consider the case ∆ = 1 and S nonzero, but small. Then the effective detuning ∆0 given in Eq. (5.19) will be close to ∆ which gives the polariton states with an increasing photonic character with increasing excitation, i.e. the corresponding chemical potential µ is gradually increased to the cavity mode with the density ρ just like S = 0 case shown in Fig.5.2(b). For larger S, around ρ = −0.5, effective ∆0 is still close to ∆. As ∆0 is linear in ρx , it has maximum value at maximum ρx (= 0.5). So from Eq.(5.19), the value of S at which re-entrant occurs is given by SΩ = (∆c − ∆)/2. For ∆ = 1, the value of SΩ = 0.5 for a re-entrant to occur which is shown in Fig.5.2(a) and corresponding jumps in chemical potential µ, shown in Fig.5.2(b). Further increase in S makes the SR state unstable to form the NS which has lower energy than SR energy as shown in Fig.5.3(a) for S = 1.3. In the √ NS, the chemical potential decreases as µN S ∝ − SΩ ρ. This is because the 2 system gets filled up with only excitons in this case which in turn renormalises the exciton energy downward continuously by increasing the effective detuning
80
1.0
0.0 Red :SR Blue:NS
0.5
-1.0
0.0
-2.0
-0.5
-3.0 (a)
-1.0
(b)
-4.0 Ω=0.5, ∆=1, S=1.3
-1.5 -0.5
0.0
0.5
Ω=0.5, ∆=-2, S=7 -5.0 -0.5
1.0
ρ
0.0
0.5
1.0
ρ
Figure 5.3: (a) This shows the energy solutions for SR (Red) and NS (Blue) for ∆ = 1, Ω = .5, S = 1.3, and (b) shows the corresponding energy solutions for ∆ = −2, Ω = .5, S = 7 ∆0 with exciton density ρx ≡ ρ. So the chemical potential µ goes down with ρ, giving the negative compressibility which makes the system unstable to phase separation. As we increase the density above ρ = 0.5 we get again a photon band with fully occupied exciton band and correspondingly chemical potential µ jumps very close to to the cavity mode. For negative detuning ∆ = −2, at small excitation densities, we have an essentially photon-like condensate at small S with monotonous increase in chemical potential as shown in Fig.5.4, because the exciton-band is well above the cavity mode. But if S > 4(effective detuning ∆0 crosses the critical detuning ∆c at S = 4), there is a regime of intermediate density where a re-entrant excitondominated condensate is stable, with the effective exciton level renormalised downward by the coupling to phonons so that the effective detuning becomes positive. For large enough S, again the NS energy becomes lower than the SR state as shown in Fig.5.3(b) for S = 7 and ∆ = −2. Although for positive detuning, despite the appearance of the Mott lobe, there is always a single stable solution, but in the case of negative detuning, the NS energy solutions jumps abruptly from a higher value to a lower one at ρ ≈ 0.15 and hence a first-order transition between two normal states as shown in Fig.5.3(b). Again, when the exciton states are filled, the chemical potential jumps to near the cavity mode,
81
1.2
0.0 (b)
(a)
λ
0.8
-1.0 -1.5 -2.0 µ−ωc
1.0
-0.5
Ω=0.5, ∆=-2, T=0 S=0 S=3 S=4 S=5 S=7
0.6
-2.5 -3.0
0.4
-3.5 -4.0
0.2
-4.5 0.0 -0.5
0.0
0.5
1.0
-5.0 -0.5
1.5
ρ
0.0
0.5
1.0
1.5
ρ
Figure 5.4: (a) This shows the variation of λ with excitation density ρ for ∆ = −2. (b) This shows the corresponding chemical potential µ − ωc and the photon population rises again.
5.1.5
Finite temperature case
We calculate the inverse critical temperature βc = 1/Tc for condensation without exciton energy broadening (i.e. for homogeneous system) in the limit of λ → 0 using Eq.(5.13) and (5.16) as, 2˜ ωc
q = ∆0 ± ∆20 − 8ρ
βc =
−1
4 tanh (2ρ) p ∆0 ∓ ∆20 − 8ρ
(5.26) (5.27)
This is the same expression[181] as Dicke model with just the modified detuning ∆0 due the exciton-phonon coupling. We calculate the critical temperature Tc = 1/βc for different values of density ρ ≡ ρx asking Tc to be real and positive and calculate corresponding chemical potential µ (measured with respect to ωc ). In Fig.(5.5), we present phase boundaries calculated from Eq.(5.27) and (5.26) for the homogeneous system separating superradiant (SR) state and normal state (NS). Here again we intend to show a re-entrant behavior with ∆ = 1 as we
82
0.7
0.7 (b)
(a) 0.6
Ω=0.5, ∆=1
0.5
S=0.0 0.8 1.0 1.2
0.6 0.5 0.4
Tc
Tc
0.4
0.3
0.3
0.2
0.2 SR
0.1
0.1
0.0 0.0 -0.5
-1.6 0.0
0.5
ρ
-1.4
-1.2 -1.0 µ−ωc
-0.8
-0.6
Figure 5.5: (a) This shows critical temperature Tc − ρ second order phase boundary for ∆ = 1. (b) This shows the corresponding Tc − µ phase boundary. increase phonon coupling S. When S = 0, as usual the transition temperature monotonically increases with density (shown in Fig.5.5(a)), and for ρx > 0 the system is always condensed as it requires the chemical potential to cross the cavity mode to exceed this density. But for S ≥ 1 as discussed earlier, we find a re-entrant behavior. The saturable nature of the excitonic states causes this re-entrant behavior. This can be explained by considering the limits ρx → ±0.5 for large detuning ∆ or smaller ∆ but large S such that effective detuning ∆0 becomes large. Around ρx = −0.5, the normal state contains few excitons weakly interacting with photon and are condensed when their density crosses the critical value set by the effective interaction strength. Similarly around ρx = 0.5, the excitonic levels are constrained to be filled completely, the NS contains few holes in an otherwise saturated excitonic states. The holes again interact with the photon and condense when the density exceeds a critical value. In Fig.5.5(b), we present the corresponding T − µ phase boundary. Due to our definition of effective detuning ∆0 = ∆ for all S at the minimum density (ρmin = −0.5), we have all curves starting from same point. Here also the reentrant behavior is shown. The shrinking of SR bubble corresponds to jump in chemical potential at ρ = 0.5.
83
5.2
Part II: Analysis with Lang-Firsov transformed hamiltonian
Till now we have figured out that exciton-phonon coupling can cause a re-entrant behavior by reducing the exciton energy i.e. the optical transition energy between the upper and lower states of the molecule, even if the detuning ∆ is lower than the critical detuning ∆c = 2 for a re-entrant to occur. For negative detuning ∆ = −2 and large enough S, we find a first order transition between two normal states. But there is no indication of first order normal state (NS) to superradiant (SR) state transition. So, next we analyse the same model in Eq.(5.1) by same path-integral approach, but after a variational canonical transformation. This analysis gives indeed a first order NS to SR state transitions, not present in earlier analysis with direct hamiltonian as the phase boundaries shown in Fig.5.5 between the NS and SR state are second order boundaries and calculated in the limit of λ → 0.
5.2.1
Variational Lang-Firsov transformation
We now analyse the model given in Eq.(5.1) after a variational Lang-Firsov transformation[148] which gives an extra parameter η to adjust to minimise the free energy. This is an exact transformation preserving the properties of the the system but does change the mean-field theory. Again such an approach is valid when the vibrational states are well described by coherent states, i.e. when the typical phonon numbers, controlled by S, are large compared to 1. We want the transformed hamiltonian as, √ ˜ dh = eK Hdh e−K where K = η S(b† b − a† a)(φ† − φ) H 2
(5.28)
Under this transformation, the different operators get changed as (omitting site index n as we are again considering the homogeneous system), √ b† a → η S(φ† − φ)b† a √ φ → B − η S(b† b − a† a)
84
(5.29) (5.30)
Using these transformed operators, we get from Eq. (5.28), i ˜ dh √ 1h H † † = ωc ψ ψ + E + Ω S(1 − η)(φ + φ) (b† b − a† a) N 2 √ √ ΩS † † + g N (b aψeη S(φ −φ) + h.c) + Ωφ† φ − η(2 − η) 4
(5.31)
The form of this transformed hamiltonian confirms that the usual polaron transformation i.e η = 1 makes the exciton-phonon coupling disappear at the expense of the dressed excitonic operators with exponential phonon operators and also system gains polaronic energy ΩS/2. Here we have omitted the polaron shift in exciton energy, introduced by hand in Eq.(5.1) to compare the earlier results conveniently for different S. Free energy and energy minimisation conditions: Using the same approach outlined earlier, we get the phonon mean-field free energy for homogeneous system, F ΩS 1 =ω ˜ c λ2 + Ωα2 − η(2 − η) − ln[2 cosh(βζ)] N 4 β
(5.32)
with the parameters defined as, p ζ = δ 2 + (˜ g λ)2
√ ǫ˜ + 2Ω S(1 − η)α δ= 2
(5.33)
And now the parameters defined as ǫ˜ = E − µ and the reduced exciton-photon √ 2 coupling g˜ = g N e−η S/2 which is coming from the fact that the hamiltonian should be normal ordered in path integral formulation. So we have used, eη
√
S(B † +B)
= eη
√
√ SB † −η SB −η 2 S/2
e
e
We now minimise the free energy given in Eq.(5.32) with respect to the vari-
85
ational parameters λ, α and η. respectively, tanh(βζ) ω ˜ c λ = g˜2 λ 2ζ √ Sδ(1 − η) tanh(βζ) α = 2ζ √ √ Ω S tanh(βζ) (1 − η) = (δΩα + g˜2 λ2 Sη) 2 ζ
5.2.2
(5.34) (5.35) (5.36)
Zero temperature case
Let us again consider normal state at zero temperature i.e. tanh(βζ) = 1 and λ = 0. The minimisation conditions (5.35) and (5.36) become identical which of course has one solution as α = 0 ⇒ η = 1. So minimisation gives a global minimum of free energy which is exactly the same as the previous calculation in Eq.(5.24). This means that normal state properties are same in both formalisms. 5.2.2.1
Condensate state, λ > 0
Again for zero temperature we put tanh(βζ) = 1 in all three minimisation conditions given in Eq.(5.34), (5.35) and (5.36). Now using Eq. (5.35) we put the value of α in Eq. (5.36) and solve it for η and subsequently for α as, Ω η= 2ζ + Ω
√ δ S α= 2ζ + Ω
(5.37)
Substituting the values of η and α in Eq.(5.33) we get two coupled non-linear self consistent equations as, δ=
ǫ˜ 2ΩSδζ + 2 (2ζ + Ω)2
√ − 1 Ω2 S g˜ = g N e 2 (2ζ+Ω)2
(5.38)
From Eq.(5.34) we deduce the gap equation as, g˜2 = 2˜ ωc ζ
86
(5.39)
The free energy for condensate state is given by, FSR = ω ˜ c λ2 +
Ωδ 2 S Ω2 S(4ζ + Ω) − ζ − (2ζ + Ω)2 4(2ζ + Ω)2
(5.40)
This approach reproduces the variational mean-field results by considering an wave function ansatz[214]. The most interesting feature in this case is a first order phase transition between two superradiant states which occurs as the light-matter coupling g is varied.
5.2.3
Finite temperature case
At finite temperature, again we use the Eq.(5.35) and (5.36) to eliminate α and find the value η and subsequently α as, Ω η= 2ζκ + Ω where we define κ =
√ δ Sκ tanh(βζ) α= 2ζκ + Ω
(5.41)
g˜2 λ2 tanh(βζ) . ξ 2 −[δ tanh(βζ)]2
Now the self-consistent equations for effective molecular transition frequency δ and phonon-suppressed optical coupling g˜ become, δ=
√ − 1 Ω2 S g˜ = g N e 2 (2ζκ+Ω)2
ǫ˜ 2ΩSδζκ2 tanh(βζ) + 2 (2ζκ + Ω)2
(5.42)
And finally gap equation is gven by, g˜2 tanh(βζ) = 2˜ ωc ζ 5.2.3.1
(5.43)
Phase diagram at λ → 0
If the transition to superradiant state is continuous, we can find the transition by looking for the appearance of a nonzero value of photon field λ. Thus we consider here the condensate state in the limit of vanishingly small but nonzero λ → 0. In this limit and at finite temperature κ → 0 and ζ → |δ|. So the self-consistent
87
Eq.(5.42) becomes, δ=
ǫ˜ 2
√ g˜ = g N e−S/2
(5.44)
And using Eq.(5.43) the gap equation is given by, gc2 =
√ gc = g N
2|δ|˜ ωc eS ; tanh(β|δ|)
(5.45)
In the limit ǫ˜ = 0 i.e. as ω ˜ c is approached to detuning ∆ = ω ˜ c −˜ǫ, critical coupling constant becomes, gc2 =
2˜ ωc eS β
(5.46)
So unlike the zero temperature case where at ω ˜ c = ∆, the critical coupling gc = 0 (condensate always forms for any value of coupling g), we get gc moved up depending on temperature T = 1/β and exciton-phonon coupling S as given in Eq.(5.46) which of course gives gc = 0 at zero temperature shown in Fig.5.6. 5.2.3.2
General solutions with λ 6= 0
We now proceed to the general solutions at finite temperature with λ 6= 0 case. For this again we use the self-consistent Eq.(5.42) and the gap Eq.(5.43) along It turns out that these with the free energy expression given in Eq.(5.32). equations may have one, three or five solutions. Of course one of these is the absolute minimum. At zero temperature and for large enough S, variational calculations suggest a first order transition between two superradiant phases when ˜c we vary the exciton-photon coupling[214]. This transition happens only when ω approaches to ∆ = ω ˜ c − ǫ˜ i.e. at the critical point ǫ˜ = 0. But the question we will address is what happens to this first order transition at finite temperature and whether there is a first order transition from normal (NS) to a superradiant (SR) state , as discussed below. Let us first recapitulate the first order transition between two SR states at zero temperature case[214] which is reproduced by this path-integral approach. At the critical point ω ˜ c = ∆, one SR branch with lower energy than NS, always
88
gc
exists as the critical coupling strength gc = 0 at this point shown in Fig.5.6. Here gc gives the second order phase boundary between NS and SR states. So if there exists another SR branch starting above NS energy and moves 6 T=0.0 0.1 down below NS as well as the previous 5 0.2 0.5 SR energy as we vary g, then we can 4 S=1, ∆=4 have a crossing of two SR states and 3 hence a first order transition between 2 two SR states. But at finite tempera1 ture, the second order phase boundary 0 -6 -5 -4 -3 -2 -1 0 is moved up enough (from gc = 0) acµ-ωc cording to Eq.(5.46) by the temperaFigure 5.6: The critical exciton-photon ture and phonon coupling, making one coupling separating normal and superraSR branch disappear and so there can diant (SR) states at different temperabe the crossing between remaining SR tures by second order phase boundaries. branch and NS energy and hence 1st order NS to SR transition. The existence of a first order NS to SR state transition can be physically understood by considering the extent of polaron formation at large S, i.e. entanglement of the phonon and electronic states of the molecule. The equation for η in Eq.(5.41), describing the extent of polaron state formation is instructive. At small λ or large Ω, η → 1, and one has fully developed polarons (vibrational state fully entangled with the electronic state, as in the absence of any applied field). If, on the other hand, λ is large then η → 0 and the polaron formation is suppressed. However, an additional level of self-consistency appears in the current problem, not present for normal variational polaron approaches. The photon field λ depends on the polarisation of the molecules, and the effective coupling strength g˜. Therefore, when the bare coupling g is small, the photon field is small, and polarons are well developed, further suppressing the effective coupling g˜. At larger couplings, polaron formation is suppressed, producing a stronger coupling. At zero temperature, this leads to a jump within the condensed phase, between a weakly and strongly polarised phase[214]. At finite temperature the same effect leads to the first order normal to condensed phase transitions, as a competition
89
Figure 5.7: The free energy is plotted at very low temperature T = 0.0001, Ω = 1, S = 6, ∆ = 4, to discuss qualitative behaviour of the system. The blue horizontal lines represent the NS energy. Left panel: For ωc = 3.9, the gap in figure is due to the condition ζ ≥ |δ|. Right panel: For ωc = 3.99, the energy gap is almost closed and the inset shows the zoomed-in portion around very small ζ, where the minima below NS energy are developed. between states with different values of η. For quantitative analysis of this transition at finite temperature, it is very convenient to express the free energy in Eq.(5.32) as a function of g and ζ with the help of the self-consistent Eq.(5.42) and the gap Eq.(5.43) as follows, v u u 2ζκ + Ω = t
Ω2 S ln
h
g 2 tanh(βζ) 2˜ ωc ζ
i
(5.47)
Once we have κ(ζ, g), we can subsequently calculate δ(ζ, g), η(ζ, g) and α(ζ, g) to investigate finally the free energy as a function of ζ and g. Fs (ζ, g) = ω ˜ c λ2 + Ωα2 −
ΩS 1 η(2 − η) − ln[2 cosh(βζ)] 4 β
(5.48)
Just to remember that NS free energy is given by Fn = −
ΩS 1 − ln(2 cosh(β|δ|)) 4 β
90
(5.49)
Fs
This is basically an energy minimisation problem and we find the value of ζ which gives the global minimum of free energy in Eq.(5.48), for a given coupling g and other parameters. At very low temperature T → 0, we plot the free energy Fs (ζ, g) as a function of ζ for different values of g, shown in Fig.5.7. The solutions are usually the global minima of each curve corresponding to different g. The gaps in the energy curves are due to the condition ζ ≥ |δ|. For ω ˜ c = 3.9 in Fig.5.7(Left panel), the solutions corresponding to global minima appear above the NS energy and start moving down with increasing g but stop at some energy well above the NS energy. The global minima reappears again just below the NS energy at around g ∼ 4.4 and continues to move down with increasing g. This corresponds to a second order phase transition. Again for ω ˜ c = 3.99 in Fig.5.7(Right panel) i.e. very close to critical point ǫ˜ = 0, the description is the same as earlier except now the gap in energy is reduced to almost zero, which will be the case exactly at ǫ˜ = 0. But now there develops another minima below the NS energy at very small ζ, which minimise the energy globally upto the value of g ∼ 4.4 and after that the global minima corresponds to big ζ. That means there is a crossing of two SR states around g ∼ 4.4, hence a first order phase transition between two SR states. Now at finite temperature, the sec-1.2 ond order boundary gc near critical ∆=4, ωc-µ=4.0, Ω=1, S=6, T=0.1 -1.4 point ǫ˜ = 0 (where gc ∼ 0 at T = 0), -1.6 is moved up according to Eq.(5.46). 6.0 5.6 So at sufficiently high temperature and -1.8 5.2 g1 for large S, critical coupling gc can be 4.8 -2.0 T 4.4 pushed far away (i.e. zero temperature 0.0 0.5 1.0 -2.2 minima corresponding to small ζ below 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 g NS energy disappear at finite temperature) and the energy minima correFigure 5.8: The typical free energy exsponding to big ζ gradually cross the trema plot at T = 0.1 with other paramNS energy from above and continue to eters given in the figure. The first order move down. So there exists a first orphase boundary where the SR state and der transition from NS to SR state as NS energies cross, is shown here. we very g. In Fig.(5.8), the typical free
91
√ Figure 5.9: Comparison between critical g N from exact on-site diagonalisation (left) and phonon mean-field path-integral approach (right) for S = 6, Ω √= 1, ∆ = 4, temperatures as indicated. The colour scale shows photon field λ/ N at the phase boundary. Light (yellow) colours imply second order transition whereas dark (blue) first order transition. NB: around resonance the transition is always first order, though sometimes weakly. This figure is taken from Ref.[215] energy extrema are shown as a function of g for the parameters indicated in the figure. While the red curve indicates the local maxima, the green curve representing the global minima that crosses the NS energy at g ∼ 4.57, hence a first order transition from NS to a SR state. In the inset, the transition point g1 for different temperature, are shown. The global minima of the free energy curve for a given g, occur at the values of ζ large enough to keep the factor tanh(βζ) to be very close to unity for the temperature range we are considering. So the global minima curves (green) do not change much with the temperature. But from Eq.(5.49), the NS energy is decreased linearly with temperature meaning by the SR and NS energy crossing point g1 increases with temperature. In the limit S = 0, the phonons are irrelevant, and the both phonon meanfield theory reduces to the Dicke model. As this mean-field approach assumes the phonon to be in coherent state, it is valid for large exciton-phonon coupling S. Figure 5.9 shows the g − µ phase boundary for different temperature, and how our mean-field approach matches with the exact on-site diagonalisation in the large S limit[215]. The exact on-site diagonalisation calculations are carried out by Justyna Cwik. The colour scale represents the value of the photon field
92
4 3 g√N 2 1 0
∆=4, Ω=1.0, S=3 -5
-4 -3 µ-ωc
-2
0
0.4 0.3 0.2 0.1 0
0.2
4 3 2 1 0
0.1 0
S=1
0.2 0.1 T -5
-4
-3
-2
0
0.2 0.1
Figure 5.10: The g − µ phase diagram for S = 3 (Left) and S = 1 (right) for the parameters Ω = 1 and ∆ = 4. The colour code represents the photon field λ at the transition, and hence the strength of first order transition.
Figure 5.11: This shows a colour map for the photon field λ on T − µ plane for the parameters S = 2, g = 2, ∆ = 4 and Ω = 0.1. λ at the transition, and hence measures the strength of the first order transition. The light (yellow) colours indicate the second order transition whereas the dark (blue) colours imply first order transition. In Fig.5.10, we present the same g − µ phase diagram for smaller S = 3 and 1 as indicated in the figure with same parameters Ω = 1 and ∆ = 4. Our phonon mean-field theory still predicts first order transitions but getting weaker as the first order jump in λ shown in the colour scale is reduced with the decrease in S and becomes second order as S → 0. At small T the strength of the first order jump is the largest at points near to, but not exactly at ω ˜ c = ∆. It is clearly visible in Fig.5.10. We show in Fig.5.11 the re-entrant behavior by presenting a T − µ phase
93
diagram with a colour map for the photon field λ for the parameters S = 2, g = 2, ∆ = 4 and Ω = 0.1. The analysis with direct hamiltonian also gives the reentrant behavior but the phase boundaries are of second order. But in this case we get a first order T − µ phase boundaries with the λ colour map determining the strength of first order transitions.
5.3
Conclusions
We have explored the effects of strong exciton-phonon coupling on the collective behavior of organic molecules (saturable absorbers as two-level systems) interacting with a single photon mode in an optical microcavity. After including the coupling of two-level systems with the molecular vibrations localized at each molecule, to regular Dicke model, we analyse what we call the Dicke-Holstein model. Firstly, we analyse this model using a mean-field theory within coherentstate path integral approach and find the re-entrant behavior producing the Mott lobes, even if the detuning is less than the critical detuning. This is due to the effect of exciton-phonon coupling which renormalises the exciton energy downward, giving an effective detuning increasing with the exciton density and the coupling constant S. For large enough S, the SR state becomes unstable and system gets filled up with excitons only, giving a negative compressibility and hence a phase separated state for both positive and negative detuning. However for large S and negative detuning, we find a first order transition between two normal states. But we find no indication of the first order NS to SR state transition in this approach. Secondly, we again investigate the model with the same approach but after a variational Lang-Firsov transformation of the hamiltonian. In this approach, we indeed find a first order NS to SR state transition in strong exciton-phonon coupling limit. We have shown that our results with mean-field approach match with the on-site exact diagonalisation for large S = 6. Our method continues to predict the first order transition with smaller S but with decreasing strength of first order jump in photon field λ at the transition. Only at S = 0 is the phase boundary everywhere second order.
94
Chapter 6 Conclusions This dissertation presents a discussion of two independent models featuring a common ingredient − the lattice vibrational modes (localised) i.e. the phonons, interacting strongly with electrons in one-dimensional Hubbard-Holstein model (which includes onsite e-e interaction too) and with excitons in Dicke-Holstein model (which also includes exciton-photon interactions). This chapter provides a brief summary of the conclusions of the dissertation and suggestions for possible future directions related to this work.
6.1
Hubbard-Holstein model in one-dimension
Chapter 3 presents a phase diagram, shown in Fig.3.8, at quarter-filling for the one-dimensional Hubbard-Holstein (HH) model in both the strong e-e and the strong e-ph interaction limit and in non-adiabatic regime. A correlated NN singlet phase occurs at various fillings in the phase diagram. Chapter 4 gives a detailed analysis of the correlated NN singlet phase. In contrast to earlier approaches, we utilise a controlled analytic approach
(that takes into account dynamical quantum phonons), to derive an effective electronic hamiltonian given in Eq.3.10 by averaging out the vibrational degrees of freedom within second order perturbation theory. This method uses both the strong e-e interaction and the strong e-ph cou-
pling limit and generates the longer range interactions (hopping to NNN
95
sites) in the effective hamiltonian (see Eq.3.10). It is shown that while the e-e interaction produces NN spin antiferromag-
netic (AF) interactions which encourage singlet formation; the e-ph interaction generates NN repulsion which is expected to promote CDW order. By analysing the probability of the occurrence of different cluster sizes and
by studying various correlation functions that result from our effective HH hamiltonian, we deduced the ground state phase diagram shown in Fig.3.8 at quarter filling and in the non-adiabatic regime We find that strong e-ph coupling stabilises a correlated NN singlet phase
in different fillings. This phase is quite distinct from a Peierls-like ( bondorder) wave and would be absent in the pure Hubbard model. We analyse this correlated NN singlet phase by essentially mapping the
effective electronic hamiltonian in Eq.3.10 onto the well-understood onedimensional t-V model with large repulsion. This also gives a distinct advantage to handle the bigger system sizes which would not be possible with effective electronic hamiltonian; furthermore, it also helped identify the numerically elusive KT transition. Because the physics is dictated by the t-V model, we find that CDW order
and superfluidity occur mutually exclusively with a CDW resulting only at 1/3 filling while superfluidity manifests itself at all other fillings. So CDW and superconductivity are incompatible in one-dimensional HH model. We have developed an easy way to calculate BEC occupation number n0
after modifying the usual WQMC method which is benchmarked against modified Lanczos algorithm. We show that n0 for our model scales as √ N , N being the system size, similar to the n0 for a HCB tight binding model. Additionally, we demonstrate numerically (using our modified WQMC method and a modified Lanczos algorithm), at a filling not equal to 1/3, that the n0 for our model is smaller than the n0 for a HCB tight binding model.
96
6.2
Dicke-Holstein model
Chapter 5 describes how the strong local exciton-phonon coupling S modifies the phase transition to the superradiant phase at finite densities in Dicke-Holstein model given in Eq.5.1, using a mean-field theory within path-integral approach. We employ two approaches to analyse the model. Firstly: The Dicke-Holstein model in Eq.5.1 is analysed directly with a mean-field
theory within coherent state path-integral approach. We find a re-entrant behavior producing a Mott lobe, even if the detuning is less than the critical detuning. This is due to the renormalisation of exciton energy downward due to exciton-phonon coupling, giving an effective detuning which depends on exciton density and phonon coupling S (see Eq.5.19) For large enough S, the SR state becomes unstable and the system gets
filled up with excitons only, giving a negative compressibility and hence a phase separated state for both positive and negative detuning. However for large S and negative detuning, we find a first order transition between two normal states. And secondly: We again consider the model with same approach but after a canonical vari-
ational Lang-Firsov transformation, which gives an additional variational parameter to adjust to minimise the free energy. We find a first order NS to SR state transitions at large exciton-phonon coupling, when we vary the exciton-photon coupling. Our results within mean-field approach match with onsite exact diagonalisa-
tion at large coupling S where the phonon state can be treated as coherent state, shown in Fig.5.9. Our method continues to predict the first order transitions with smaller S, but with decreasing strength of first order jump in photon field λ at the transition, shown in Fig.5.10. Only at S = 0 is the phase boundary everywhere second order.
97
6.3
Future work
This section gives a number of possible extensions to this work. Firstly regarding the HH model: As our formulation of the effective HH hamiltonian can be easily extended
in higher dimensions and given the immense interest in understanding the coexistence of or competition between diagonal and off-diagonal long-range orders, we could study two-dimensional HH model to see whether it permits the coexistence of CDW and superconductivity. In this respect we could also study other but related models such as t-J-Holstein model, variants of Hubbard-Holstein model which would include long-range interactions, etc. Furthermore, we could study systems with coexisting SDW and supercon-
ductivity such as FeAs-based high Tc superconductors. Additionally, the study of the possibility of coexistence of diagonal and off-diagonal longrange orders at the interface of heterostructures made of one material with diagonal long-range order while the other has off-diagonal long range order, would be interesting. Also the WQMC method that we have developed could be generalised to
study the 2D HH model, and it would be interesting to see if the correlated singlet phase we discovered earlier, survives. Secondly regarding the Dicke-Holstein model: We could study the non-equilibrium properties of the Dicke-Holstein model
for organic polaritons using Keldysh formalism which allows one to find the equation of motion for the non-equilibrium Green’s function of a system coupled to external baths. Along with the fundamental interests in room temperature Bose-Einstein condensation, organic polaritons provide a novel route to electrically pumped organic lasers with ultra-low thresholds[124]. This vibronic coupling indeed can play a complex and crucial role in energy transfer in light harvesting complexes[121; 122; 123].
98
The model for entangled light-matter states, namely polaritons, is also rel-
evant in describing the properties (Bose-Einstein condensation and lasing) of electronic excitations of trapped atoms, interacting with trapping photon field in an optical lattice. Also the formation of molecules from atoms in optical lattices manifests itself in modified polariton properties, e.g. an anisotropic optical spectrum. And of course the study of light-matter interaction is fundamental to quantum entanglement and decoherence phenomena. And finally regarding the exciton-phonon coupling, it would be very inter-
esting to study the effect of an exciton introduced into an electron gas, in a situation where the presence of the exciton can − via electron-phonon coupling − introduce a large shift in the fermionic energy level. Such a shift, if done by hand, would simply be a Kondo impurity, but here we have the opportunity to follow the dynamics of such a process under realistic conditions.
99
Appendix A The formulation of perturbative correction in electronic operator form
In this appendix, we will outline our approach to carrying out perturbation theory and obtaining the ground state energy. We assume a Hamiltonian of the form H = H0 +H1 where the unperturbed H0 has separable eigenstates |n, mi = |niel ⊗ |miph with |0, 0i being the ground state with zero phonons; the eigenenergies, (0) ph corresponding to |n, mi, are En,m = Enel + Em . Furthermore, the perturbation H1 is the electron-phonon interaction term of the form given in Eq. (3.6). After a canonical transformation[200], we obtain: ˜ = eS He−S H 1 = H0 +H1 +[H0 + H1 , S]+ [[H0 +H1 , S], S] + · · · . 2
(1)
In the ground state energy, we know that the first-order perturbation term is zero by construction (in fact, hn1 , 0|H1 |n2 , 0i = 0). To eliminate the first-order term in H1 , we set H1 + [H0 , S] = 0. Consequently, we obtain the matrix elements hn1 , m1 |S|n2 , m2 i = −
hn1 , m1 |H1 |n2 , m2 i . (En1 ,m1 − En2 ,m2 ) 2
(2)
We now assume that both NN hopping integral te−g and the Heisenberg spin
100
interaction strength J are much smaller compared to the phononic energy ω0 (0) which is true at large couplings g. Hence, we make the approximation (En1 ,m1 − (0) ph ph En2 ,m2 ) ≃ (Em − Em ); then, using Eqs. (1) and (2), we obtain: 1 2 ˜
phhm1 |H|m2 iph
+
1X ¯ ph phhm1 |H1 |mi 2 m¯
≃ phhm1 |H0 |m2 iph 1 1 ¯ 1 |m2 iph + ph . phhm|H ph ph ph Em E m1 − E m 2 − Em ¯ ¯
(3)
(2)
Next, it is important to note that the second order correction En,m , corresponding (0) to the unperturbed eigenenergy En,m , can be expressed as follows: (2) En,m =
X hn, m|H1 |mi ¯ ph m ¯
ph Em
¯ 1 |n, mi phhm|H ph − Em¯
˜ mi − hn, m|H0 |n, mi. (4) ≃ hn, m|H|n,
˜ 0i is the total energy that Furthermore, since hn1 , 0|H1 |n2 , 0i = 0, hn, 0|H|n, resulted from performing second order perturbation theory on the unperturbed (0) energy En,0 . Our procedure for finding ground state amounts to obtaining the ˜ 2 , 0i; this is equivalent lowest eigenvalue for the matrix with elements hn1 , 0|H|n to finding the ground state of the effective Hamiltonian He (as was done in Ref. [200]): He = phh0|H0 |0iph + H (2) ,
(5)
where H (2) =
X ph h0|H1 |mi ¯ ph × phhm|H ¯ 1 |0iph ph E0ph − Em ¯
m ¯
.
(6)
This procedure amounts to considering the restricted subspace spanned by eigenstates |n, 0i1 obtained from carrying out first order perturbation theory on |n, 0i: |n, 0i1 = |n, 0i +
X |mi ¯ ph
¯ 1 |n, 0i phhm|H , ph ph E0 − Em ¯
m ¯
101
(7)
It is important to recognize that the state |n, 0i1 is not separable, i.e., cannot be expressed as a product of an electronic wavefunction and a phononic wavefunction. We have restricted ourselves to the subspace of the states |n, 0i1 because the states |n, m 6= 0i1 correspond to higher energy states due to the fact that the electronic excitation energy is much smaller than the phononic energy, i.e., 2 te−g j), is reached. Since the size n of the unbroken cluster of electrons between the
105
two empty sites j and k is given by n = k − j − 1, add a2i to CP(k − j − 1). Then, again start searching for electrons from site k + 1 onwards until the next empty site is reached and again obtain the next unbroken cluster size n1 . Similar to the previous case, add a2i to CP(n1 ). Continue the searching process for the whole system, i.e., from site j to site (j +N ) with site (j +N ) being equivalent to site j. 6. Repeat steps 3 to 5 successively for all the basis states φi with corresponding coefficients ai to get CP(n) where 1 6 n 6 N/2. 7. Finally, by normalization, calculate NCP for n−particle cluster using the N/2 P expression NCP(n) = CP(n)/ CP(n) for 1 6 n 6 N/2. n=1
106
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