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Polaron physics beyond the Holstein model Marchand, Dominic 2011

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Polaron physics beyond the Holstein model by Dominic Marchand  B.Sc. in Computer Engineering, Université Laval, 2002 B.Sc. in Physics, Université Laval, 2004 M.Sc. in Physics, The University of British Columbia, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2011 c Dominic Marchand 2011  Abstract Many condensed matter problems involve a particle coupled to its environment. The polaron, originally introduced to describe electrons in a polarizable medium, describes a particle coupled to a bosonic field. The Holstein polaron model, although simple, including only optical Einstein phonons and an interaction that couples them to the electron density, captures almost all of the standard polaronic properties. We herein investigate polarons that differ significantly from this behaviour. We study a model with phonon-modulated hopping, and find a radically different behaviour at strong couplings. We report a sharp transition, not a crossover, with a diverging effective mass at the critical coupling. We also look at a model with acoustic phonons, away from the perturbative limit, and again discover unusual polaron properties. Our work relies on the Bold Diagrammatic Monte Carlo (BDMC) method, which samples Feynman diagrammatic expansions efficiently, even those with weak sign problems. Proposed by Prokof’ev and Svistunov, it is extended to lattice polarons for the first time here. We also use the Momentum Average (MA) approximation, an analytical method proposed by Berciu, and find an excellent agreement with the BDMC results. A novel MA approximation able to treat dispersive phonons is also presented, along with a new exact solution for finite systems, inspired by the same formalism.  ii  Preface A number of publications covering the work presented in this thesis are being prepared at the moment. A short manuscript with the main results of chapter 4 has already been published in 2010 as “Sharp Transition for Single Polarons in the One-Dimensional Su-Schrieffer-Heeger Model”, in Physics Review Letters 105 (25), 266605 [60]. A more detailed breakdown of the author’s contributions to this Letter and to the work that will be included in the forthcoming publications is presented below.  Chapter 1 — Polaron Physics This chapter is essentially a review and does not present any original work done by the author. Figures 1.2 and 1.3 were produced from data provided by A. Macridin and were originally published in his PhD thesis [59]. Figure 1.4 was similarly produced from data contributed by M. Berciu, and originally published in [9].  Chapter 2 — Bold Diagrammatic Monte Carlo This chapter starts by reviewing very standard material. The development of the adaptation of Σ-DMC and Bold Diagrammatic Monte Carlo (BDMC) to lattice polarons was a collaborative effort between N. Prokof’ev, B. Svistunov and the author, and the resulting computer code was used to generate the results presented in chapter 4 and the Letter [60]. A short description of the BDMC technique written by the author is published in the supplementary online material accompanying the Letter [9]. Most key ideas of Σ-DMC and Bold Diagrammatic Monte Carlo (BDMC) have been introduced by N. Prokof’ev and B. Svistunov in references [77, 81, 82] for different models, and are simply reviewed here. The actual computer codes used in this work were entirely written from scratch by the author without access to other implementations. The Σ-DMC code was developed under the supervision of N. Prokof’ev and B. Svistunov. This supervision took the form of discussions in front of a black board, but the code was written independently. This served as a kernel to develop a BDMC code, with occasional suggestions from N. Prokof’ev. The  iii  Preface main elements of the BDMC algorithm, including the estimators for the energy and quasiparticle weight, the restrictions to avoid double-counting, and the use of a Fast Fourier Transform (FFT) to apply Dyson’s equation, had already been laid down by N. Prokof’ev and B. Svistunov and have been implemented by the author for the specific case of a lattice polaron. Many technical elements such as the exact choice of the set of updates, the probability distributions used, the approximation to smoothen the results, etc., were chosen by the author. Original contributions by the author include the use of a momentum-dependent chemical potential µ(k) to allow faster convergence at all momenta, the use of Σ(τ → 0) for normalization, and a number of subtleties related to the use of FFTs.  Chapter 3 — Momentum Average Approximation for Models with Dispersive Phonons This chapter also starts by reviewing previous work in section 3.1. The specific type of the Momentum Average (MA) approximation used to produce the results presented in chapter 4 and in the Letter [9] is described in section 3.1.3. Both the technique and the results are the work of M. Berciu. Extensions to the MA technique to treat acoustic phonons, presented in sections 3.2, 3.3 and 3.4, are original and independent contributions of the author, and are being prepared for publication. They are used in chapter 4 to produce some of the results presented. The new techniques borrow some elements from the MA method introduced by L. Covaci and M. Berciu [17] for multiple Holstein branches (section 3.1.4).  Chapter 4 — Phonon-Modulated Hopping The work contained in this thesis is part of a multi-frontal study of the SSH model with optical Einstein phonons (SSHo). Work on this project started independently with BDMC and MA, and was already under way when we learned that another group was also investigating the SSHo model with two other techniques. Instead of reporting similar findings separately, we decided to collaborate. The short manuscript [9] mentioned above summarizes the findings of both our group and our collaborators. A longer manuscript more focused on our work is currently being prepared. The final version of the Letter [60], with the exception of the short sections on the numerical techniques found in the supplementary material, was by and large written by P. Stamp, with help from A. Mishchenko. The short discussion of the sign problem, of the DMC and the BDMC techniques were provided by the author. All results obtained with BDMC were produced by iv  Preface the author, while the MA results were produced by M. Berciu, the DMC results were produced by A. Mishchenko, and the Limited Phonon Basis Exact Diagonalization (LPBED) results were produced by G. De Filippis and V. Cataudella. The interpretation of the results came from discussions between all. In the thesis, all BDMC and perturbative results for the SSHo model, and the related computer programs, were produced by the author. MA results presented in Figure 4.2, 4.3 and 4.6 were produced by M. Berciu. The MAωq results of Figure 4.7 were produced by the author. All work and results on other models presented in this chapter (SSH coupling to acoustic phonons (SSHa), SSH coupling to both acoustic and Einstein phonons (SSHoa), SSH coupling and Holstein coupling to Einstein phonons (SSHod)) were carried out by the author and will be included in the long manuscript.  v  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiv  Abstract Preface  Dedication Foreword  1 Polaron Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Early days: Pekar’s large polaron . . . . . . . . . . . . . . . . . . . 1.1.1 Pekar’s polaron . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The effective mass of Pekar’s polaron . . . . . . . . . . . . . 1.2 Fröhlich’s large polaron, a first microscopic model . . . . . . . . . . 1.2.1 The Fröhlich Hamiltonian . . . . . . . . . . . . . . . . . . . 1.2.2 Weak coupling regime . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Other treatments of the Fröhlich polaron . . . . . . . . . . . 1.3 The lattice polaron . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The lattice polaron Hamiltonian . . . . . . . . . . . . . . . . 1.4 The Holstein model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Weak coupling perturbation . . . . . . . . . . . . . . . . . . 1.4.2 Strong coupling perturbation . . . . . . . . . . . . . . . . . . 1.4.3 Ground state properties as a function of the coupling . . . . 1.4.4 Momentum-dependent properties of the Holstein polaron . . 1.4.5 Spectral properties of the Holstein polaron . . . . . . . . . . 1.5 From large polaron to small polaron: transition versus crossover . . 1.6 Measuring polaron properties experimentally . . . . . . . . . . . . . 1.6.1 ARPES — Angle-Resolved Photoemission Spectroscopy . . . 1.6.2 Effective mass and mobility . . . . . . . . . . . . . . . . . . . 1.6.3 Optical conductivity and Photo-Induced Infrared Absorption  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  1 1 2 5 7 7 10 11 14 15 17 18 19 21 22 22 26 28 28 29 31 vi  Table of Contents 1.7  Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2 Bold Diagrammatic Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . 2.1 Generic Monte Carlo technique . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Weighted averages . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Size of the configuration space and Monte Carlo average . . . . . 2.1.3 The stochastic sum as a random walk . . . . . . . . . . . . . . . . 2.1.4 Exploration of a graph . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Metropolis algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Continuous variables . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Sign problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Diagrammatic Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Imaginary time and imaginary frequency . . . . . . . . . . . . . . 2.2.4 Diagrammatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Ground state energy, quasiparticle weight and effective mass . . . 2.2.6 Monte Carlo sampling and normalization . . . . . . . . . . . . . . 2.2.7 Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.8 Chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.9 Orthonormal functions . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Self-Energy Diagrammatic Monte Carlo . . . . . . . . . . . . . . . . . . . 2.3.1 Self-energy and self-energy diagrams . . . . . . . . . . . . . . . . . 2.3.2 Ground state energy and quasiparticle weight from the self-energy 2.3.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Bold Diagrammatic Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Bold line and double-counting . . . . . . . . . . . . . . . . . . . . 2.4.2 Bold algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Restrictions to avoid double counting . . . . . . . . . . . . . . . . 2.4.4 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  32  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  34 34 34 35 36 37 38 41 42 42 43 47 47 48 50 51 56 57 57 59 59 61 61 65 66 67 68 68 69 70 71 73 77  3 Momentum Average Approximation for Models with Dispersive Phonons 3.1 Review of previous MA techniques . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 MA for Holstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 MA for models with phonon-momentum-dependent coupling g(q) . . . 3.1.3 Variational interpretation of MA . . . . . . . . . . . . . . . . . . . . . . 3.1.4 MA for Holstein with two phonon branches . . . . . . . . . . . . . . . . 3.2 Exact solution for finite systems . . . . . . . . . . . . . . . . . . . . . . . . . .  79 80 81 85 86 87 90 vii  Table of Contents 3.3  3.4 3.5  MA with dispersive phonons (MAωq ) . . . . . . . . . . . . . . 3.3.1 MA for a single momentum-restricted sub-branch . . . 3.3.2 Dividing one phonon branch into multiple sub-branches: 3.3.3 Interpretation of MAωq and remarks . . . . . . . . . . Multigrid extension of MA with dispersive phonons (mgMAωq ) Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . MAωq . . . . . . . . . . . .  4 Phonon-Modulated Hopping . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Phonon-modulated hopping and non-diagonal coupling . 4.1.3 SSH Hamiltonian for a simple monoatomic chain . . . . . 4.1.4 Sign problem . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Other variants of the SSH Hamiltonian . . . . . . . . . . 4.2 SSHo — SSH coupling to one dispersionless optical branch . . . 4.2.1 Hamiltonian, parameters and normalization . . . . . . . 4.2.2 Previous studies . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Results and general comments . . . . . . . . . . . . . . . 4.2.4 From weak coupling to strong coupling . . . . . . . . . . 4.2.5 Sharp transition . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Spectral function . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Perturbation theory . . . . . . . . . . . . . . . . . . . . . 4.2.8 Relevance to real systems and experimental validation . . 4.2.9 SSHod — SSHo with Holstein diagonal coupling . . . . . 4.3 SSHa — SSH coupling to one longitudinal acoustic branch . . . 4.3.1 Hamiltonian, parameters and normalization . . . . . . . 4.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . 4.3.3 SSHoa — SSH coupling to longitudinal and acoustic and stein) branches . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . .  . . . . . .  . 93 . 94 . 99 . 102 . 105 . 108  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . optical (Ein. . . . . . . . . . . . . . . .  110 110 110 111 114 117 118 119 119 121 122 124 130 133 133 135 138 139 139 140 145 145  5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Appendix A: Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.1 Rayleigh-Schrödinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A.2 Wigner-Brillouin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160  viii  List of Figures 1.1  Schematic plot of the Holstein polaron states when g = 0. The solid line shows the free electron dispersion and the dashed lines are the free electron plus one phonon states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ground state as a function of coupling for 1D Holstein from DMC for ω0 = 0.1 and ω0 = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum-dependent properties of the lowest polaron state for the 1D Holstein model from DMC for ω0 = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holstein spectral function at k = 0 for small and large coupling λ = 0.6, 1.2 from MA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13  Random walk as a graph exploration. . . . . . . . . . . . . . . . . . . . . . . . Update pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suggestion and acceptance probabilities . . . . . . . . . . . . . . . . . . . . . . Balance equation for a pair of update that adds or remove a continuous variable First order diagram contributing to the imaginary time Green’s function . . . . Second order diagrams contributing to the imaginary time Green’s function . . Diagrams contributing to the real frequency Green’s function up to third order Diagrams contributing to the real frequency Green’s function using the self-energy Diagrams contributing to the real frequency self-energy . . . . . . . . . . . . . Drawing diagrams with a more complicated propagator . . . . . . . . . . . . . Forbidden and allowed self-energy bold diagrams up to third order . . . . . . . Algorithm to check if a self-energy diagram is allowed in BDMC . . . . . . . . Relation between the Fourier transform and the Discrete Fourier Transform . .  38 39 40 43 54 55 62 63 64 74 75 75 76  3.1  n-fold convolution of uniform probability densities on [−1/2, 1/2] for n = 2, 3, 4, 5 and associated color: red, blue, green and cyan. . . . . . . . . . . . . . . . . . .  96  1.2 1.3 1.4  4.1 4.2 4.3 4.4  18  24 25  SSHo with ω0 = 0.5 from 1st order Σ-DMC and 1st order WB perturbation theory125 SSHo with ω0 = 0.5 from BDMC, MA, and 1st and 2nd order RS and WB perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 SSHo with ω0 = 3.0 from BDMC, MA, and 1st and 2nd order RS and WB perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 SSHo with ω0 = 4.0 from BDMC and 1st and 2nd order RS and WB perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128  ix  List of Figures 4.5 4.6 4.7 4.8 4.9  SSHo with ω0 = 100 from BDMC and 1st and 2nd order RS and WB perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 SSHo phase diagram in the two-dimensional plane (ω0 , c λo ) from BDMC and MA132 Results for the SSHo spectral function for ω0 = 3 and λo = 1 . . . . . . . . . . 134 SSHa with ω0 = 3 from BDMC, MAωq and 1st and 2nd order RS and WB perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Results for the SSHa spectral function for ω0 = 3 and λa = 1 . . . . . . . . . . 144  A.1 Denominator of RS perturbation theory correction for optical and acoustic models160  x  Acknowledgements I would like to express my gratitude to my supervisors, Mona Berciu and Philip Stamp, for their invaluable help and guidance over the last 5 years. Special thanks to them both for going beyond the call of duty and spending numerous hours reading this thesis and suggesting corrections and improvements. Sincere thanks also to the members of my supervisory committee and of my examining committee, George Sawatzky, Joerg Rottler, Nando de Freitas, Joshua Folk, Boris Svistunov and Joel Feldman, for their insightful suggestions. I am very much indebted to Nikolay Prokof’ev for guiding me in learning and developing Diagrammatic Monte Carlo algorithms. His help and that of Boris Svistunov, as well as many students and postdoctoral fellows in their group, was most appreciated. In particular, I am grateful to Evgeni Burovski, Felix Werner, Kris Van Houcke and Lode Pollet. I am also grateful to Alexandru Macridin for giving me copies of his thesis [59] and his data on the Holstein model. A completely unrelated work also completed during this PhD, but not presented herein, consisted in a study of the Giant Proximity Effect (GPE) with an extension of the WORM algorithm [80] adapted to the inhomogeneous XY model. I am indebted to Lucian Covaci for suggesting this project in the first place, and to Matthew Choptuik for allowing me to pursue this topic as my final project for his class 555B: Computational Physics. It was published under “Giant Proximity Effect in a Phase-Fluctuating Superconductor” in Physics Review Letter 101(9), 097004 [61]. The extension of the WORM algorithm to the inhomogeneous XY model, and the computational work were done by myself, but I gratefully acknowledge Marcel Franz, Mona Berciu and Lucian Covaci, for writing the manuscript and for their major contributions in interpreting the data. Over the course of my PhD, I have had the pleasure of many fruitful discussions about my project and interesting physics in general with Rodrigo Pereira, Igor Tupitsyn, Marcel Franz, Lucian Covaci, Glen Goodvin, Bayo Lau, Sanaz Vafaei, Lara Thompson, Alejandro Gaita, Justin Malecki, Conan Weeks, Gili Rosenberg, Jason Penner, Ashley Cook, Ryan McKenzie, Mya Warren, Dan Mazur, Anindya Mukherjee, Zhen Zhu, Hadi Ebrahimnejad, Jean-Sébastien Bernier, Simon Savary, Holger Fehske and many more. I am extremely grateful to many people who allowed me to commandeer their computers, xi  Acknowledgements from time to time, for my simulation needs, be they clusters, personal desktops or laptops. I thank Mona Berciu, Marcel Franz, Joerg Rottler, Matthew Choptuik, Geneviève Brisson, Glen Goodvin and Mathieu Marchand, for the use of their computing resources. Thank you to the UBC Physics and Astronomy department’s staff for their support. Special thanks to Swarn Rai, Bridget Hamilton, Oliva Dela Cruz-Cordero and everyone on the IT staff. And last but not least, thanks to my parents, Jacques and Hélène, my brother and sister, Mathieu and Julie, their spouses, children and the rest of my family and friends, for their support and encouragement. No words can convey the extent of my gratitude and love to my partner, Geneviève, for her constant support and love, and to our soon-to-be-born daughter Chloé, to whom this thesis is dedicated.  xii  To Chloé, for whom all begins.  xiii  Foreword Ever since Landau pointed out the possibility of an electron becoming self-trapped in its own lattice distortion, people have looked for sharp transitions in polaronic states and, in many cases, have predicted one where there was none, due to technique-related artifacts. Gerlach and Löwen have settled this issue in the very restrictive case of a gapped (i.e. optical) phonon branch and an electron-phonon coupling g(q) depending only on the phonon momentum q. They proved that non-analyticities cannot exist in this specific case, but whether a sharp transition could still be found in other models has remained an open question for the last twenty years. By presenting unbiased DMC results supported by other numerical and analytical approximations, we believe this work to be the first unequivocal demonstration of a sharp transition in a polaronic system. The present thesis aims to contribute to our understanding of the effects of phononmodulated hopping on the renormalization of the properties of a single particle. Phononmodulated hopping arises in a tight-binding Hamiltonian when the hopping integral depends on the actual positions of the atoms, and not simply on their equilibrium positions. Such dependency means that any tight-binding model has to be phonon-modulated to some extent, despite the fact that it is often neglected. This phenomenon, and by association our work, has therefore a wide-ranging applicability. It may well be that this effect can be neglected in some materials, but ignoring it in all cases can lead to incorrect predictions. A theory with any hope of achieving a quantitative comparison to experiments will most likely need to take this dependency into account. One of the more widely used models to study phonon-modulated hopping is the Su-SchriefferHeeger (SSH) model. In this Hamiltonian the hopping integral is expanded up to linear order in the distance separating two sites. We point out that due to phonon-modulated hopping, one of the salient features of this model is the presence of an effective next-nearest neighbour hopping term with a negative hopping constant. This leads to a new contribution to the polaron energy that favours a minimum at a finite momentum of π/2, as opposed to the usual contribution from the bare particle hopping, that favours a minimum at zero momentum. The competition between the two contributions leads in turn to the above-mentioned sharp transition, and a number of other very surprising properties, including a diverging effective mass at the critical xiv  Outline coupling. Above this transition the polaron exhibits a finite-momentum degenerate ground state, a sizeable quasiparticle weight with an unusual feature near π/2, and a decreasing mass with increasing coupling. Although we only briefly touch on the subject of acoustic phonons in this thesis, we believe that work on this subject will become equally important to achieve a quantitative comparison to experiments. The universality of acoustic phonon branches directly follows from the fact that, strictly speaking, all physical solids are compressible. Our preliminary investigation suggests that the properties of polarons with acoustic phonons differ significantly from models with optical phonons. We also point out that few exact numerical methods outside of DMC can consider those models. On the more technical side, the development of new methods remains a very important area of research, as what can be achieved too often dictates what is studied. Some problems, especially in strongly correlated many body problems, are outside the reach of even the most sophisticated analytical methods. In other cases, analytical methods are available but need to be validated. There is therefore an important need for unbiased exact methods like DMC techniques. They are versatile and can be applied to any dimension, any number of phonon branches, and all types of couplings. They are, however, strongly hampered by the so-called sign problem. The extension of the faster BDMC algorithm, introduced by Prokof’ev and Svistunov, to lattice polaron models is an important step forward achieved by the present work. On the other hand, convergence of DMC methods can be slow and are akin to heavy artillery. Obtaining information on excited states is also no straightforward task, and require analytic continuation techniques. There is therefore a need for lighter and faster analytic methods such as the MA approximation. Once proven and tested against an exact method, it can be used to probe a wide range of parameters efficiently to single out the interesting physics. This work makes use of MA results for the SSH model, but also proposes a number of extensions to study models with dispersive phonons. It is hoped that our current work on generalizing these techniques will clarify how other numerical methods, such as Exact Diagonalization, might be extended to efficiently treat acoustic phonons.  Outline The content of this work is organized as follows. 1. We open chapter 1 with a selective review of polaron physics, and starting with the early days of Pekar’s large polaron (section 1.1), the first microscopic model proposed by Fröhlich xv  Outline (section 1.2), and a very condensed survey of various treatments of the Fröhlich’s polaron (section 1.2.3). This brief review has no pretension of exhaustivity, and very important contributions such as those of Feynman are only briefly mentioned. The type of polaron that will interest us, the lattice polaron, is then introduced (section 1.3.1), and a more complete presentation of the most well-studied example, the Holstein polaron, is reviewed (section 1.4). This section concludes by discussing the question of a transition versus a crossover in such models (section 1.5). In this work, we are interested in studying polaron physics not considered in the simple Holstein model. We focus more specifically on the effects of phonon-modulated hopping and acoustic phonons on polaron properties. To proceed we rely on two broad categories of techniques. 2. The first category covers stochastic Diagrammatic Monte Carlo (DMC) algorithms. Chapter 2 will review basic Monte Carlo techniques, before presenting a few of these diagrammatic techniques. The Bold Diagrammatic Monte Carlo (BDMC) method, presented in section 2.4, is an improved variant of this category. We adapted it to study lattice polarons, and this work constitutes the first use of BDMC on this type of problems. 3. The second category consists in a type of analytical approximation called the Momentum Average (MA) approximation. Chapter 3 reviews a few MA variants, before introducing three new extensions that allow for models with dispersive phonons. The models studied and their properties are grouped at the end of the thesis in one chapter: 4. Chapter 4 will look at the Su-Schrieffer-Heeger (SSH) Hamiltonian to study the effect of phonon-modulated hopping. Coupling to both acoustic and optical phonons is considered separately. The latter case is especially interesting, and exhibits some striking differences from the standard polaronic behaviour of the Holstein model. The last few sections of Chapter 4 show that the sharp transition found in the SSH model with optical phonons is robust, and survives when other phonon branches or types of coupling are included. 5. We conclude with a summary and a short list of topics to be investigated in future work, in Chapter 5. Finally, some material is relegated to the appendix: Appendix A. We present a brief review of perturbation theory for polaron models. xvi  Suggested reading  Suggested reading The reader less numerically inclined can safely skip chapters 2 and 3, and come back if needed. Unless familiar with this topic, we do suggest a quick review of sections 1.3, 1.4 and 1.5, before jumping to chapter 4, where the new physics beyond the Holstein model is presented. If interested in numerical methods, chapter 2 and 3 are more relevant. The interested reader will find the newer material in the later sections of those chapters, in sections 2.3.1 2.4.1 for DMC, and in sections 3.2 to 3.4 for MA.  xvii  Chapter 1  Polaron Physics In this chapter, we first give a brief overview of the semiclassical approach to the large polaron by following Landau and Pekar’s work on the Pekar polaron. We then present the microscopic model proposed by Fröhlich, Pelzer and Zienau for the large polaron, followed by a very brief survey of the relevant techniques. This section is more historical in nature and is provided to set the stage, and show how the concept of polaron was first introduced. We then turn to the small polaron with the ubiquitous Holstein model. This is a prime example of what is usually considered the standard polaronic behaviour, and we shall spend some time presenting the main properties of the Holstein polaron. It will serve as our baseline comparison for the various models studied in later chapters. Contrary to the large polaron which we treat in its usual three-dimensional case, we choose to focus on the one-dimensional Holstein polaron here. The main features of the Holstein polaron do not depend strongly on the dimensionality of the problem, and this choice will allow for a more direct comparison with the one-dimensional models that we study in this thesis. We conclude this chapter by addressing the question of the existence of a phase transition between the small and large polaron regimes.  1.1  Early days: Pekar’s large polaron  The introduction of the concept of an electron self-trapping itself in a strongly distorted crystal lattice can be found in Landau’s short 1933 paper titled “Über die Bewegung der Elektronen im Kristallgitter” [50]. Landau points out that the most energetically favourable state of an electron in a polar lattice can be either a freely moving particle in an undistorted (or slightly distorted lattice), or a trapped particle in a strongly distorted lattice. The implication is that a sharp transition could exist in such systems. Landau’s work was aimed at explaining crystal defects like F-centers in NaCl. Pekar pushed this concept and developed a macroscopic model where the polar lattice is described as a dielectric continuous medium [71, 72]. In this semiclassical model, the polarization self-consistently supports the electron in its trapped state. The coupling for this mechanism is inversely proportional to the Pekar factor ǭ and depends on experimentally measurable quantities. The Pekar factor relates the polarization P to the 1  1.1. Early days: Pekar’s large polaron electric displacement field D, and represents an effective dielectric constant, or permittivity. These papers were focused on this new quasiparticle, composed of a localized electron and the potential well of the distorted lattice, for which Pekar coined the term polaron. Up to this point, the concept of self-trapping was fully equivalent to localization. The idea of the polaron as a mobile free charge carrier was presented in a subsequent paper [73], and the effective mass of the polaron was calculated in the oft-cited paper by Landau and Pekar [51]. We review briefly the Pekar polaron below, before turning to a microscopic model of the large polaron in the next section.  1.1.1  Pekar’s polaron  The following overview follows loosely the review by Alexandrov and Devreese in [2], as well as Landau and Pekar’s original work [73]. We restrict ourselves to the three-dimensional case as was originally studied. Pekar’s polaron describes an electron interacting with optical phonons of frequency ω0 , when the size of the trapped state is large enough that the discreteness of the lattice can be ignored, and a continuous dielectric medium can be used instead, hence a large or continuous polaron. We will therefore need to obtain a dielectric function ǫ(ω) that relates the polarization P (r, ω) of the medium to the electric displacement field D(r, ω) due to the electron’s electric field E(r, ω): D(r, ω) = ǫ(ω)E(r, ω) = ε0 (1 + χe (ω))E(r, ω) = ε0 E(r, ω) + P (r, ω),  (1.1)  where ε0 is the permittivity of free space and χe is the electric susceptibility. The medium is assumed isotropic and uniform such that ǫ only depends on frequency. The dielectric function of a polar crystal has a resonance at ωopt in the optical or ultraviolet range corresponding to the oscillation of the electronic cloud, and a resonance at ωir ≈ ω0 in  the infrared range associated with the oscillations of the ions themselves. Let us assume that the electron is moving slowly enough such that we can use the static dielectric function ǫ(0),  which we will denote simply as ǫ0 , and which is readily measured experimentally. This holds if the electron moves a distance smaller than the polaron’s size rp , to be obtained shortly, during the characteristic time of the lattice relaxation ≈ ω0−1 . The electron velocity should therefore  be |v| ≪ ω0 rp . For the Pekar polaron, one is concerned with the polarization of the lattice  only, also called the inertial polarization. The electronic polarization is taken care of in the  Hartree-Fock approximation by renormalizing the bare electron’s mass me . Pekar refers to this as the effective mass approximation. It should be noted that this is an effective mass m for the electron due to interactions with other core electrons, not the effective mass of the polaron. We 2  1.1. Early days: Pekar’s large polaron will refer to it as the bare effective mass approximation to avoid any confusion. The electron dispersion relation is assumed parabolic after renormalization. Since the polarization  ε0  P tot = 1 − D, ǫ0  (1.2)  does include the electronic contribution, we need to remove it explicitly. This contribution can be measured experimentally by going to large enough frequencies, i.e. for ω > ωir, where the contribution from the lattice vanishes because the ions are too heavy and move too slowly to respond. Only the contribution from the electrons is left, provided the frequency is kept below the optical or ultraviolet resonance ωopt . Denoting the dielectric constant in this range by ǫ∞ we find the inertial polarization as P inertial = P tot − P ∞ =  ε  0  ǫ∞  −  ε0  D, ǫ0  (1.3)  and we define the effective dielectric constant, or Pekar factor, as 1 1 1 = − . ǭ0 ǫ∞ ǫ0  (1.4)  We now search for the ground state by minimizing the kinetic energy and the potential energy of the electron in the polarized lattice, together with the lattice deformation energy Ud (kinetic and potential energy of the lattice) E[ψ] = where  Z       ∇2 1 dr ψ (r) − ψ(r) − P inertial (r) · D(r) + Ud , 2m ε0 ∗  Z e |ψ(r ′ )|2 D(r) = ∇ dr ′ , 4π |r − r ′ |  (1.5)  (1.6)  and with the deformation energy of ionic crystals 1 Ud = 2ε0  Z  dr P inertial (r) · D(r).  (1.7)  The deformation energy of the crystal is the total energy of the crystal under the assumption that the electron is suddenly removed. We look for the electron state ψ(r) for a given polarization of the lattice. Using the R normalization condition dr|ψ(r)|2 = 1, substituting for D, and taking a functional derivative 3  1.1. Early days: Pekar’s large polaron on both sides while keeping P inertial fixed, we get the equation of motion Eψ(r) =    −  ∇2 e − 2m 8πε0  Z  dr ′ P inertial (r ′ ) · ∇′   1 ψ(r). |r ′ − r|  (1.8)  Substituting (1.3) and (1.6) and integrating by parts yields Eψ(r) =    e2 ∇2 − − 2m 8πǭ0  Z   |ψ(r ′ )|2 dr ψ(r), |r ′ − r| ′  (1.9)  where we also used ∇2 (1/r) = −4πδ(r). The resulting energy functional is obtained by multiplying by ψ ∗ (r) and integrating E[ψ] =  Z  dr    |∇ψ(r)|2 1 − 2m 2maB  Z   |ψ(r)|2 |ψ(r ′ )|2 dr , |r ′ − r| ′  (1.10)  where we define an effective Bohr radius as aB =  4πǭ0 . me2  (1.11)  Finally we look for a solution for the ground state by using the variational ansatz proposed by Pekar    r 2 ψ(r) = A 1 + + βr e−r/rp . rp  (1.12) 3/2  Minimizing with respect to the polaron radius rp leads to A = 0.12/rp , β = 0.45/rp2 and rp = 1.51aB with a ground state energy E0 = −0.0547/ma2B . The most important feature of this ansatz is the exponential factor. Using ψ(r) = Ae−r/rp , and dropping the two other  terms of (1.12) only leads to small corrections (one finds an energy of E0 = −0.0487/ma2B  and a polaron radius roughly 2.12 times larger). Again this is the ground state energy for the polaron state and the lattice deformation. The breakdown of the various contributions to the energy given in (1.6) is as follows: the kinetic energy of the electron is −E0 , the potential  energy of the electron in the deformed lattice is 4E0 , while the lattice deformation energy is −2E0 . In a photoexcitation process where the polaron is suddenly excited to the conduction  electron band, while leaving the lattice deformation behind, the photon energy needed would be 3|E0 | = 0.164/ma2B . The thermal activation energy of the polaron to the electron conduction band, on the other hand, can be as low as |E0 |, provided the lattice distortion disappears in  the process due to thermal fluctuations.  4  1.1. Early days: Pekar’s large polaron  1.1.2  The effective mass of Pekar’s polaron  As mentioned above, it was first thought that Pekar’s polaron solution implied self-trapping, and people have been looking for signs of self-trapping for a long time. It is now known that self-trapping, or localization, does not occur in clean systems. Pekar and Landau indeed quickly realized that nothing prevented this object to move, and the stationary Pekar polaron presented in section 1.1.1 is only relevant when the velocity of the electron v → 0. If the polaron is mobile,  it therefore has an effective mass which we define as ∗  m = h̄  2    dE(k) dk2  −1  ,  (1.13)  k=0  where E(k) is the energy of the polaron with momentum k. It is this quantity which we now consider for Pekar’s polaron. To calculate the mass of the polaron, Laudau and Pekar [51] consider the case where the polaron moves as a whole such that ψ = ψ(r − vt) and similarly D = D(r − vt) and P =  P (r − vt). We drop the subscript inertial here for convenience. The translational invariance  suggests we Fourier transform the polarization and the electric displacement field D(r, t) =  X  Dk ei(k·r−ωk t) ,  (1.14)  P k ei(k·r−ωk t) ,  (1.15)  k  P (r, t) =  X k  with ωk = k · v. We should point out that these harmonics are not the usual electromagnetic  waves. They have a different velocity than light, a different dispersion relation and they are not transverse. In the long wavelength approximation, the frequency-dependent version of (1.3) allows us to relate them through P k (ω) =  ε  0  ǫ∞  −  ε0 ε0  Dk (ω) = Dk (ω). ǫ(ω) ǭ(ω)  (1.16)  We consider the case of a slow moving electron with velocity |v| ≪ ω0 rp , or similarly |v| ≪ ω0 aB . We can approximate ǭ−1 (ω) by the following expansion 1 1 1 2 ω + ..., = + ǭ(ω) ǭ0 ǭ0 ω02  (1.17)  provided that we neglect the imaginary part of the dielectric function, i.e. we assume the absorption is negligibly small. This is a very reasonable assumption for ionic crystals as the 5  1.1. Early days: Pekar’s large polaron imaginary part of the dielectric function is strongly peaked at ωopt , but otherwise much smaller than the real part (see for example Ashcroft and Mermin, Chapter 27 [4] and references therein). The first correction to the static part is quadratic since ǫ(ω) is even. This term correspond to the maximum of the inertial polarization which reaches roughly twice the static polarization at ω0 . Since multiplying Dk (ω) by −ω 2 is equivalent to taking the second time derivative of the same quantity, we get the following after the inverse Fourier transform: P (r, t) =  ε0 ε0 ∂ 2 D(r, t) D(r, t) − . ǭ0 ∂t2 ǭ0 ω02  (1.18)  Putting this back in the energy functional (1.8), we gain one more term. Upon replacing r by r − vt, we find that the first non-vanishing correction to the stationary polaron is equal to v2 6ω02 ǭ0  Z  dr D(r) · ∇2 D(r).  (1.19)  The effective mass associated with the polaron kinetic term is thus 1 m = 3ω02 ǭ0 ∗  Z  dr D(r) · ∇2 D(r).  (1.20)  Using Gauss’s law we finally find e2 m = 3ω02 ǭ0 ∗  Z  dr |ψ(r)|4 .  (1.21)  Using (1.12) we find m∗ = 0.02α4 m,  (1.22)  with the dimensionless coupling constant e2 α= 4πǭ0  r  1 m = 2ω0 aB  r  1 . 2mω0  (1.23)  As we already mentioned, the polaron radius should be large compared to the lattice constant to justify the use of a continuous medium, but also for the bare effective mass approximation of the electron to hold, and for the parabolic electron dispersion to be applicable. Since the polaron radius rp is of the same order as the effective Bohr radius aB , (1.23) leads to rp ∝ α−1 ,  and α should not be too large . On the other hand the classical description of the polarization  is only justified if the number of phonons is large. This number can be estimated by dividing  6  1.2. Fröhlich’s large polaron, a first microscopic model the energy of the cloud by the energy of a single phonon, thus finding a number of phonons of the order of Ud /ω0 = −2E0 /ω0 . Using (1.11) and (1.23), we find E0 = −0.219α2 ω0 for the immobile polaron. Requiring that the number of phonons be much larger than one, translates  into α2 ≫ 5 approximately. These two conflicting conditions imply that the strong-coupling  Pekar polaron can be hard to realize in real systems.  1.2  Fröhlich’s large polaron, a first microscopic model  Although a generalization of Pekar’s semiclassical approach to a quantum theory was being developed at a similar time on the Soviet side of the iron curtain by Pekar himself [75] and other collaborators such as Rashba, Bogoliubov and Tyablikov, we will focus next on the better known work of Fröhlich, Pelzer and Zienau [31] on a microscopic model for the large polaron.  1.2.1  The Fröhlich Hamiltonian  The Fröhlich model has a starting point similar to the Pekar polaron, with the electron in the periodic potential of the three-dimensional lattice treated through the bare effective mass approximation and the ionic crystal approximated by a continuous dielectric medium. However, this time we want to treat the polarization in a proper quantum mechanical way. In the second quantized form, the ionic displacement will be described by phonons. Here we only consider the longitudinal optical phonon branch. As we shall see shortly, the coupling is such that it is much stronger at long wavelengths. For the acoustical mode in this regime, the positive and negative ions move in phase. As the phonon momentum q → 0, the motion approaches the limiting case of the translation of the entire crystal. The polarization created by this motion  is very small. Moreover the transverse optical modes, being perpendicular to q, also have a negligible effect since they do not create any polarization along the wave vector. This leaves the longitudinal mode of the optical branch which is almost dispersionless, allowing us to set ω(q) = ω0 . We consider the effective charge density ρ(r) associated with the inertial polarization P (r). In this context, Gauss’s law becomes ρ(r) = ∇ · P =  ε0 ∇ · D(r). ǭ0  (1.24)  The electric displacement field can in turn be related to the electric potential Φ, defined as E(r) = −∇Φ(r),  (1.25) 7  1.2. Fröhlich’s large polaron, a first microscopic model and to the electric potential energy V . The latter is the potential energy associated with the electron placed in the electric potential produced by the effective charge density ρ(r), such that V = eΦ(r). We find ∇ · D(r) = ε0 ∇ · E(r) = −ε0 ∇2 Φ(r) =  (1.26)  ε0 2 ∇ V. e  (1.27)  Ultimately we need to find V to write down the Hamiltonian. We know that the polarization density P is the dipole moment per unit volume and the dipole moment due to the charge displacement is simply the product of the charge and the displacement. With the lattice constant a and denoting the effective mass of the ions by µ and using phonon creation and annihilation operators, the displacement along x̂ is 1 X xj = √ L qx  s  i h̄ h bqx eiqx (aj) + b†qx e−iqx (aj) , 2µω0  (1.28)  and similarly in the other directions. Assuming two unit charges e per unit cell, the dipole moment is 2er and the polarization density is the dipole moment per unit volume. Using N as the total number of sites, the polarization is 2e P (r) = 3 a  s  X q   h̄ bq eiq·r + b†q e−iq·r . 2µN ω0 q |q|  (1.29)  Going to a continuous description by replacing sums by integrals and factors of N by (2π)3 , we find  s  P (r) = 2e and therefore  h̄ 2µω0  s  ρ(r) = i2e  Z  h̄ 2µω0  Z  Solving for V we find −i2e2 V (r) = ε0  s  h̄ 2µω0   dq q  iq·r √ bq e + b†q e−iq·r , 3 8π |q|  dq  iq·r √ q bq e − b†q e−iq·r . 8π 3 Z   dq 1  iq·r √ bq e − b†q e−iq·r . 3 8π q  (1.30)  (1.31)  (1.32)  To make a comparison to the Pekar polaron more convenient, we need to relate µ to ǭ0 . More 8  1.2. Fröhlich’s large polaron, a first microscopic model specifically we want to find the spring constant κ = µω02 . We know that the force between the ions on a unit cell is  2e 2eǭ0 D = 2 P, ε0 ε0  F = κr = 2eE =  (1.33)  and the polarization density, i.e. its dipole moment per unit volume, is 2e r. a3  (1.34)  4e2 ǭ0 , ε20 a3  (1.35)  P = Together, equations (1.33) and (1.34) yield κ= allowing us to write s  V (r) = −ie  h̄ω0 a3 2ǭ0  Z   dq 1  iq·r √ bq e − b†q e−iq·r . 8π 3 q  (1.36)  Using the dimensionless coupling constant defined earlier e2 α= 4πǭ0 h̄  r  m , 2h̄ω0  (1.37)  equivalent to the one defined for the Pekar polaron when we set h̄ = 1, we find √ 2h̄5 ω03 V (r) = −i 2παa3 m  !1/4 Z   dq 1  iq·r √ bq e − b†q e−iq·r . 8π 3 q  (1.38)  Now that we have the form of the interaction, we can write the Fröhlich Hamiltonian by including the electron kinetic energy and the phonons themselves. We set h̄ = 1 and a = 1 and find  ∇2 Ĥ = − + ω0 2m  Z  dq b†q bq  +  Z  where we defined the electron-phonon coupling   dq  iq·r √ gq e bq + gq∗ e−iq·r b†q , 8π 3  2ω0 i√ gq = − 2παω0 q m  !1/4  .  (1.39)  (1.40)  This coupling is known to as the Fröhlich coupling. Considering only the first two terms, the state can be specified by the momentum of the 9  1.2. Fröhlich’s large polaron, a first microscopic model electron and the set of phonon occupation numbers {nq }, where nq = b†q bq . At zero temperature the unperturbed state is the phonon vacuum |0i and the electron plane wave: |k, 0i = p  1 eik·r |0i. (2π)3  (1.41)  The third term of (1.39) is the electron-phonon interaction which can create or absorb a phonon of momentum q while the electron loses or gains the same momentum. The correp sponding matrix element is hk − q, (n + 1)q |Ĥe-ph |k, nq i = gq∗ (nq + 1)/8π 3 .  1.2.2  Weak coupling regime  For small α we treat the electron-phonon interaction as a perturbation and, since it has no diagonal matrix element, the first non-vanishing correction is of second order. We find Z |gq |2 dq k2 − , 2m (2π)3 (k − q)2 /2m + ω0 − k2 /2m k2 αω0 qp k = − arcsin , 2m k qp  Ek =  with qp =  √  2mω0 .  (1.42) (1.43)  (1.44)  This solution is valid for k < qp otherwise the polaron energy gains an imaginary component. In the limit k ≪ qp of a slow-moving polaron we have k2 − αω0 , 2m∗  (1.45)  m ≈ m(1 + α/6). 1 − α/6  (1.46)  Ek = with an enhanced effective mass m∗ =  The perturbed ground state energy corresponds to a polaron at rest and is given by E0 ≈ −αω0 .  The first order correction to the ground state is (1)  |k = 0i  =−  Z  |gq |2 dq |−q, (1)q i, (2π)3 q 2 /2m + ω0  (1.47)  where |−q, (1)q i is the unperturbed state composed of one electron with momentum −q, and  one phonon with momentum q. Taking the expectation value of the resulting perturbed state  10  1.2. Fröhlich’s large polaron, a first microscopic model for the number of phonons, given by nph =  X  b†q bq ,  (1.48)  q  shows that the coupling constant also measures the number of phonons in the cloud for this model, as nph = α/2. Equation (1.27) can be used to get the charge density ρ(r) = −  eε0 qp2 e−qp r . 4πǭ0 r  (1.49)  Taking rp = qp−1 we find an exponential factor as in (1.12), but the dependence on r is rather different. The weak coupling results are valid for α ≪ 1, but perturbative results for the energy are found to be surprisingly accurate in the range 0 < α < 6, well beyond the range  of validity of perturbation theory [49]. This is not necessarily true for the wave function and √ other quantities. Since α ∝ 1/ ω0 , this corresponds to the non-adiabatic, or dynamic, regime.  1.2.3  Other treatments of the Fröhlich polaron  To extend the range of validity of the results of the previous section to intermediate coupling, Lee, Low and Pines (LLP) proposed to put them on a variational basis using two canonical transformations [55]. They first apply a unitary transformation to remove the electron momentum k, such that the transformed state is specified only by the total momentum of the system, a conserved quantity defined as p=k−  Z  dq qb†q bq .  (1.50)  The LLP transformation takes the state |ψi and transforms it into |ψ ′ i = eSLLP |ψi, where SLLP  Z i h = −i p − dq qb†q bq · r.  (1.51)  The transformed Hamiltonian is given by Ĥ′ = eSLLP Ĥe−SLLP  2 Z Z Z  1 dq  † † =− i∇ + dq qbq bq + ω0 dq bq bq + √ gq bq + gq∗ b†q . 2m 8π 3  (1.52)  The state left after the transformation is the phonon cloud only, since the electron has been removed. Unfortunately, the number of phonons is already quite large for intermediate cou-  11  1.2. Fröhlich’s large polaron, a first microscopic model pling; perturbation theory should therefore not be trusted, whether for Ĥ, or for Ĥ′ . One can  circumvent this problem by applying a second canonical transformation to the phonon cloud, called the displacement transformation. It takes care of the part of the lattice polarization which follows the electron instantly. Applied on the phonon vacuum, it creates a specific type of phonon cloud. At zero temperature, this state is called a phonon coherent state (PCS), and it is used in various polaron variational studies. It describes a cloud of phonons moving coherently as a gaussian wave packet of minimum uncertainty. It also represents a superposition of an ensemble of independent coherent states, one for each phonon momentum. We point out that a coherent state is a solution to the quantum harmonic oscillator, whose dynamics most closely resemble the classical harmonic oscillator. It is therefore a good candidate to describe the state of the lattice oscillations, as we go from a regime where the semiclassical description of the Pekar polaron was valid, to a regime where the lattice needs to be described quantum mechanically. The displacement operator is given by eSPCS , where Z i h SPCS = dq f (q)bq − f ∗ (q)b†q .  (1.53)  f (q) is a form factor determined by local electron-phonon correlations and is the quantity minimized in the variational treatment. We note that f (q) is the complex eigenvalue of the annihilation operator bq , associated with the normalized coherent state e−  |f (q)|2 2  |f (q)i = ef (q)bq −f  ∗ (q)b† q  |0i.  (1.54)  When determining f (q), the minimization of the ground state energy is done assuming eSPCS |ψi = |0i, where ψ is the state of the phonon cloud in the ground state and |0i is the phonon vacuum.  The specific result of this minimization is not stated here, but can be found in [55]. Let us simply mention that for a slow polaron with total momentum p ≪ qp , the effective mass of the polaron is the same as (1.46) and the energy is Ep =  p2 − αω0 . 2m∗  (1.55)  Of course the next order correction differs. Another approach, valid in the strong coupling regime of Pekar’s polaron, is to do a perturbative expansion in powers of α−1 . We only mention this technique for the sake of completeness, and refer the reader to a lecture given by Allcock and published in [49]. This review draws upon  12  1.2. Fröhlich’s large polaron, a first microscopic model the work of Pekar [74, 75], Bogoliubov and Tiablikov [10, 92]. This expansion is valid in the adiabatic region where ω0 is small. Here it is the electron that follows the lattice polarization adiabatically, since the characteristic time of the lattice relaxation ≈ ω0−1 is much larger. The polaron state is rather complex due to the large number of phonons, and closed-form expressions for the energy or the eigenstates are not available. In the limit ω0 → 0 one can, however,  recover the energy of the stationary polaron obtained by Landau and Pekar’s semi-classical  calculation. It can also calculate a polaron mass and a mobility in agreement with Feynman’s path-integral calculation [27]. This brings us to Feynman’s treatment of the Fröhlich polaron [27, 28]. His path-integral calculation is superior as it covers the entire coupling range. Feynman formulates the polaron problem in the Lagrangian formalism and then integrates out the phonons before summing over all possible trajectories of the electron. A subsequent refinement, also by Feynman, introduced a variational ansatz based on this path integral calculation. In the weak coupling regime, he finds  and  while in the strong coupling,  and    α2 E0 = ω0 − α − − ... , 81   α m∗ = m 1 + + . . . . 6  (1.56)  (1.57)    E0 = ω0 − 0.106α2 − 2.83 − . . . ,  (1.58)    m∗ = m 0.0202α4 + . . . .  (1.59)  Prokof’ev and Svistunov [79] were the first to used the Diagrammatic Monte Carlo method to study this model. This work was later expanded in collaboration with Mishchenko and Sakamoto [65]. They not only obtained a ground state energy and an effective mass that agrees remarkably well with Feynman’s result at all couplings, but they also obtained momentumdependent properties of the lowest lying polaron eigenstate such as its energy and its quasiparticle weight (defined later in this chapter). Since the main features of the Fröhlich polaron obtained therein are similar to those obtained for the simpler Holstein model, we simply refer the reader to the two papers mentioned here for more details and we will instead allot more space to describe the Holstein polaron below.  13  1.3. The lattice polaron  1.3  The lattice polaron  If we increase the coupling to phonons such that the polaron radius becomes comparable with the lattice constant a, a continuum model is no longer justified. In this regime, the bare effective mass approximation for the electron and the use of a parabolic dispersion are also not valid anymore. Instead, the result is a small, or lattice, polaron where the electron is spread over a small region of space and is accompanied by a strong local distortion of the lattice. The small polaron, not the electron, is expected to be the charge carrier in strongly polarizable media such as oxides and is therefore essential to describe transport phenomena is these systems. We should also point out that the following could similarly be extended to describe a polaron constituted of a hole in a full band and dressed by phonons. To study the small polaron, we start by considering the Hamiltonian in the zero-coupling limit, that is, in the absence of interactions between the lattice distortion and the electron. The electron travelling through the crystal is then in a Bloch state. This Bloch state is obtained through a sequence of approximations that we will collectively call the Born-Oppenheimer approximation. We note that this name usually refers to the approximation introduced by Born and Oppenheimer [12] to calculate the state of a molecule, but what follows has the same essential features. The state of the system is assumed separable into its electronic and its lattice parts. We first solve for the electronic part while keeping the lattice in its equilibrium position, and then solve for the state of the lattice, i.e. the phonon state. We further treat the Bloch state in the tight binding approximation, where the electron kinetic energy is represented by a hopping term. Next, the interaction between the lattice and the electron is added. It is assumed proportional to the lattice displacement, and can take various forms depending on the type of system described and the approximations used. One essential difference we will find between the small and large polaron is the effective mass of the polaron. The mass of the large polaron is only slightly renormalized with an increase proportional to the coupling α in the weak coupling regime (Fröhlich polaron in the weak coupling regime) and a mass proportional to ≈ 0.02α4 in the strong coupling regime  (Fröhlich polaron or Pekar polaron in the strong coupling regime). The small polaron, as exemplified below by the Holstein polaron in the strong coupling regime, has an exponential mass enhancement with increasing coupling.  14  1.3. The lattice polaron  1.3.1  The lattice polaron Hamiltonian  The textbook electron-phonon interaction derivation starts with a Hamiltonian in the following form Ĥ = Ĥa + Ĥe + Ĥe-a ,  (1.60)  where the first term is for the atoms or the lattice, the second is for the electrons and the last is the interaction between the electrons and the atoms. We are only concerned with single electron systems here, but for a many-body system we would assume that the electron-atom interaction has already been renormalized by screening, and the electron-electron interaction is now implicitly included in the three effective terms above. The lattice degrees of freedom can be cast into the usual phonon Hamiltonian Ĥa = Ĥph .  In our derivation of the Fröhlich Hamiltonian above, we assumed the specific case of a polar  crystal where the electron was most strongly coupled to longitudinal acoustic phonons. Here we start from a slightly more generic premise where we leave the type of phonon unspecified and use a general form of the electron-atom interaction potential Ve-a (r − R) with r and R the position of the electron and atom respectively. We rewrite Ĥ = −  h̄2 2 X ∇ + Ve-a (r − Rj ) + Ĥph , 2m e  (1.61)  j  where m is the mass of the electron, ∇e is the gradient taken with respect to the electron position, and the position of the j th atom,  Rj = R0j + uj ,  (1.62)  has a deviation uj from its equilibrium position R0j . Since the deviation from the equilibrium position is small, we approximate the potential by expanding it to first order, such that Ĥ = −  X h̄2 2 X ∇e + Ve-a (r − R0j ) − uj · ∇Ve-a (r − R0j ) + Ĥph . 2m j  (1.63)  j  At this point the Hamiltonian remains fairly general. As advertised above, the Born-Oppenheimer approximation only considers the first two terms to find the Bloch wave functions of the electrons, such that they depend only on the equilibrium position of the atoms. Next we apply a tight-binding approximation, where the Bloch state is replaced by a localized state around each atom. Tunnelling can occur between neighbouring sites and is proportional to the wave function overlap measured by the hopping integral t. This gives rise to the effective hopping 15  1.3. The lattice polaron term, which approximates the electron kinetic energy in the lattice. Next nearest neighbour hopping and higher order hopping is also possible, but the overlap of the wave function decreases rapidly such that each correction is exponentially smaller than the previous one. The effective Hamiltonian is now Ĥ = −t |   X † X cj cj ′ + c†j ′ cj + Ĥph − uj · ∇Ve-a (r − R0j ) .  hjj ′ i  (1.64)  j  }|  {z  Ĥ0  {z  Ĥel-ph  }  The exact form of the electron-phonon interaction term Ĥel-ph in its real space second quantization form depends on the potential Ve-a . As long as the total momentum is conserved,  however, its form in Fourier space is the same with only an electron-phonon coupling g(q) to be specified. For a system of size N , we now have Ĥ =  X  ǫ(k)c†k ck +  q  k  |  X  {z  Ĥ0   1 X ω(q)b†q bq + √ g(q)c†k+q ck b†−q + bq , N k, q } | {z }  (1.65)  Ĥel-ph  where ω(q) is the phonon dispersion and the electron dispersion is ǫ(k) = −2t  d X  cos(ki ),  (1.66)  i=1  for the simple d-dimensional hypercubic lattice. For now we focus on optical phonons approximated by dispersionless Einstein phonons with ω(q) = ω0 . The possible forms of electronphonon couplings include the polar, or Fröhlich, coupling discussed above with g(q) ∝ 1/|q|, the deformation coupling with g(q) ∝ |q|, and the simpler Holstein coupling g(q) = g.  We define two dimensionless parameters to describe the various regimes of the lattice po-  laron, as opposed to the single dimensionless coupling constant α of the large polaron. First is the adiabaticity ratio  ω0 . t  (1.67)  The second parameter is the dimensionless electron-phonon coupling constant λ=  Ep , zt  (1.68)  where Ep is the lattice deformation energy due to the electron presence, zt is the electron half16  1.4. The Holstein model bandwidth, and z is the lattice coordination number. λ corresponds to the ratio of the ground state energy of a stationary polaron when t → 0, to the bare electron half-bandwidth. We will see shortly that Ep in the thermodynamic limit is found to be Ep =  Z  dq |g(q)|2 , (2π)d ω(q)  (1.69)  using the Lang-Firsov transformation [52].  1.4  The Holstein model  We now turn to the simpler Holstein model as our main baseline comparison for the polaron model studied in this work. This model captures all the quintessential polaronic features found in the Fröhlich model while being amenable to various analytical approaches both in the weak coupling and strong coupling regimes. More recently Goodvin, Berciu and Sawatzky introduced an approximation able of covering the entire parameter range accurately [9, 35, 37]. On the numerical side, the Holstein model has been studied using Exact Diagonalization (ED) methods on a small cluster [1, 15, 23, 62, 95–97], ED methods on a variationally determined subspace of the Hilbert space [11], Density-matrix renormalization-group (DMRG) [45], dynamical mean field theory [16], variational ansätze such as the Global-Local (GL) approximation [58, 86] and of course using the Diagrammatic Monte Carlo techniques (DMC) [24, 47, 48, 59]. Since the models presented in the following chapters are one-dimensional, we will now focus more closely on the one-dimensional Holstein model. In most cases, the Holstein model does not depend strongly on dimensionality, and the general polaronic features we want to underline remain the same. The reader interested in a more quantitative comparison to the Holstein model in higher dimensions will find a wealth of information in the many works referenced above. As for comparing to the large polaron results of section 1.1 and 1.2, we will limit ourselves to a qualitative comparison that will not depend on the dimensionality. The coupling g is constant, and the dimensionless coupling, defined in the previous section, is given by λ=  g2 . 2tω0  (1.70)  Perturbative results in the weak and strong coupling are derived below, before summarizing all-coupling results from DMC and MA techniques for later comparison purposes.  17  1.4. The Holstein model  1.4.1  Weak coupling perturbation  In the weak coupling limit, where λ → 0, we can treat the electron-phonon interaction as a  perturbation. The unperturbed states and energies are simply those of the bare electron ǫ(k),  with a number of phonons nph present with an energy cost of ω0 each. Since the phonons are dispersionless, the ensemble of states with one phonon of momentum q ∈ [0, 2π] form a  continuum of states. Figure 1.1 depicts the lowest eigenstate with nph = 0, and a few of the first excited states forming the continuum. Ek  k  ω0 k′  Figure 1.1: Schematic plot of the Holstein polaron states when g = 0. The solid line shows the free electron dispersion and the dashed lines are the free electron plus one phonon states.  When turning on the coupling, the electron first becomes lightly dressed by phonons spread over a large region of space, and the effective mass increases slightly. This regime is the already discussed large polaron for which the quasi-classical approximation of the continuum polaron is valid. Rayleigh-Schrödinger (RS) perturbation theory provides accurate results, but only at small momentum. As discussed more fully in Appendix A on perturbation theory, for an adiabaticity ratio ω0 /t ≤ 4, the RS energy displays an unphysical maximum and goes to −∞ at k = π. For the energy in the thermodynamic limit we find Ek = −2t cos(k) −  Z  dq 2tω0 λ . 2π 2t cos(k) + ω0 − 2t cos(k − q)  (1.71) 18  1.4. The Holstein model Since a phonon has a fixed energy cost of ω0 , we expect to find the one polaron plus one phonon continuum at E0 + ω0 . The picture is the same as in Figure 1.1, but the bare electron dispersion need to be replaced by the polaron energy Ek . RS finds the continuum at ǫ(0) + ω0 instead, as it only considers one phonon processes, and more phonons need to be considered in the cloud for an accurate energy for the continuum. RS also fails at evaluating the number of phonons in the cloud or the quasi-particle weight at larger momentum (see Appendix A). Wigner-Brillouin perturbation theory does not suffer from many of these limitations, but it underestimates the denominator of (1.71) and produces less accurate results than RS generally. As an aside, the effective mass can be obtained analytically [56] for the RS perturbation theory, and is found to be m∗ = m0  λ 1− 2  r  t 1 + ω0 /2t  ω0 1 + ω /4t3/2 0  !−1  ,  (1.72)  with m0 being the mass of the bare electron. For λ → 0, the mass enhancement is thus  proportional to g2 in accordance with the large polaron results.  1.4.2  Strong coupling perturbation  In the opposite limit, when λ ≫ 1, the electron kinetic term can be treated as a perturbation. In this limit, we use the Lang-Firsov canonical transformation [52] to remove the part of the polaronic cloud corresponding to the stationary small polaron, i.e. the zeroth order result for t = 0. With the transformed Hamiltonian eSLF Ĥe−SLF , the ground state in the limit λ → ∞ is  a new phonon vacuum. Applied on a state, the transformation is given by |ψ ′ i = eSLF |ψi with SLF = −  g X n̂j (b†j − bj ). ω0  (1.73)  j  where n̂j is the electron number operator. Physically, it represents an average shift of 2g/ω0 in the atom position where the electron is present. The transformed phonon annihilation operator is given by B j = bj +  g n̂j , ω0  (1.74)  and the transformed annihilation electron operator is given by g  Cj = cj e ω0  (b†j −bj )  .  (1.75)  19  1.4. The Holstein model The unperturbed Hamiltonian for the one-electron case is Ĥ0 = ω0  X j  Bj† Bj −  g2 , ω0  (1.76)  where the second term is the lattice deformation energy of (1.69) equal to the ground state energy of the unperturbed Hamiltonian. We note that this term is momentum-independent as it should be, since the electron is trapped (t = 0) and the Fourier transform of a Kronecker delta is a constant. The general form for Ep given by (1.69) was obtained with B q = bq +  g(−q) X † √ c c , ω(q) N k k−q k  (1.77)  which is simply the Fourier transform of (1.74). In the unperturbed state, the electron is fully localized and its state is N-fold degenerate. Its energy is −g2 /ω0 . When we turn on the hopping term through t, the electron is allowed  to move again, but the transformed electron operators changes the ion equilibrium position by moving the cloud along. Using (1.75) to rewrite the hopping term, and using the vacuum state |0′ i of the transformed phonon Hamiltonian to trace out the phonon degrees of freedom, we  find a first order correction to the energy corresponding to an exponentially suppressed hopping term h0′ |Ĥt′ |0′ i = −t  X  hjj ′ i  h0′ |Cj† Cj ′ |0′ i = −t∗  X  hjj ′ i  c†j cj ′ = −te2λt/ω0  X  c†j cj ′ ,  (1.78)  hjj ′ i  due to the polaron’s exponentially enhanced mass m∗ = 2tm∗ = e2λt/ω0 . m0  (1.79)  The number of phonons in the cloud of the ground state is easily calculated to be nph = h0′ |  X j  b†j bj |0′ i =  2λt g2 = 2. ω0 ω0  (1.80)  The quasiparticle weight of the ground state, a measure of the overlap between the bare electron ground state in the phonon vacuum |ǫ0 i × |0i, and the polaron state in the transformed phonon  vacuum |E0 i × |0′ i, is defined as  Z0 = |hǫ0 |E0 ih0|0′ i|2 ,  (1.81)  20  1.4. The Holstein model and is found to be exponentially suppressed as well: Z0 = e−2λt/ω0 .  (1.82)  The next correction comes from virtual processes where the electron hops to a neighbouring site leaving the cloud behind and returning to its original position. The correction to the energy is given by −t/λ, for a resulting energy for the lowest lying polaron state given by Ek = −2λt −  t − 2t∗ cos k. λ  (1.83)  We now compare the small and large polarons in their strong coupling limit. More specifically, we consider the Holstein model and the strong coupling regime of the Fröhlich polaron as obtained by Feynman. We note that the results above are for the one-dimensional case, but the results for higher dimensions remain essentially the same, up to a multiplicative constant, for the effective coupling λ. We already pointed out the difference in the effective mass: an exponential function of the effective coupling (m∗ ∝ e2tλ/ω0 ) for the small polaron, and a power  law (m∗ ∝ α4 ) for the Fröhlich polaron. This will lead to a very different dependence of the  mobility and transport properties on the coupling. On the other hand, we find that the zeroth  order perturbation result for the energy of the Holstein polaron scales as λ ∝ g2 , whereas Feyn-  man finds the Fröhlich polaron energy scales as α2 . Despite this similarity, other corrections are very different.  1.4.3  Ground state properties as a function of the coupling  We start by looking at the effect of the coupling strength on the properties of the ground state. Figure 1.2 shows the ground state energy E0 (λ), the quasiparticle weight Z0 (λ), the average number of phonons nph (λ), and the effective mass m∗ /m0 , as a function of the dimensionless coupling. We observed that both the ground state energy E0 (λ), and the number of phonons nph (λ), vary linearly with the coupling at both small and large coupling, albeit with a very different slope. The crossover between these two regimes happens around λ ≈ 1. Also around this value, the quasiparticle weight drops sharply while the effective mass starts to increase.  In this region, the ground state is a mixture of both a large polaron-like state, with its main contribution from the bare electron state, and a small-polaron-like state, with a larger number of phonons.  21  1.4. The Holstein model  1.4.4  Momentum-dependent properties of the Holstein polaron  Accurate results for the Holstein polaron with large momentum had to wait for the advent of numerical techniques such as DMC, due to the failure of perturbation theory at large momentum (see Appendix A). The large momentum Holstein polaron state is very different from the small momentum state at small and intermediate coupling. Instead of a lightly dressed electron with a large quasiparticle weight and a dispersion very similar to that of the bare electron, the polaron state starts to mix with the one-electron plus one-phonon continuum. The polaron dispersion flattens and looks more like the phonon dispersion. Consequently, the quasiparticle weight approaches 0 as the energy nears the continuum edge. At small coupling the number of phonons at large momentum is related to the number of phonons of the ground state by nph (k ≫ 0) ≈ nph (k = 0) + 1. As explained above, the continuum is found at exactly E0 (k = 0) + ω0 , not ǫ(k = 0) + ω0 . Exactly as the polaron state is pushed down below the energy of the bare electron by interaction with the continuum states, so are the states of the one-phonon continuum pushed down by interaction with the two-phonon continuum. It can be shown that this protects the polaron state and ensures that, in the one-dimensional case at least, the polaron dispersion does not have end points [35]. By end point, we mean a case where the polaron state would enter the continuum, at some point ke of the dispersion. Past this point, we find a resonance in the spectral function with a finite lifetime instead of the infinitely-lived quasiparticle peak. The three-dimensional case does have end points, however. Since this is not a universal feature of the small polaron, and since it depends on both the model and the dimensionality of the problem, we do not reproduce here the more rigorous proof presented by Goodvin and Berciu [35] for the absence of end points in the one-dimensional Holstein polaron. Figure (1.3) presents the one-dimensional polaron dispersion, the quasiparticle weight, and the average phonon number as a function of momentum. The polaron bandwidth is seen to decrease rather fast with increasing coupling.  1.4.5  Spectral properties of the Holstein polaron  A look at the spectral weight confirms that the continuum is at E0 (k = 0) + ω0 . We also note the presence of the so-called second bound state (an excited discrete state) below the continuum for a large enough coupling.  22  1.4. The Holstein model  -2  E0(k=0)  ω=0.1 ω=0.5  0.8  -3  0.6 -4 0.4 -5 -6  0.2  (a) 0  0.5  1  λ  1.5  2  2.5  30  0  Z0(k=0)  1  (b) 0  0.5  1  0.5  1  λ  1.5  2  2.5  1.5  2  2.5  1  20  0.4  10  0.2  (c) 0  *  0.6  m0/m  nph(k=0)  0.8  0  0.5  1  λ  1.5  2  2.5  0  (d) 0  λ  Figure 1.2: Ground state as a function of coupling λ for 1D Holstein for ω0 = 0.1 and ω0 = 0.5. DMC results reproduced from A. Macridin’s thesis [59] with his permission. (a) Ground state energy E0 (k = 0); (b) Quasiparticle weight Z0 (k = 0); (c) Average number of phonons in the cloud nph (k = 0) and (d) Logarithm of the ratio of the effective mass to the bare effective mass of the electron, (m∗ /m0 ). (a)-(d) share the same legend with red being for ω0 = 0.1 and green for ω0 = 0.5.  23  1.4. The Holstein model  0.5 (a)  E0(k)  0.4 0.3 0.2  λ=0.25, ω=0.5 λ=1.00, ω=0.5  0.1 0  0  0.2  0.4  0.6  0.8  1  k/π 0.8  (b)  Z0(k)  0.6 0.4 λ=1.00, ω=0.5 λ=0.25, ω=0.5  0.2 0  0  0.2  0.4  0.6  0.8  1  k/π 5  nph(k)  (c)  λ=0.25, ω=0.5 λ=1.00, ω=0.5  4 3 2 1 0  0  0.2  0.4  0.6  0.8  1  k/π Figure 1.3: Momentum-dependent properties of the lowest polaron state for the 1D Holstein model obatained with the DMC method. These results are for ω0 = 0.5, and are reproduced from A. Macridin’s thesis [59] with his permission. Figure (a) shows E0 (k) − E0 (0); Figure (b) shows Z0 (k); and Figure (c) shows nph (k).  24  1.4. The Holstein model  10  A(k=0, ω)  8  ground state  λ=0.6, ω=0.5  6  polaron + one phonon continuum  4 2 0 10 ground state  A(k=0,ω)  8  λ=1.2, ω=0.5  st  1 polaron excited state  6  polaron + one phonon continuum  4 2 0 -3  -2.75  -2.5  -2.25  -2 ω/t  -1.75  -1.5  -1.25  -1  Figure 1.4: Holstein spectral function at zero momentum A(k = 0, ω) for small coupling λ = 0.6 (red) and large coupling λ = 1.2 (green). These results are reproduced from [9] with permission from M. Berciu.  25  1.5. From large polaron to small polaron: transition versus crossover  1.5  From large polaron to small polaron: transition versus crossover  We will refrain from using the expression phase transition as we are oftentimes interested in the behaviour of a single polaron. A single polaron, or even a small number of polarons, would be invisible in any of the thermodynamic properties. To encompass both the case of a single particle and that of a macroscopically large number of particles, we will simply talk about transitions. Such transitions, if they exist in polaronic systems, would be accompanied by an abrupt or sudden change, or non-analyticity, in one or more quantities. The possibility of a self-trapping transition was introduced alongside the concept of the polaron from the very beginning. As mentioned earlier, the first techniques to span the gap from the weak coupling regime to the strong coupling regime were variational in nature. Unfortunately, an important shortcoming of variational calculations is that it is not possible to distinguish between a non-analytical behaviour of a solution due to an artifact of the ansatz used, and an intrinsic non-analytical property of the model. As it happens, the first study of this kind by Buckingam [14] found a discontinuity in the ground state wave function, supporting Landau’s original hypothesis. Despite Fröhlich’s warning [30] regarding the inconclusiveness of these results, the existence of a self-trapping transition of the polaron became a wide-spread belief. The seminal contribution of Gerlach and Löwen [34] settled this issue by presenting a three-part formal proof showing that this type of polaron does not have a sharp transition, selftrapping or otherwise, but a smooth crossover for all quantities of interest. We focus here on their work on what they call the free polaron, that is a polaron with no external potential such as one coming from an impurity. More specifically we consider a Hamiltonian as in (2.33) with a coupling g(q) that depends on the phonon momentum only. In fact, they actually assumed a parabolic electron dispersion, but later showed that under some conditions the proof can be carried out in exactly the same way for a more general band structure ǫ(k). Let us simply state here that this condition holds for the electron dispersion (1.66). Both g(q) and ω(q) are considered isotropic. We briefly present the assumptions made and the results, but we refer to the review [34] for the detailed proofs. Gerlach and Löwen start by considering the domain of analyticity of the polaron energy E(λ, k) and the polaron wave function ψ(λ, k) as functions of momentum and dimensionless coupling. Assuming ω(q) ≥ ω > 0, ω(q 1 ) + ω(q 2 ) ≥ ω(|q 1 + q 2 |),  (1.84)  26  1.5. From large polaron to small polaron: transition versus crossover Z  dq  |g(q)|2 < ∞, 1 + (rp q)2  with the polaron radius rp =  r  h̄ , mω  (1.85)  then the lowest polaron state energy E(λ, k) exists and is an isolated and simple eigenvalue for 0 ≤ λ < ∞. E(λ, k) and ψ(λ, k) are real analytic functions of λ and k. The three-dimensional  case has an extra restriction on the momentum given by h̄2 k2 /2m < h̄ω. The first condition is necessary to guarantee the existence of an energy gap above the lowest lying polaron state.  The second condition is needed to ensure a lower bound for the edge of the continuum. The last condition is needed to avoid E(λ, k) going to −∞. This statement implies that observables of the polaron state, computed either as the derivatives of E(λ, k) or as expectation values of an operator independent of λ and k, are smooth functions of λ and k. Examples include the polaron mass, the polaron radius, and the number of phonons in the cloud. The second problem considered is whether E(λ, 0) < E(λ, k 6= 0). Under the assumptions  made above, this is proven to hold for all 0 ≤ λ < ∞. This excludes any quantum-mechanical  symmetry breaking and the existence of a localization-delocalization transition [33].  The last proof is related to the free energy F (λ, β), where β is the inverse temperature, and the partition function Z(λ, β) = e−β[F (λ, β)−F (0, β)] .  (1.86)  F (λ, β) − F (0, β) is shown to be real and analytic, or equivalently Z(λ, β) to be bounded, for 0 ≤ λ < ∞ and 0 < β < ∞ provided that one of the following is satisfied: Z  dq  or ω(q) ≥ ω > 0  |g(q)|2 < ∞, ω(q) const and |g(q)| < p . q d−1  (1.87)  (1.88)  Gerlach and Löwen also pointed out that it is still unclear whether the assumption ω(q) ≥  ω > 0 is really needed. There does not seem to be any analyticity proof available as to now that would cover acoustic phonons. Various variational studies of polaron problems with acoustic  phonons claim non-analyticities, but they cannot rule out an ansatz-related artifact. This is clearly reminiscent of the optical case before Gerlach and Löwen’s contribution.  27  1.6. Measuring polaron properties experimentally  1.6  Measuring polaron properties experimentally  We want to briefly mention some of the experimental techniques used to study polarons. There is a vast literature on the subject, and we select only a small subset here. The models studied herein are for a single-polaron in the zero temperature limit, which reduces somewhat the selection of relevant methods. We start our short review with what is surely the most useful technique for those models, namely Angle-Resolved Photoemission Spectroscopy (ARPES). We then touch on the subject of transport properties and optical conductivity.  1.6.1  ARPES — Angle-Resolved Photoemission Spectroscopy  ARPES is a type of Photoemission Spectroscopy (PES), and therefore relies on the photoelectric effect, where an electron is ejected from a solid after absorbing a photon [20, 21]. By measuring the direction and speed of the photoelectron, ARPES can obtain momentum-dependent properties of the electronic band structure. Due to the intricacies of the photoemission process in solids, the analysis of the experimental results is often done under two simplifying assumptions: the independent particle picture and the sudden approximation. The first neglects many-body interactions, while the sudden approximation assumes that the system relaxes instantaneously during the photoemission. The neglected many-body effects arise because the initial state of the system is an N -electron eigenstate, while the final state is an (N − 1)-electron eigenstate. The latter can be significantly different from the initial state. Both states need to obey the proper boundary conditions, which are somewhat complicated by the presence of the surface. Furthermore, the final state of the whole system includes a propagating plane wave component to account for the escaping electron. Energy and momentum conservation during the photon absorption also need to be enforced. In a fully quantum-mechanical treatment, the entire photoemission process is a one-step process characterized by the overlap of the initial and final states, and the energies of the entire system before and after. The probability of a specific process is approximated by Fermi’s golden rule. The full Hamiltonian is not trivial, and to further simplify the problem the photoemission event is sometimes viewed as a three-step process. The electron is first excited by absorbing the photon, then it travels to the surface, and, finally, overcomes the surface barrier to escape. While the electron is travelling to the surface, it can become dressed by phonons, and when the electron is analyzed, single-polaron properties can be extracted. The more rigorous one-step treatment might be needed when this three-step description fails, but it implies solving a much more complicated Hamiltonian. Other complications can also arise when probing insulators. For instance, charges that are removed by photoemission from an insulator leave holes behind that can take a long time to 28  1.6. Measuring polaron properties experimentally recombine. This can further complicate the ARPES measurement in the type of systems that will be relevant to chapter 4 such as organic semiconductors. Let us assume that the difficulty of the analysis of ARPES data can be overcome. We then obtain the momentum-resolved spectral function, which tells us the probability of finding the particle with energy ω and momentum k. It is usually defined as  1 A(k, ω) = − ℑ G(k, ω) , π  (1.89)  where ℑ returns the imaginary part of the argument, and G(k, ω) is the retarded Green’s function. The Green’s function describes the system by giving the probability amplitude that  the electron is found in the same state after being created and having interacted with phonons. The Green’s function is −1 |ki, (1.90) G(k, ω) = k|(ω − Ĥ + iη where η is an infinitesimal number. The spectral function can also be defined using the self-  energy: A(k, ω) = −  Σ′′ (k, ω) 1 , π [ω − ǫ(k) − Σ′ (k, ω)]2 + [Σ′′ (k, ω)]2  (1.91)  where we have used a prime and double-prime to identify the real and imaginary part of the self-energy, i.e. Σ(k, ω) = Σ′ (k, ω) + iΣ′′ (k, ω). The self-energy can be related to the Green’s function by G(k, ω) =  1 , ω − ǫ(k) − Σ(k, ω) + iη  (1.92)  and with ǫ(k) being the dispersion of the bare particle. (See also section 2.3.1 for a diagrammatic definition.) From the spectral function we can get everything we need, including the polaron dispersion relation E(k), the quasiparticle weight Z(k) and the effective mass m∗ (k) (equation (1.93)). E(k) is given by the position of quasiparticle peaks, the weight under the peak is related to Z(k), and the effective mass can be obtained from the k-dependence of E(k) near the ground state.  1.6.2  Effective mass and mobility  As we have seen, the isotropic effective mass (1.13) is given by the inverse of the second derivative of the dispersion relation with respect to momentum. We can define a momentumdependent effective mass m∗ (k), but in most cases in the literature, m∗ is used to refer to the k = 0 effective mass. For most polaron models the ground state is at kgs = 0, and thus 29  1.6. Measuring polaron properties experimentally comparing to a quadratic dispersion relation makes more sense at this point. In this thesis we assume ∗  2  m (k) = h̄    dE(k) dk2  −1  ,  (1.93)  but we will use m∗ to refer to the effective mass of the ground state m∗ (kgs ), except where the momentum dependence is explicitly shown. It should be noted that we will always consider a fully isotropic system (and usually, a one-dimensional system). Here, we will consider the effective mass to be a constant. In theory, the effective mass can depend on frequency, which would be relevant in ac measurements. The effective mass can also be related to the self-energy using ω) 1 − ∂Σ(k, m∗ ∂ω ω=E(kgs ) = , 2 Σ(k, E(k )) ∂ gs m 1+m ∂k 2 k=kgs  (1.94)  where m = 1/2t is the bare electron mass. We thus need to know the value of the hopping integral or, equivalently, the bare effective mass (or band mass) to determine the mass enhancement due to the electron-phonon coupling. We point out that the effective mass is not necessarily related to the inverse bandwidth. In the usual polaron picture, the effective mass diverges at infinite coupling, while the bandwidth is reduced to zero and the particle becomes dispersionless and localized. A more complicated dispersion relation might, however, lead to an infinite effective mass, while preserving a finite bandwidth and quasiparticle weight, provided that the dispersion relation has an inflexion point. (See section 4.2.5 for such an unusual example.) We have already seen that the effective mass can be obtained from ARPES. We now look at other ways to get this quantity experimentally. Cyclotron resonance experiments have been done for large polarons in metals [42]. The Fröhlich Hamiltonian can be used to evaluate the bare effective mass and the coupling α from the cyclotron measurement if we measure also ǫ0 , ǫ∞ and ω0 . In insulators, on the other hand, one usually measures the mobility. An example of a typical measurement where the charge carrier is a large polaron can be found in [87]. For a charged particle of mass m, the mobility is the average velocity per unit of applied electric field. It is approximated by µ = −e  τ , m  (1.95)  where τ is the average scattering time, or electron lifetime, −e is the charge, and m is the mass. This equation is only valid for a constant mass and if we assume that the effect of all interactions can be approximated with this single scattering time τ . 30  1.6. Measuring polaron properties experimentally We have to distinguish between the large and small polaron mobility. For the large polaron mobility, the lifetime is calculated for scattering by impurities and phonons. The essential feature of the large polaron mobility is the exponential increase of the mobility with decreasing temperature, due to the scattering with phonons. The phonons’ average thermal density increases exponentially with temperature. On the other hand, measurements of the mobility at low temperature are complicated by the presence of impurities. As a consequence, the measured mobility at low temperatures will depend greatly on the sample. The Fröhlich polaron’s mobility in the weak coupling shows a simple and intuitive behaviour with a zero-temperature mobility µ0 = −eτ0 /m [53] expressed as a function of the effective bare mass m. The lifetime  due to phonon scattering is given by τ0 = (eβω0 − 1)/(2αω0 ). The Frölich polaron mobility is  then found to be ballistic for weak coupling α [54], with µ = −eτ /m∗ and τ /τ0 = 1 + O(α2 )  and m∗ /m = 1 + α/6 + O(α2 ).  For the small polaron, a different behaviour is observed. The mobility increases with increasing temperature. At temperatures below the characteristic phonon energy, the mobility is determined by Boltzmann kinetics. At larger temperatures, the polaron band collapses and the transport is diffusive with thermally activated hopping events [43]. The effective mass is momentum-dependent and is not trivially related to the mobility. In the high-temperature limit, Holstein shows that the polaron motion is a random walk, and he calculates a hopping mobility µh , which is proportional to the square of the hopping integral. Usually, the mobility is measured as a function of temperature [87]. One can then compare the temperature dependence of the mobility to the predicted value by the model, and in some cases, like in the small coupling limit, deduce what the effective mass is [53]. Since we focus on the low-temperature limit in this work, we do not calculate any temperature dependence, and ARPES experiments remain better suited for our purpose.  1.6.3  Optical conductivity and Photo-Induced Infrared Absorption  Photo-Induced Infrared Absorption (PIA) [64] is a technique in which a source of infrared light is shone on a sample, while a detector on the other side measures the transmission (absorption). A laser is then pointed at the surface of the sample, and the measurement is repeated. Photoinduced polarons will modify the absorption spectrum, and quantities such as the polaron binding energies, lifetimes and information on the electron-phonon coupling can be obtained. The infrared spectrum of the optical conductivity can be calculated and compared to computational or analytical results. For the small polaron in the zero-temperature limit, the optical  31  1.7. Summary conductivity was shown by Reik [83] to be √ 2 2   2πt e n ω/ω0 −nph ω0 ω/ω0 σ(ω) = , e nph ω h̄3 ω02  (1.96)  where t is the hopping integral, and e is the charge. This form was confirmed by experiments and numerical calculations [3]. It was also shown to be more or less temperature independent [46]. The temperature independence suggests that it would be a useful tool to compare zerotemperature calculations to finite-temperature experiments, but we will not use this quantity in the present work.  1.7  Summary  1. We distinguish between a large or continuous polaron (sections 1.1 and 1.2), valid when the polaron spans many lattice sites, and a lattice polaron (sections 1.3 and 1.4), where the lattice discreteness is important. 2. Pekar’s large polaron (section 1.1) is a semiclassical model for a slow moving electron in an ionic crystal, where the crystal is described as a continuous polarizable medium. The polaron state is obtained by minimizing (1.5), the energy of the electron and of the lattice deformation together. We distinguish between the electronic polarization and the ionic polarization. The first is accounted for by renormalizing the bare electron mass, and using an effective parabolic dispersion. The ionic polarization is parametrized by an effective dielectric function given in (1.17). An upper bound for the energy is obtained with the ansatz (1.12). An effective coupling constant is defined in (1.23) that the energy and the number of phonons both scale as α2 , while the mass scale as α4 . It is valid in a strong coupling regime with α ≫ 5 and rp ≫ a. 3. Fröhlich’s large polaron (section 1.2) is again a continuous model, but with a proper quantum treatment of the lattice. The electronic polarization is treated as for Pekar’s polaron. The ionic polarization, however, is related to the displacement of the ions and expressed as a function of phonon operators. The Fröhlich Hamiltonian is given by (1.39), and finds a coupling ∝ 1/q given in (1.40). Many techniques have been applied to the Fröhlich model including perturbation theory in the weak coupling (section 1.2.2),  canonical transformation in the intermediate coupling (section 1.2.3), Feynman’s path integral techniques at all couplings (section 1.2.3), and numerical techniques. In the very weak coupling regime, we find a light polaron with the mass enhancement, the energy 32  1.7. Summary and the number of phonon, proportional to the coupling. In the strong coupling regime the energy scale in the same way as the Pekar polaron. 4. The lattice polaron (section 1.3) description is needed when the polaron radius becomes comparable with the lattice constant. The Hamiltonian (1.64) is obtained in the BornOppenheimer approximation and the tight-binding approximation. The kinetic energy of the electron is given by a hopping term, while the lattice deformation is described by the phonon Hamiltonian. The electron-phonon interaction can take various form depending on the model. In the weak coupling regime, the lattice polaron is large and light and behaves like the continuous large polaron, at least at small momentum. As the coupling is increased, we find a much heavier small polaron, with an effective mass that increases exponentially with the coupling strength, instead of the power law found at small coupling. 5. The Holstein model (section 1.4) is a simple model for the lattice polaron with optical phonons and a momentum-independent coupling. Despite its simplicity, it captures the essential features considered standard polaronic behaviour, including those summarized above. We do find an effective mass as a function of coupling increasing as a power-law at small coupling, and as an exponential function in the heavy small polaron regime. Other generic features of lattice polarons with optical phonons include a polaron plus one phonon continuum starting at E0 + ω0 , a very small quasiparticle weight at large coupling and an average number of phonons increasing rapidly at large coupling. 6. Gerlach and Löwen’s seminal contribution (section 1.5) was to prove formally that polaron models with optical phonons and couplings dependent only on the phonon momentum do not exhibit any sharp transition or non-analyticities in their properties. 7. Angle-Resolved Photoemission Spectroscopy (section 1.6) is the best suited technique to study the momentum-dependent and coupling-dependent properties of the single lattice polaron at zero temperature. It yields the angle-resolved spectral function A(k, ω) from which the energy, quasiparticle weight and effective mass can be extracted.  33  Chapter 2  Bold Diagrammatic Monte Carlo The Monte Carlo (MC) method is an extremely general stochastic numerical technique encompassing a plethora of different algorithms and variants. It is used in numerous fields of physics, but also in engineering, mathematics, finance, etc. Many of its applications consist essentially in evaluating a multidimensional integral with complicated boundary conditions. Its ubiquity stems from a universal need in physics and mathematics to study systems with a large number of coupled degrees of freedom. This number is usually so prohibitively large that an exact answer simply cannot be computed and an approximate value based on a statistical sampling of those degrees of freedom is the best that can be obtained. This work is more specifically concerned with a limited class of methods referred to as Diagrammatic Monte Carlo (DMC). The quantities calculated in this case are various propagators or Green’s functions that can be expressed in a Feynman diagrams expansion. We review two such techniques before presenting a faster DMC sampling method called Bold Diagrammatic Monte Carlo (BDMC).  2.1  Generic Monte Carlo technique  In this section we shall first review the basic concepts used in any Monte Carlo calculation.  2.1.1  Weighted averages  In its most generic form the Monte Carlo technique evaluates the ratio of two N -dimensional sums or integrals. Restricting ourselves to the case of a weighted average of an observable over an ensemble of configurations {(i1 , . . . , iN )} in the n-dimensional Hilbert space spanned by the  set of summation indices and integral variables {i1 , . . . , iN }, we define the ensemble average hAi = Z −1 Z=  X i1  X i1  ···  ···  X  X  Ai1 ,...,iN Wi1 ,...,iN ;  (2.1)  iN  Wi1 ,...,iN ,  (2.2)  iN  34  2.1. Generic Monte Carlo technique where Ai1 ,...,iN and Wi1 ,...,iN are the value and weight of each configuration. We use a summation P symbol here to represent indiscriminately a sum over a discrete variable in or an integral over R a continuous variable din . To simplify we define a configuration index ν as the collection of  all summation indices with ν ≡ (i1 , . . . , iN ). We can then rewrite hAi =  X Aν Wν ν  Z  ;  Z=  X  Wν .  (2.3)  ν  We also can define the probability of a configuration ν as pν =  Wν . Z  (2.4)  In statistical mechanics, Z is simply the partition function and the probability of a con-  figuration is related to the energy Eν of the configuration with pν = Z −1 exp(−Eν /kB T ) and  Wν = exp(−Eν /kB T ) given by the Boltzmann factor.  A demonstrative exercise is to calculate the volume of a multidimensional body whose surface is defined as Fν = 0 and where Fν > 0 holds inside the body, while Fν ≤ 0 outside.  One can then write an expression for the volume using the step function and an hypercube V0 of side L and volume LN as the configuration space: Z  Z  L  dxN Θ F (x1 , . . . , xN ) dx1 . . . 0 0 X   = Θ Fν = LN Θ Fν ν ∈ V0 .  Vbody =  L  (2.5)  ν  So the volume of the body is written in the form of (2.3) with Wν = 1 ∀ν. If Wν is not constant,  the problem can be viewed as evaluating the mass of a N-dimensional body where the mass  density is given by Wν . Most problems can be mapped onto one of these two examples.  2.1.2  Size of the configuration space and Monte Carlo average  When N is large, it proves impossible to add all the terms in the sum in a reasonable amount of time. Just consider the case where each variable can take 2 values as in the magnetization of an Ising spin. If N = 100, this makes 2100 ∼ 1030 terms to sum, making it impossible to complete  the sum over one’s lifetime. We further note that if the configuration space is continuous, the number of configurations is infinite and cannot be sampled completely, once again forcing us to forgo an exact answer. A good estimate can still be reached with high accuracy in many cases provided that the following holds: 35  2.1. Generic Monte Carlo technique 1. hAi does not contain much information about {ν} and only depends on the average of  some quantity. A representative set of such an ensemble would roughly contain the same information about hAi as the complete set. A very accurate estimate for hAi can then be calculated. For the magnetization h|M |i for example, a system of N Ising spins can only N 2  − 1 different values. A representative set for this system will be of order N 2 only √ since we will see shortly that the statistical error ∝ 1/ N . have  2. Despite the fact that we are summing the product of two quantities Aν and Wν , the structure of Wν is often such that it is finite only in a small region. Even if the number of configurations outside this region is very large, if their weight is exponentially small the answer will depend almost exclusively on a tiny fraction of the parameter space. MC algorithms consist in replacing the unmanageably large set of terms to sum by a random subset. The sums in (2.3) are thus replaced by stochastic sums  where  P′  P P′ Aν Wν ν ν Aν Wν /pν hAi = P , ≈ P ′ ν Wν ν Wν /pν  (2.6)  selects terms according to pν . In the limit of an infinitely long simulation, each term  is chosen Nν times, where Nν is itself proportional to pν such that Nν = cpν . In this limit we recover the exact answer P′ P P Aν Wν Nν /pν ν Aν Wν /pν ν cAν Wν ν lim = P = hAi, (2.7) P′ = P time→∞ ν Wν Nν /pν ν cWν ν Wν /pν The actual efficiency of a specific algorithm will depend greatly on how terms are selected.  If a representative set can be chosen and furthermore terms with larger weight are more likely to be sampled, the answer will be more accurate and converge faster.  2.1.3  The stochastic sum as a random walk  The central idea of the Metropolis algorithm [63] is to implement the stochastic sum  P′  as a  random walk in the configuration space such that configurations are added as they are visited. This is also known as the Markov Chain Monte Carlo method, since each term depends on the previous one only, a property of Markov Chains. When the number of indices of the configuration space is large, this correlation between consecutive elements allows for a faster sampling since a previous configuration can be reused and only be modified slightly, or updated, to get a new one, instead of building it from scratch every time. Even more importantly, the 36  2.1. Generic Monte Carlo technique random walk can be implemented with the properties mentioned above to ensure a faster convergence than that of a uniform sampling. The random walk is then defined by the set of all possible updates that can be applied to a configuration, and the associated probabilities according to which they are chosen. This set of updates and probabilities must satisfy two criteria to ensure the solution converges toward the correct limit (2.7): 1. The stochastic generation of configurations must be ergodic. That is, in a very long simulation, all possible configurations must be generated eventually. 2. The probability pν must be known for each ν, in order to properly weight each term added in the sum. The latter can be greatly simplified if the probabilities are taken to be directly proportional to their weight, such that (2.7) reduces to  2.1.4  P′ P′ Aν ν Aν Wν /pν hAi ≈ P′ = Pν′ . ν Wν /pν ν1  (2.8)  Exploration of a graph  To introduce the remaining features of the Metropolis algorithm, it is useful to describe the random walk as a graph exploration. The graph is constructed so that each node represents a configuration, and each update between two configurations is represented by an arrow. We attach a probability pν→ν ′ to each arrow leaving the node. The random walk will explore this graph by selecting, at each step, one of the arrows leaving the current node. The selection is done according to the attached probabilities and the visited configurations are summed. The ergodicity condition also states that there should be enough updates so that any two nodes of the graph are linked by a path in both direction. The probabilities pν→ν ′ only need to be calculated to ensure that the probability to visit a node is proportional to its weight. The full construction of such a graph is at least as complicated as calculating hAi itself, so we will want instead to calculate probabilities on the fly, that is, at the time of selecting an arrow. This should be done using only local information ( i.e. the weight of the current configuration and possibly the weight of the neighbours). A fully connected graph for example would obviously require a global knowledge of the configuration space. Choosing a limited set of updates which are the same for each node will simplify greatly balancing the probabilities, but is not sufficient by itself to deal away with the need for a global knowledge of the graph, as will be shown below. 37  2.1. Generic Monte Carlo technique  p1 p2  1  p4 p5  5 4  7  p5  7 6  5  p7  4  3  p6 p7  6  p7  p6  4  p6  5 7  p4  p3  6  5  p4  4  2  4 3  p2  2  p4  p5  4  p3 p4 6  3  1  4  p2  1  p3  2  1  p1  3  p4  p1  2  7  Figure 2.1: Random walk as a graph exploration. Nodes νi are configurations and arrows between node νi and node νj are updates with the associated probabilities pνi →νj . The set of updates is not unique and need only be ergodic and balanced.  2.1.5  Balance equation  To balance, or normalize, the probabilities of the graph, we consider again an infinitely long simulation, with Nν the number of times each node is visited, and Mν→ν ′ the number of times each update is applied (i.e. the associated arrow traversed). Clearly the random walk cannot find itself leaving a node more often than it enters it; therefore X  Mν→ν ′ =  hν ′ iν  X  Mν ′ →ν ,  hν ′ iν  ∀ν,  (2.9)  where hν ′ iν is the set of neighbouring nodes {ν ′ } that can be reached by updating ν. This  current conservation at each node specifies a set of equations determining the probabilities {pν→ν ′ } as a function of the weights {Wν } of the initial problem. Starting with lim  time→∞  Mν→ν ′ = Nν pν→ν ′ ,  ∀ν,  (2.10)  38  2.1. Generic Monte Carlo technique and using (2.7) and (2.8) lim  time→∞  Mν→ν ′ =  lim  time→∞  cWν pν→ν ′ ,  ∀ν,  (2.11)  where c is some constant, we can put this back in (2.9) to obtain Wν  X  pν→ν ′ =  X  Wν ′ pν ′ →ν ,  hν ′ iν  hν ′ iν  ∀ν.  (2.12)  This equation can have many complicated solutions depending on the topology of the graph. The simplest approach is to balance the updates by pairs as shown in Figure 2.2. We require each update to have an inverse update so that the current conservation is respected for each pair. If each pair has a null net current, then so will each node. Equation (2.12) becomes Wν pν→ν ′ = Wν ′ pν ′ →ν ,  ∀hν, ν ′ i,  (2.13)  where hν, ν ′ i represent two neighbouring nodes linked by a pair of updates.  p  p Figure 2.2: Balancing pairs of updates ensures current conservation while keeping the balance equation simple. Here ν and ν ′ is a pair of nodes and the update ν → ν ′ has a probability pν→ν ′ , while the inverse update has a probability pν ′ →ν . The balance equation between these updates is given by (2.13).  Our goal of implementing the random walk without any global knowledge of the graph is not yet reached, however. The probabilities of the updates leaving a node should be normalized. These probabilities depend on the weights of all the neighbouring nodes, but also on the probabilities of the inverse updates. Those need in turn to be chosen so that the probabilities of the updates leaving the neighbouring nodes are also normalized, and so on. The solution, which we explain below, is to divide the update process into two stages by factoring the up39  2.1. Generic Monte Carlo technique date probability into a suggestion probability psuggest and an acceptance probability paccept ν→ν ′ ν→ν ′ . From the graph point of view, an arrow leaving a node bifurcates, with an arrow going back  to the original node with probability psuggest 1 − paccept ν→ν ′ ν→ν ′ , and one reaching the new node with  accept probability psuggest ν→ν ′ pν→ν ′ . This is shown in Figure 2.3.  The key point is that the suggestion probability can be arbitrarily chosen, as long as the probabilities of the updates leaving the node are normalized. They need not depend on any global information, and the set of updates and probabilities can be the same for every node. The acceptance probability will of course depend on the weight Wν ′ of the destination node and the suggestion probabilities leaving it, but the dependence stops there. There is, therefore, no recursive dependence on all the other nodes of the graph. We will see later that psuggest ν→ν ′ can be chosen to make the algorithm more efficient by extracting a limited amount of local information from the neighbouring nodes.  paccept psuggest  psuggest  p accept Figure 2.3: Dividing the update ν → ν ′ into two stages by factoring the probability into a accept suggestion psuggest ν→ν ′ and an acceptance probability pν→ν ′ and similarly for the inverse update ν ′ → ν. The current conservation law for a pair of updates needs to be modified into suggest accept accept ′ Wν psuggest ν→ν ′ pν→ν ′ = Wν pν ′ →ν pν ′ →ν ,  ∀hν, ν ′ i.  (2.14)  This last equation is often what is meant by the balance equation. The suggestion probabilities suggest psuggest ν→ν ′ and pν ′ →ν are assumed to be known or easily calculated on the fly during the suggestion  process at each step. We now define the acceptance ratio R=  paccept ν→ν ′  paccept ν ′ →ν  suggest Wν ′ pν ′ →ν = . Wν psuggest ν→ν ′  (2.15) 40  2.1. Generic Monte Carlo technique The acceptance ratio can be calculated with local information about the current and the sugaccept gested updated configurations. The choice of paccept ν→ν ′ and pν→ν ′ is not unique, but a convenient  choice, which will ensure a high acceptance probability, is =  (  paccept ν ′ →ν =  (  paccept ν→ν ′  R 1 1  R≤1 R>1 R≤1  1/R R > 1  (2.16)  accept Many other choices are possible, but they will usually have paccept ν→ν ′ ∝ pν ′ →ν . The above scheme,  on the other hand, always has an optimal acceptance probability for one of the two updates.  We note that in order for the algorithm to be as efficient as possible, the acceptance ratio should be close to one. This means that the optimal suggestion probabilities are given by the ratio of the weights  psuggest ν→ν ′  psuggest ν ′ →ν  =  Wν ′ . Wν  (2.17)  Allowing for R to deviate from this ideal value was the crucial step in designing a random walk that does not require the full knowledge of the configuration space, but we shall later see that the ideal value can be reached for some pairs of updates.  2.1.6  Metropolis algorithm  We have covered all the ingredients of the basic Metropolis algorithm, so we summarize it in the pseudocode below.  Metropolis Algorithm 1. Initialize Sum A = 0 and Z = 0. 2. Generate a first configuration ν. 3. Include the configuration ν in the sum: Sum A = Sum A + Aν , Z = Z + 1. 4. Suggest another configuration ν ′ from ν (update) 5. Accept ν ′ on average  Wν ′ Wν  times.  If accepted: ν → ν ′ . If rejected: ν → ν.  6. Repeat from step 3 until desired convergence. 41  2.1. Generic Monte Carlo technique  2.1.7  Continuous variables  Often an index of the configuration space will be a continuous variable. Then, the balance equation (2.17) needs to be modified to use probability densities accept suggest suggest Wν paccept ν→ν ′ Pν→ν ′ δν→ν ′ = Wν ′ pν ′ →ν Pν ′ →ν δν ′ →ν ,  (2.18)  where we balance the updates between a volume of the configuration space δν→ν ′ and a volume δν ′ →ν . P suggest is the product of all the probability densities associated with the change in value of the variables affected by the pair of updates between ν and ν ′ .  A more problematic case is when ν and ν ′ do not have the same set of degrees of freedom as we can no longer simply balance updates in pairs. Consider for simplicity an update which adds a single degree of freedom x and the inverse update that removes it. As depicted in Figure 2.4, we are balancing an infinite and continuous set of updates leaving ν, with a single update suggest coming back from each node of the infinite set {ν ′ (x)}. Let Pν→ν ′ (x) (x) be the probability  distribution of suggesting the specific update ν → ν ′ (x), psuggest ν→ν ′ the probability to suggest the  update ν → ν ′ in general and P(x) be the probability distribution used to pick x. The balance  equation becomes  suggest accept suggest Wν paccept ν→ν ′ (x)Pν→ν ′ (x) (x)δ(x) =Wν pν→ν ′ (x)pν→ν ′ P(x)δ(x)  (2.19)  suggest =δ(x)Wν ′ (x)paccept ν ′ →ν (x)pν ′ →ν (x),  where δ(x) on the right hand side comes from the integral over all the possible configurations {ν ′ (x)} in the current conservation equation (2.12). The acceptance ratio (2.20) is now R(x) =  2.1.8  paccept ν→ν ′ (x)  paccept ν ′ →ν (x)  =  suggest Wν ′ (x) pν ′ →ν (x) Wν psuggest ν→ν ′ P(x)  (2.20)  Sign problem  A sign problem is a generic feature of many systems, and one that hinders greatly our ability to solve them with Monte Carlo methods. If the weight Wν is not a positive-definite function we have to separate the sign and the weight to use the Monte Carlo procedure presented above. A Monte Carlo estimator would now be calculated with P ν Aν signν |Wν | hAi = P , ν signν |Wν |  (2.21)  42  2.1. Generic Monte Carlo technique  p accept (x)  est  P  sugg  (x) (x) ) st ugge (x ps  (x)  x  p accept (x) Figure 2.4: Updates between the single node ν and a continuum of nodes {ν ′ (x)} with an added continuous degree of freedom x. The update ν → ν ′ is suggested with a probability suggest suggest accept distribution Pν→ν ′ (x) (x) = pν→ν ′ P(x) and accepted with probability pν→ν ′ (x), while each of the individual inverse update of the continuum of updates {ν ′ (x) → ν} has a probability of accept psuggest ν ′ →ν (x) of being suggested and pν ′ →ν (x) of being accepted.  where signν = sign(Wν ). We then replace  P  by  P′  which selects terms with a probability  proportional to the positive-definite weight |Wν |, such that P′ P′ P′ 1 hA · signi ν Aν signν ν Aν signν = ≈ . hAi ≈ P′ P′ P′ ν hsigni ν signν ν1 ν signν  (2.22)  In the best of cases the estimator will simply converge slower with a sign problem. This is what we would call a weak sign problem. If hsigni goes to zero, however, there is little that can  be done. hAi is the ratio of two quantities both converging to zero, such that all we see is the ratio of the statistical errors fluctuating wildly.  A problem might have a sign problem in some formulation, while being sign-problem free in another. Choosing the right representation, however, is problem-dependent. The sign problem has kept MC simulations from being applied to many-body fermionic systems, for example. We will see later that, in the in the case of a weak sign problem, the proper choice of an estimator can play an important role if some quantities converge faster than others. Schemes that allow to sum more configurations at once can also help to alleviate a weak sign problem.  2.1.9  Error analysis  There is no meaningful computational calculation without at least an estimate of the error. Rounding errors and the limited precision of floating point numbers is an unavoidable part  43  2.1. Generic Monte Carlo technique of any numerical technique, but the stochastic nature of the Monte Carlo method implies that the error on computed observables is mostly statistical in nature. From the statistical point of view, the configuration space is a population for which the observable A is a random variable distributed according to a probability distribution, with mean hAi and a characteristic  2 = hA2 i − hAi2 , or by the standard dispersion measured either by the variance Var(A) = σA  deviation σA . We want to obtain an estimate for hAi from a random sample of size N of  the population, along with an estimate of the statistical error on this value. This is a very common problem which is treated in any undergraduate-level course on statistics (see [41] for example). The first part of this task is fairly obvious as we can simply use the sample mean as an approximation with hAi ≈ Ā =  N X Ai i=1  N  ,  (2.23)  which is exactly what MC calculates. To obtain the error we want to look at each element of the sample as a random variable, with probability distribution characterized by the mean hAi  2 . From the central limit theorem we know that Ā is also a random variable and the variance σA  with an approximately normal distribution of mean hĀi = hAi, and variance Var(Ā) =  (2.24)  2 σA . N  (2.25)  The distribution of Ā is referred to as the sampling distribution. Its standard deviation is given the meaningful name of standard error, as it describes the dispersion of the sample averages Ā’s of all such samples. This is the error estimate we were looking for, but unfortunately σA is unknown. We can however approximate it with the sample standard deviation s and define thereby the sample standard error s ∆= √ = N  sP  N i=1 (Ai  − Ā)2 = N (N − 1)  sP  N 2 i=1 (Ai  − Ā2 ) . N (N − 1)  (2.26)  Using the standard error as an error bar on hAi is somewhat optimistic. The sampling distribu√ tion being normal, Ā has 68.26% probability of being found in the range [hAi − σA / N , hAi + √ √ σA / N ]. A more conservative estimate would be Ā ± 3σA / N ≈ Ā ± 3∆ with a 99.78%  probability for Ā to be in this range.  A very important observation is that errors in Monte Carlo simulations should decrease as 44  2.1. Generic Monte Carlo technique the square root of the sample size, or equivalently as the square root of simulation time. In theory, a MC estimator can always be made more precise by running the code for a longer time. In practice, other sources of error will impose a limit on the achievable precision. Furthermore, the value of σA determines the achievable precision in a reasonable amount of time. Equation (2.26) for the error assumes that samples are statistically independent from each other. We, however, relinquished this property when we chose to use local updates applied to the current configuration, instead of generating new configurations from scratch. The sequence of configurations generated are thus correlated, and (2.26) will underestimate the error. This correlation dies off as we apply more updates and the current configuration differs even further from the original one. Typically we will need to apply a number of accepted updates at least comparable to the number of degrees of freedom. This number will likely need to be even bigger if the updates used are more likely to change an updated variable by a small value, instead of reselecting a new value uniformly on their complete allowed range. And again we are talking about the number of accepted updates, such that the typical simulation time before correlations die off will be a complicated function of the estimator considered, the configuration space, the set of updates, the acceptance ratios, and by extension the probabilities and probability distributions used in the update process. The correlation between samples effectively reduces the size of the sample we have available. To see this we need to evaluate how correlations decrease with simulation time or with the sample size. We again look at each element Ai of the sample as a random variable, and calculate the variance of Ā with correlation given by  N 1 X Var(Ā) = Var Ai = N   =  i=1  *  N 1 1 X Var(Ai ) + 2 2 N N i=1  2 N 1 X Ai N  i=1 N N XX  +  −  *  Cov(Ai , Aj ),   N 1 X Ai N i=1  +2  ,  (2.27)  i=1 j=1 j6=i  where the covariance is defined as Cov(Ai , Bj ) = hAi Bj i − hAi ihBj i,  (2.28)  also called the autocovariance when B = A. A dimensionless, but equivalent quantity called the autocovariance coefficient can also be defined: ρj−i =  Cov(Ai , Aj ) , Var(A)  (2.29) 45  2.1. Generic Monte Carlo technique such that (2.27) becomes "  # N −1  2 X σA k Var(Ā) = 1− 1+2 ρk . N N  (2.30)  k=1  If we assume that the correlation decreases to zero after M ≪ N values, with N ≫ 1, we can  write  # " ∞ 2 X σ2 σA ρk = A 2τA , 1+2 Var(Ā) ≈ N N  (2.31)  k=1  thereby defining the integrated autocorrelation time τA . By comparing this to (2.25) we see that the correlation indeed effectively reduces the size by a factor of 1/(2τA ). Typically the autocovariance coefficient, i.e. the correlations, decrease exponentially as e−N/τA such that τA is simply the relaxation time of the system. Integrating the autocorrelation coefficient then yields τA directly. It can be estimated from the MC sample with N −1 N −1 PN −j 1 X 1 X i=1 (Ai − Ā)(Ai+j − Ā) τA ≈ + . ρj ≈ + PN −j 2 2 2 i=1 (Ai − Ā) j=1  (2.32)  j=1  The error of a MC estimate can thus be calculated using (2.32) and (2.26). In practice, however, the error bar is determined by dividing the MC sample into blocks. Since Ai ’s are correlated, groups of these values will also be correlated. We can thus organize the full sample in a sequence of smaller samples or blocks of size M , and calculate a block average Āi . Since correlations die off with simulation time, the larger we make M the less correlated those blocks averages are, and the smaller the sample standard error will be. This suggests that an honest error estimate can be obtained by making blocks of increasing size and observing how the standard error saturates with increasing M . Various techniques based on this principle suggest different recipes for building larger blocks from smaller ones. These include the bootstrap and jackknife methods, but the simplest and the only one we present briefly here is the blocking method. The blocking method simply defines the smallest block size as M , and groups the MC configurations generated into these blocks. The sample mean of each of those blocks is kept for error analysis. We perform the error analysis only if a statistically significant number of blocks is collected, say larger than 16. As the calculation progresses, we keep forming new blocks and perform the error analysis every time the number of blocks doubles. Superblocks are formed by combining smaller blocks together and averaging their block averages. The sample standard deviation of a sample of superblocks will increase with the block size as they get less correlated. 46  2.2. Diagrammatic Monte Carlo By plotting the sample standard deviation as a function of the block size we should observe a saturation for large enough, and therefore uncorrelated, blocks.  2.2  Diagrammatic Monte Carlo  The Diagrammatic Monte Carlo (DMC) method presented here is a Monte Carlo sampling of a Feynman diagram expansion. It was first introduced by Prokof’ev, Svistunov and Tupitsyn [78] and further developed to study the Fröhlich polaron [65, 79]. It has since been applied to a variety of polaron problems such as the Holstein model [59] and the spin polaron [66], to name a few. The details of the algorithm depend on the type of diagrams being summed, and on the quantities being calculated. This section will consider only the special case of a polaronic quasiparticle.  2.2.1  Hamiltonian  The most general Hamiltonian considered in the following work is that of a single electron on a d-dimensional tight-binding lattice with any number of phonon branches, and an electronphonon interaction gα (k, q) depending on the phonon branch α and on both the electron momentum k and the phonon momentum q. Taking c (c† ) and b (b† ) to be the electron and phonon annihilation (creation) operators, the Hamiltonian in momentum space is then given by Ĥ =  X k  |  X  ǫ(k) − µ c†k ck + ωα (q)b†α, q bα, q α, q  {z  }  Ĥ0   1 X gα (k, q)c†k+q ck b†α, −q + bα, q , +√ d N α, k, q | {z }  (2.33)  Ĥint  where µ is the chemical potential, ωα (q) is the phonon dispersion for branch α and the electron dispersion is ǫ(k) = −2t  d X  cos(ki ).  (2.34)  i=1  The chemical potential of (2.33) does not have any physical meaning here, since we only have one electron, but is simply an adjustable zero of energy which will prove convenient for the computation later on. 47  2.2. Diagrammatic Monte Carlo For convenience we divided (2.33) into an interaction Hamiltonian Ĥint , referring to the  electron-phonon interaction, and the non-interacting Ĥ0 , for the electron and the phonon parts. Slightly more general Hamiltonians with more than one orbital per site for the electron  could also be considered with little modification to the treatment below. A generalization to more than one electron is however much more involved and will not be covered in the present thesis. We shall first concern ourselves with the case of a single phonon branch. The modifications needed to consider more than one branch will be mentioned when needed. Also, the dimensionality of the problem does not change the form of the algorithm in any nontrivial way so the rest of this chapter will assume d = 1.  2.2.2  Green’s function  The main quantity of interest to us when describing the polaron is the T = 0 Green’s function, or propagator, in the momentum representation, by which we mean the amplitude to inject one electron in a known momentum state |ki at time t0 and to find it in the same state at some  later time t. The system is assumed to be initially in its ground state so there are no phonons  present. Denoting the vacuum state by |0i, we write in the Heisenberg picture G(t, t0 , k) = −ih0| ck (t) c†k (t0 ) |0i,  (2.35)  where the factor −i in front is a matter of convention. From this function we will be able  to extract most features of interest pertaining to the polaron state, such as its energy and its effective mass. Before we proceed with modifying the above expression for the Green’s function into something that we can work with, we need to describe how the state of the system evolves with time. We recall that the evolution of the system is given by the Schrödinger equation ih̄  ∂ |ψ(t)i = Ĥ(t)|ψ(t)i, ∂t  (2.36)  with initial condition |ψ(t0 )i = |ψ0 i. It is convenient to define a time-evolution operator Û to  time evolve an initial state at time t0 to the corresponding state at time t: |ψ(t)i = Û (t, t0 )|ψ0 i.  (2.37)  48  2.2. Diagrammatic Monte Carlo From (2.38) we see that this operator is given by ih̄  ∂ Û (t, t0 ) = Ĥ(t)Û (t, t0 ). ∂t  (2.38)  This operator must be unitary to insure the conservation of probability, and Û (t0 , t0 ) = 1. The  solution for a time-independent Hamiltonian like (2.33) is simply a time-ordered exponential  i  Û (t, t0 ) = exp − Ĥ(t − t0 ) . h̄  (2.39)  Using the time-evolution operator we rewrite (2.35) in the Schrödinger picture G(t, t0 , k) = −ih0, t| ck Û (t, t0 ) c†k |0, t0 i = −ihk, t| Û (t, t0 ) |k, t0 i,  (2.40)  but we shall drop the explicit time dependence of the state from here on. Strictly speaking we will be interested in the retarded Green’s function where t > t0 . Since Û can evolve a state for  arbitrary times −∞ < t < ∞, we define the retarded Green’s function using the step function G(t, t0 , k) = −ihk| Θ(t − t0 )Û (t, t0 ) |ki.  (2.41)  We should mention that we refer to the Green’s function as the expression defined above, but it can also refer to the operator. For the retarded Green’s function operator we have Ĝ(t, t0 ) = −iΘ(t − t0 )Û (t, t0 ).  (2.42)  Setting h̄ = 1, t0 = 0 and rewriting (2.41) with (2.39) we now have G(t, k) = −iΘ(t)hk|e−itĤ |ki.  (2.43)  If we next define the set of eigenstates of the full Hamiltonian as |n, ki and the corresponding  eigenvalues En (k), we find  G(t, k, µ) = −iΘ(t)  X  e−it    En (k)−µ  n  |hn, k|ki|2 ,  (2.44)  where we chose to keep the dependence on µ explicit for reasons that will become clear later in this chapter. We can further define Zn (k) = |hn, k|ki|2 ,  (2.45) 49  2.2. Diagrammatic Monte Carlo which measures on a scale from 0 to 1 how similar the eigenstate of the polaron is to the bare electron eigenstate. For a discrete eigenstate, Zn (k) is called the quasiparticle weight. However, we will usually be interested in the lowest lying eigenstate such that will refer to Z0 (k) simply as the quasiparticle weight unless specified otherwise. We define a similar quantity for the bare electron, without the electron-phonon interaction, using Ĥ0 :    −it ǫ(k)−µ  G0 (t, k, µ) = −iΘ(t)e  .  (2.46)  Finally, using the usual definition for the Fourier transformations as given by f (ω) =  Z  ∞  iωt  dt e  f (t)  and  −∞  f (t) =  Z  ∞ −∞  dω −iωt e f (ω), 2π  (2.47)  we define the corresponding frequency bare Green’s function from (2.46), G0 (ω, k, µ) =  1 , ω − ǫ(k) + µ + iη  (2.48)  Zn (k) , ω − En (k) + µ + iη  (2.49)  and frequency Green’s function from (2.44), G(ω, k, µ) =  X n  where η ≥ 0 is infinitesimally small.  2.2.3  Imaginary time and imaginary frequency  The aforementioned Green’s functions were expressed in terms of real time and real frequency. One problem with these quantities is that they are complex-valued which means that a Monte Carlo sampling of terms contributing to these quantities will involve a sign problem. We will see that this can be avoided by analytic continuation of the Green’s functions to imaginary time, defined by τ = it. This substitution makes the exponential of the Green’s function (2.44) and (2.46) real and positive definite, and is therefore very well suited for a Monte Carlo computation. Similarly, an imaginary frequency can be defined by ξ = −iω. This allows us to define an imaginary time Green’s function Gi (τ, k, µ) and an imaginary frequency Green’s  function Gi (ξ, k, µ). We cannot, however, simply substitute τ and ξ in the expressions above and still preserve the usual Fourier transforms because they do not commute with the analytic continuation. We thus need to decide if we do the analytic continuation in time or in frequency first. Following [77], let us do the analytic continuation in frequency and obtain a definition 50  2.2. Diagrammatic Monte Carlo for the imaginary time Green’s function from (2.47). This choice will simplify the later use of some properties that apply only to the frequency Green’s function. By analytic continuation we have the following substitution in frequency: Gi (ξ, k, µ) = G(ω = iξ, k, µ) or such that  G(ω, k, µ) = Gi (ξ = −iω, k, µ),  X  Gi (ξ, k, µ) =  n  Zn (k) , iξ − En (k) + µ  (2.50)  (2.51)  where we can drop η. For the bare propagator we have Gi0 (ξ, k, µ) = We can then look at the Fourier transform Z i G0 (τ, k, µ) =  1 . iξ − ǫ(k) + µ  (2.52)  ∞  dξ e−iξτ Gi (ξ, k, µ)  = − Θ(τ )e− ǫ(k)−µ τ . −∞  (2.53)  For the full Green’s function we have Gi (τ, k, µ) = −Θ(τ )  X  Zn (k)e−  n    En (k)−µ τ  .  (2.54)  Therefore, we have the following transformation rules between real and imaginary time Green’s functions: Gi (τ, k, µ) = −iG(t = −iτ, k, µ) or  2.2.4  G(t, k, µ) = iGi (τ = it, k, µ).  (2.55)  Diagrammatics  We now show that the imaginary time Green’s function can be expressed as an infinite sum of Feynman diagrams. The diagrammatics are the rules to draw and calculate these diagrams. We start with the imaginary-time evolution operator for time-independent Hamiltonians Û (τ, 0) = e−τ Ĥ ,  (2.56)  51  2.2. Diagrammatic Monte Carlo and the Schrödinger equation −  ∂ Û (τ ) = Ĥ Û (τ ). ∂τ  (2.57)  Since H0 is simple to diagonalize, we switch from the Schrödinger representation to the  interaction representation. In this picture both the states and the operators carry part of the time-dependence. Defining a time-evolution operator for Ĥ0 only as Û0 , and denoting states and operators in the interaction representation by primes, a state is now given by |ψ ′ (τ )i = Û0 (0, τ )|ψ(τ )i,  (2.58)  Ô′ (τ ) = Û0 (0, τ ) Ô Û0 (τ, 0).  (2.59)  and an operator by  Since Ĥ0 commutes with itself we have Ĥ0′ (τ ) = Û0 (0, τ ) Ĥ0 Û0 (τ, 0) = eτ Ĥ0 Ĥ0 e−τ Ĥ0 = Ĥ0 ,  (2.60)  ′ Ĥint (τ ) = Û0 (0, τ ) Ĥint Û0 (τ, 0) = eτ Ĥ0 Ĥint e−τ Ĥ0 .  (2.61)  but  Rewriting Schrödinger’s equation in the interaction picture we find − −  ∂ ′ ′ |ψ (τ )i = Ĥint |ψ ′ (τ )i, ∂τ  ∂ ′ ′ ′ Û (τ ) |ψ0′ i = Ĥint (τ ) Ûint (τ ) |ψ0′ i, ∂τ int ∂ ′ ′ ′ (τ ) = Ĥint (τ ) Ûint (τ ), − Ûint ∂τ  (2.62)  where we defined a time-evolution operator in the interaction picture. We now solve for this operator knowing that we can get back ′ ′ Û (τ ) = Û0 (τ ) Ûint (τ ) Û0 (0) = Û0 (τ ) Ûint (τ ).  Next we solve ′ Ûint (τ )  =1−  Z  (2.63)  τ 0  ′ ′ dτ1 Ĥint (τ1 ) Ûint (τ1 ),  (2.64)  and by iteration ′ Ûint (τ )  =1−  Z  τ 0  ′ dτ1 Ĥint (τ1 ) +  Z  τ 0  ′ dτ1 Ĥint (τ1 )  Z  τ1 0  ′ dτ2 Ĥint (τ2 ) + . . .  (2.65) 52  2.2. Diagrammatic Monte Carlo which we rewrite as an infinite sum ′ Ûint (τ ) =  Z ∞ X (−1)j j!  j=0  τ  0  dτ1 · · ·  Z  τ  0  ′ ′ dτj T̂ [Ĥint (τ1 ) · · · Ĥint (τj )].  (2.66)  The integrals have been forced to be on [0, τ ] by using the time ordering operator T̂ , but we shall revert back to the previous notation shortly as it is more convenient for writing diagrams. Putting this back into (2.63) and (2.54) yields X ∞  Gi (τ, k, µ) = −Θ(τ )e−τ  ǫ(k)−µ  j=0  (−1)j j!  Z  Z τ ′ ′ (τ1 ) · · · Ĥint (τj )|ki]. (2.67) dτ1 · · · dτj T̂ [hk|Ĥint  τ  0  0  Next we go back to the Schrödinger picture using the inverse of (2.61) and we insert a full P ′ (τ ), where k basis k, z, {q} |k, q1 , . . . , qz ihk, q1 , . . . , qz | of eigenstates of Ĥ0 between each Ĥint  is the momentum of the electron, z is the number of phonons and qj are the phonon momenta. The zeroth order term of (2.67) is simply the bare Green’s function th  Gi, 0 (τ, k, µ) = −Θ(τ ) e−τ    ǫ(k)−µ  = Gi0 (τ, k, µ)  (2.68)  We further note that the odd terms of (2.67) vanish because Ĥint either adds or removes a phonon and an odd power of Ĥint applied to an eigenstate of Ĥ0 cannot have any overlap with  another eigenstate. In other words, we cannot contract, or pair, an odd number of phonon creation and annihilation operators in a way that leaves the number of phonons unchanged. The first non vanishing term is thus the second term of (2.67). We will label this contribution 1st because it involves one phonon which is created and later absorbed. We find i, 1st  G  Z  τ  −τ ǫ(k)−µ  (τ, k, µ) = − e  Z  dτ1 0  0  τ1  dτ2 hk| Û0 (−τ1 ) Ĥint  X  k1 , q 1   X |k2 , q2 ihk2 , q2 | Ĥint Û0 (τ2 ) |ki. · Û0 (τ1 − τ2 )   |k1 , q1 ihk1 , q1 | · (2.69)  k2 , q 2  Since bq |ki = 0, only the creation operator contributes at τ2 and 1 X Ĥint |ki = √ g(k, −q)|k − q, qi. N q  (2.70)  From the orthogonality of the eigenstates we find that q2 = q = q1 and k2 = k − q = k1 . This 53  2.2. Diagrammatic Monte Carlo time only the annihilation operator contributes at τ1 and i, 1st  G  Z τ1 Z  1 X τ dτ1 dτ2 e−(τ −τ1 ) ǫ(k)−µ g(k − q, q)· (τ, k, µ) = − N q 0 0   · e−(τ1 −τ2 ) ǫ(k−q)+ω(q)−µ g(k, −q)e−τ2 ǫ(k)−µ .  (2.71)  This can be represented diagrammatically very simply if we separate the electron and the phonon contribution of Û0 . For each electron created at time τ2 and annihilated at time τ1 , we  draw a straight line of length τ1 −τ2 with an associated index k, and its value is e−(τ1 −τ2 ) ǫ(k)−µ .  Similarly a phonon created at τ2 and annihilated at τ1 is represented by a dashed line above the  electron line, going from τ2 to τ1 with an associated index q, and a value of e−(τ1 −τ2 )ω(q) . The interaction is a vertex with an electron line going in and one going out plus a phonon line going √ either in or out. The value of a vertex is g(kin , kout − kin )/ N with kin being the momentum  of the incoming electron and kout the momentum of the outgoing electron. Momentum is thus  conserved at the vertex with q = kout − kin for an absorbed phonon and q = kin − kout for an  emitted phonon. The values of each of these parts are multiplied together and then multiplied by −1 to get the value of the diagram. The topology of the diagram in (2.72) is shown in Figure 2.5. We still need to sum over, or sample, all values of q, τ1 and τ2 to obtain Gi, 1st (τ, k, µ) from this topology.  q k  k 0  2  q  k 1  st  Figure 2.5: First order diagram Gi, 1 (τ, k) contributing to the imaginary time Green’s function.  The next non-vanishing term in (2.67) is the fourth order term in the interaction, i.e. second order in the number of phonons. Once again for the first (last) vertex, this time at τ4 (τ1 ), only the phonon creation (annihilation) operator contributes. For the two other vertices we need to have one of each again, but we will have different types of diagrams depending on their order and their specific pairing, for a total of three different topologies. Figure 2.6 shows the  54  2.2. Diagrammatic Monte Carlo diagrams corresponding to the following equation " X Z τ Z τ1 Z τ2 Z τ3 1 dτ1 dτ2 dτ3 dτ4 e−(τ −τ1 ) ǫ(k)−µ Gi, 2nd (τ, k, µ) = − 2 N q ,q 0 0 0 0 1 2  g(k − q2 , q2 )e−(τ1 −τ2 ) ǫ(k−q2)+ω(q2 )−µ g(k − q1 − q2 , q1 )·   · e−(τ2 −τ3 ) ǫ(k−q1−q2 )+ω(q1 )+ω(q2 )−µ g(k − q1 , −q2 )e−(τ3 −τ4 ) ǫ(k−q1 )+ω(q1 )−µ g(k, −q1 )  +g(k − q1 , q1 )e−(τ1 −τ2 ) ǫ(k−q1)+ω(q1 )−µ g(k − q1 − q2 , q2 )·   −(τ2 −τ3 ) ǫ(k−q1 −q2 )+ω(q1 )+ω(q2 )−µ −(τ3 −τ4 ) ǫ(k−q1 )+ω(q1 )−µ ·e g(k − q1 , −q2 )e g(k, −q1 )  +g(k − q2 , q2 )e−(τ1 −τ2 ) ǫ(k−q2)+ω(q2 )−µ g(k, −q2 )·   · e−(τ2 −τ3 ) ǫ(k)−µ g(k − q1 , q1 )e−(τ3 −τ4 ) ǫ(k−q1)+ω(q1 )−µ g(k, −q1 ) #  e−τ4 ǫ(k)−µ . (2.72)  q1 q2 0  4  3  2  1  0  4  q1  q2  3  2  q1 1  0  4  q2 3  2  1  nd  Figure 2.6: Second order diagram Gi, 2 (τ, k) contributing to the imaginary time Green’s function. The full expansion for Gi (τ, k, µ) can thus be calculated by generating all diagrams admitted by (2.67). We conclude this section with a number of remarks. We note that in the thermodynamic limit we can of course replace the sums over momenta by integrals such that a vertex now is √ g(kin , kout − kin )/ 2π. We should also point out that the topology of the diagrams derived here is not affected by the Fourier transform or the analytic continuation so that all the Green’s functions defined above admit a diagrammatic expansion with the same topology. The main difference lies in the value of each part of the diagram and the indices associated with them. We also note that what is essentially a convolution of multiple terms in time becomes a simple 55  2.2. Diagrammatic Monte Carlo product in frequency.  2.2.5  Ground state energy, quasiparticle weight and effective mass  It will be convenient to work with a positive-definite Green’s function G̃i (τ, k, µ) = −Gi (τ, k, µ).  We have already defined the quasiparticle weight of the polaron of momentum k as Z0 (k) and we define the polaron energy as the lowest eigenstate E0 (k) of the set {En (k)}. We now show how  these can be extracted from the long time behaviour of the imaginary-time Green’s function given by (2.54). For large τ all other contributions to the Green’s function vanish exponentially faster, and we are left only with G̃i (τ, k, µ) = lim Θ(τ ) τ →∞  X  Zn (k)e−    En (k)−µ τ  n  = Θ(τ )Z0 (k)e−    E0 (k)−µ τ  .  (2.73)   We can thus extract the polaron energy from the slope at large τ of the ln G̃i (τ, k, µ) e−µτ ,  while the quasiparticle weight can be extracted by exponentiating the offset of the same linear  approximation. The chemical potential had to be removed to allow comparison with other results as it is used only for simulation purposes here. The effective mass is another standard way of measuring how strongly the interaction modifies the properties of the dressed particle compared to the bare electron. It is defined as ∗  m =    ∂ 2 E0 (k) ∂k 2  −1  ,  (2.74)  k=0  and is to be compared with 1/2t for the bare electron. Another interesting quantity is the average number of phonons nph in the cloud. There are some direct ways of calculating the number of phonons in a DMC code, but they either involve calculating more complicated Green’s functions (than the ones presented above) with phonons present in the initial and final states, or they do not generalize well to the more advanced techniques presented below. We instead shall rely on the Hellmann-Feynman theorem [26] which states that nph (k) = h0, k|  X q  b†q bq |0, ki =  ∂E0 (k) . ∂ω(q)  (2.75)  This of course involves running at least two simulations, one with phonon dispersion ω(q) and the other with ω(q) + δ and carrying a numerical derivative. One can also look at the average number of phonons in a specific momentum or momentum range by making δ(q) momentumdependent.  56  2.2. Diagrammatic Monte Carlo  2.2.6  Monte Carlo sampling and normalization  Since the configurations under consideration here are diagrams, ν will stand for the position of all vertices τi and the momenta of all the phonon lines; Wν is evaluated according to the diagrammatic rules presented above. Different values of the electron momentum need to be carried separately. In theory different values of τ could also be calculated separately but this would not be very efficient. Instead, we will consider the enlarged configuration space of all diagrams with any length τ and sample τ as one more index. To calculate the Green’s function we will not need to consider the denominator of (2.3). Instead, we will normalize the answer knowing that G̃i (0, k) = 1. This follows directly from the properties of the time-evolution operator. We now have hG̃i (τ, k)i =  X δ(τ − τν )Wν , Z(k)  (2.76)  {ν}  where Z(k) is the partition function.  2.2.7  Updates  All that is needed now is a set of updates that will satisfy the ergodicity principle. A minimum set of updates is given by • Changing the length of the diagram τ ∈ [0, τmax ] if no phonons are present, or between τ ∈ [τlast , τmax ] where τlast is the time of the last vertex if there are phonons  • Inserting a phonon with momentum q ∈ [0, 2π[ with vertices at τ1 ∈ ]0, τ ] and τ2 ∈ [0, τ1 [ • Deleting one of the phonons For simplicity we assume that each update is chosen with equal probability of 1/3. Updates are balanced in pairs: inserting and deleting a phonon are balanced together, while changing the length of the diagram can be easily balanced with itself by pairing each specific update with its inverse update. The probability distribution can be chosen to be uniform Pchange τ (τ ) = =  1 τmax  ,  1 , τmax − τlast  if nph = 0 if nph > 0  (2.77)  where τlast is the time for the last vertex of the diagram. A better choice, because it is exactly the same as the ratio of weights of the diagrams, would be to distribute the new length according 57  2.2. Diagrammatic Monte Carlo to the value of the bare propagator [ǫ(k) − µ]e−[ǫ(k)−µ]τ , 1 − e−[ǫ(k)−µ]τmax [ǫ(k) − µ]e[ǫ(k)−µ]τ , = −[ǫ(k)−µ]τ last − e−[ǫ(k)−µ]τmax e  Pchange τ (τ ) =  if nph = 0 if nph > 0  (2.78)  Since this update is balanced with itself, the normalization factors of this distribution cancel and the acceptance ratio to go from ν to ν ′ is simply Rchange τ =  Wν ′ [ǫ(k)−µ](τ ′ −τ ) ′ ′ e = e−[ǫ(k)−µ](τ −τ ) e[ǫ(k)−µ](τ −τ ) = 1. Wν  (2.79)  This update is maximally efficient since the suggested update is always accepted. There are many ways in which the phonon insertion and deletion can be balanced together. We choose to sample the momentum uniformly over [0, 2π[. We then choose the time of the first vertex τ1 uniformly on ]0, τ ]. For the second vertex we use the value of the phonon propagator ∝ e−ω(q)(τ2 −τ1 ) to sample the time of the second vertex. This will keep the probability distri-  bution closer to the ratio of weights, but ignores the more complicated values of the electron propagators. The resulting probability distribution for inserting a specific phonon is then Pinsert =  1 1 1 ω(q)e−ω(q)(τ2 −τ1 ) . · · · 3 2π τ 1 − e−ω(q)(τ −τ1 )  (2.80)  The inverse update, deletion, simply selects one phonon out of the nph + 1 phonons in the diagram ν ′ . Remember that the diagram ν ′ has one more phonon than the diagram ν, and since we are calculating the acceptance ratio of the insertion update, the number of phonons used to do this calculation is the number of phonons of the diagram ν. The probability to select the specific phonon corresponding to the current insertion is thus 1/(nph + 1) and the overall probability is  1 1 · , 3 nph + 1  (2.81)  Wν ′ 2πτ 1 − e−ω(q)(τ −τ1 ) . · Wν (nph + 1) ω(q)e−ω(q)(τ2 −τ1 )  (2.82)  Pdelete = and the acceptance ratio is Rinsert =  58  2.2. Diagrammatic Monte Carlo Similarly, for the deletion update we choose a phonon with probability 1/nph and the acceptance ratio becomes Rdelete =  2.2.8  Wν ′ nph ω(q)e−ω(q)(τ2 −τ1 ) . · Wν 2πτ 1 − e−ω(q)(τ −τ1 )  (2.83)  Chemical potential  As mentioned above, the chemical potential is simply a change of the zero of energy used for computational convenience. The large-τ behaviour of Gi (τ, k) implies that large diagrams are exponentially suppressed. The very small number of diagrams generated at large time means that we might not have a statistically good enough sampling to estimate the slope and offset at the origin of log(Gi (τ, k)). To remedy this situation we need to artificially improve the sampling of long-time diagrams by scaling their weight by a factor f (τ ) and correcting for this when collecting statistics for the MC sums by adding (1/f (τ )). The function f is called a fictitious potential. In the case of DMC we want to counter the exponential decrease by multiplying the value of a diagram by another exponential factor exp(µτ ). Since the bare propagator is simply an exponential factor, this fictitious potential is equivalent to shifting the zero of energy, hence the name chemical potential. In practice we do not remove the effect of the chemical potential in MC sums, but simply rescale the results when calculating other quantities. This chemical potential obviously needs to be chosen below E0 (k), otherwise mostly large τ diagrams will be generated, and the statistics at small τ will be poor, leading to an inaccurate normalization. The number of phonons will also increase with the length τ of the diagram, which will slow down the update mechanism. Thus µ needs to be set below, but close to E0 (k), to provide a good estimate of the asymptotic behaviour, while allowing for an adequate normalization. The maximum length τmax of the diagram will also need to be set accordingly, to easily extract the asymptotic behaviour without spending more time than needed generating very long, and computationally costly, diagrams.  2.2.9  Orthonormal functions  We need an efficient way of storing the continuous functions like the imaginary time Green’s function. A simple histogram would do if the spacing was chosen to be small enough. One problem, however, is that a small bin size means a smaller number of points in each bin, and thus a larger statistical error. Since the number of points is going to be larger for smaller τ , a histogram with bin size increasing with τ will do better. It will provide a better sampling at 59  2.2. Diagrammatic Monte Carlo small times for an accurate normalization, while helping to accumulate enough points per bin at large τ to average over statistical noise. When collecting statistics with variable-sized bins, we need to include a factor of 1/δτ , with δτ the size of the bin, so that each bin is properly normalized with respect to each other. Storing the Green’s function might be a technical detail of the algorithm, but the accuracy of the normalization will greatly depend on being able to estimate the τ = 0 limit. With a simple histogram the first few points will have to be used to estimate this value. A better way is to go beyond a simple average inside each bin, and to expand the Green’s function over the range of this bin (of size δτ ) with a set of orthonormal functions Fn (τ − τm ) centred at the bin  centre τm . For example, an arbitrary function f (τ ) can be written as f (τ ) =  X a  ca Fa (τ − τm ) + correction  for τ ∈ [τm − δτ /2, τm + δτ /2],  (2.84)  where the ca are the coefficients of the expansion. A full basis of functions would need to be used for an exact expansion, but a finite set of functions will be sufficient if the bin size δτ is not too large, and if the function is smooth on this range. Any set of orthonormal functions would work and the Gram-Schmidt orthogonalization procedure can be used to normalize the chosen set if needed. The most obvious choice is the set of Legendre polynomials Pn (τ − τm ),  but normalized such that  Z  δτ /2  dτ Fa2 (τ ) = 1,  (2.85)  −δτ /2  instead of the usual Pn (1) = 1. The first 4 orthonormal functions used in this work are r  1 δτ r 12 (τ − τm ) F1 = δτ 3 r   δτ 2 180 2 F2 = (τ − τm ) − δτ 5 12 r   12δτ 2 2800 3 F3 = (τ − τm ) (τ − τm ) − δτ 7 80 F0 =  ...  (2.86)  Collecting statistics for a specific bin is now done for each coefficient ca . After an update,  60  2.3. Self-Energy Diagrammatic Monte Carlo if the diagram has a length that falls on the bin’s range, each coefficient is updated with ca → ca + Fa (τ − τm ).  (2.87)  Reconstruction of the original function on the range of the bin is done with (2.84).  2.3  Self-Energy Diagrammatic Monte Carlo  Since DMC is a generic term that can apply to all the techniques presented in this work, we shall refer to algorithms that focus on calculating the Green’s function as G-DMC. The Self-energy Diagrammatic Monte Carlo (Σ-DMC) algorithm is conceptually very close to the G-DMC algorithm, but samples the self-energy Σ instead. Its diagram expansion has fewer terms and should therefore prove faster, provided we can get the Green’s function afterward, or extract some interesting information directly from it. This will be our stepping stone for the Bold Diagrammatic Monte Carlo (BDMC) algorithm, to which we will turn next. Some of the material presented here is covered by Prokof’ev and Svistunov in their introduction of the BDMC technique [77, 81, 82].  2.3.1  Self-energy and self-energy diagrams  In this section we will draw the diagrams in their frequency representation such that the length of the propagators used is irrelevant. The equivalent diagrams in time would have the same topology as pointed out earlier, but would need to sample all possible times of the vertices. The self-energy Σ(ω, k, µ) is a quantity that will allow us to reorganize the diagrams of the Green’s function into a much smaller set. Figure 2.7 shows the diagrams for G(ω, k, µ) up to third order. To simplify this expression we can think of these diagrams as blocks with inseparable phonons linked by bare electron propagators. We define a new type of diagram, the self-energy, as the sum of all the possible diagrams that cannot be separated into a number of pieces by cutting only an electron line. We can then rewrite the diagram expansion for G(ω, k, µ) by using a combination of self-energy blocks connected together by bare propagators G0 (ω, k, µ), as shown in Figure 2.8. Since the value of a diagram is obtained by multiplying all of its components, each diagram using the self-energy already represents an infinity of diagrams. Properly speaking, these diagrams are called proper self-energy diagrams and are shown up to third order in Figure 2.9. Figure 2.8 also shows that this diagram expansion is a simple geometric series. This means  61  2.3. Self-Energy Diagrammatic Monte Carlo  G  =  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +…  Figure 2.7: Diagrams contributing to the real frequency Green’s function G(ω, k) up to third order in the phonon number. The length of a propagator is irrelevant, each phonon propagator i has a momentum qi , and each electron propagator j has a momentum kj = P P k − i′ ∈{phonons above j} qi′ and a frequency ωj = ω − i′ ∈{phonons above j} ω(qi′ ). Only the topology is shown and the momentum and frequency indices are implicit. that the Green’s function can also be expressed in the following way G(ω, k, µ) =  G−1 0 (ω,  1 1 = , ω − ǫ(k) + µ − Σ(ω, k, µ) + iη k, µ) − Σ(ω, k, µ)  (2.88)  for real frequency. In imaginary frequency, we have Gi (ξ, k, µ) =  Gi0 −1 (ξ,  1 1 = , i iξ − ǫ(k) + µ − Σi (ξ, k, µ) k, µ) − Σ (ξ, k, µ)  (2.89)  where Σi (ξ, k, µ) is the Fourier transform of Σi (τ, k, µ), which can be calculated with the DMC algorithm. The imaginary time self-energy can be calculated using the same diagrammatic rules  62  2.3. Self-Energy Diagrammatic Monte Carlo  G =  =  =  +  +  1+  +  +  +  +…  +…  1 1  Figure 2.8: Diagrams contributing to the real frequency Green’s function G(ω, k, µ) in terms of the self-energy Σ(ω, k, µ). The length of a propagator is irrelevant and each of them has a momentum k and a frequency ω.  as those used in the G-DMC method for the propagators and the interaction vertices, but using the topology of the diagrams in Figure 2.9. Equations (2.88) and (2.89) are solutions of the self-consistent equation known as Dyson’s equation which we only wrote for the time-evolution operator before solving it by iteration. We derive it below for completeness by starting from the real frequency Green’s function operator, to simplify the notation. We have Ĝ(ω) =  1 ω − Ĥ + iη  .  (2.90)  Bringing the denominator to the left hand side and using Ĥ = Ĥ0 + Ĥint we find (ω − Ĥ + iη) Ĝ(ω) = 1 (ω − Ĥ0 + iη) Ĝ(ω) = 1 + Ĥint Ĝ(ω) Ĝ0 (ω) (ω − Ĥ0 + iη) Ĝ(ω) = Ĝ0 (ω) + Ĝ0 (ω) Ĥint Ĝ(ω) Ĝ(ω) = Ĝ0 (ω) + Ĝ0 (ω) Ĥint Ĝ(ω)  (2.91)  The last line is the Dyson identity. We shall however refer to its solutions (2.88) and (2.89)  63  2.3. Self-Energy Diagrammatic Monte Carlo  =  +  +  +  +  + +  + +  +  +  + +  +…  Figure 2.9: Diagrams contributing to the real frequency self-energy Σ(ω, k, µ). The length of a propagator is irrelevant, each phonon propagator i has a momentum qi , and each electron P propagator j has a momentum k = k − j i′ ∈{phonons above j} qi′ and a frequency ωj = ω − P i′ ∈{phonons above j} ω(qi′ ). Only the topology is shown and the momentum and frequency indices are implicit. under the same name. Going to the momentum representation G(k) = hk|Ĝ|ki we have G(ω, k) = G0 (ω, k) + G0 (ω, k) hk| Ĥint  X q   |k − q, qihk − q, q| Ĝ(ω)|ki,  X g(k, −q) √ = G0 (ω, k) + G0 (ω, k) hk − q, q|Ĝ(ω)|ki, N q  (2.92)  where |k − q, qi denotes a state with one electron of momentum k − q and one phonon of  momentum q.  Dyson’s equation gives us the means to obtain the imaginary time Green’s function from the imaginary time self-energy, by first Fourier transforming to imaginary frequency, applying Dyson’s equation and then Fourier transforming back. From there, all the usual quantities like the polaron energy, quasiparticle weight and effective mass can be calculated.  64  2.3. Self-Energy Diagrammatic Monte Carlo  2.3.2  Ground state energy and quasiparticle weight from the self-energy  Prokof’ev and Svistunov showed in [77] that most of the relevant observables can be extracted directly from the imaginary time self-energy without having to rely on Dyson’s equation to obtain the Green’s function. We first note that both the imaginary time Green’s function and the self-energy depend on the chemical potential µ exclusively through the exponential factor exp(µτ ), which means in imaginary frequency that Gi (ξ, k, µ) ≡ Gi (ξ − iµ, k)  and Σi (ξ, k, µ) ≡ Σi (ξ − iµ, k).  (2.93)  We also recall the asymptotic limit of (2.73), which implies that the Green’s function has a pole, and can be written as Gi (ξ − iµ, k) =  Z0 (k) + regular part. iξ + µ − E0 (k)  (2.94)  By setting µ = E0 (k) and comparing to (2.89) we get iξ/Z0 (k) = iξ − ǫ(k) + E0 (k) − Σi (ξ − iE0 (k), k).  (2.95)  We then consider the limit ξ → 0 and expand Σi (ξ − iE0 (k), k) to first order. We define A(k, E0 (k)) = −  ∂ i Σ (ξ − iµ, k) ∂ξ  ξ−iµ=−E0 (k)  = −i  ∂ i Σ (−iµ, k) ∂µ  .  (2.96)  µ=E0 (k)  We insert the expansion to first order using this definition back in (2.95) and find that iξ/Z0 (k) = iξ − ǫ(k) + E0 (k) − Σi (−iE0 (k), k) + iξA(k, E0 (k)).  (2.97)  The real and imaginary part of this equation yield two interesting results: E0 (k) = ǫ(k) + Σi (0, k, E0 (k)), and Z0 (k) =  1 . 1 + A(k, E0 (k))  (2.98)  (2.99)  The former can be used to calculate the polaron energy. We recognize the zero frequency component Σi (0, k, E0 (k)) as being the average over time of Σi (τ, k, E0 (k)). Assuming the DMC sampling of Σi (τ, k, µ) was done for some convenient chemical potential, we simply need  65  2.3. Self-Energy Diagrammatic Monte Carlo to correct for that by multiplying by an exponential factor and therefore E0 (k) = ǫ(k) +  Z  ∞  dτ Σi (τ, k, µ) e(E0 (k)−µ)τ .  (2.100)  0  To obtain A(k, E0 (k)) we need to use a well-known property of the Fourier transform. Taking a derivative with respect to frequency in the frequency domain is equivalent to multiplying the time-domain quantity by the time variable. Since (2.96) involves again the zeroth frequency, we can simply take the time-average of the self-energy multiplied by τ : A(k, E0 (k)) =  2.3.3  Z  ∞  dτ τ Σi (τ, k, µ) e(E0 (k)−µ)τ .  (2.101)  0  Normalization  Normalization can be achieved in many ways, but the simplest is to use the equivalent τ → 0 limit of the self-energy that we used for the Green’s function. In this case, however, the topology of the proper self-energy diagram require that we have one phonon. For any τ , we find Σi (τ, τ0 , k) = −Θ(τ )hk, τ | Ĥint Û (τ, 0) Ĥint |k, τ0 i.  (2.102)  Setting τ0 = 0 without loss of generality and dropping the explicit time dependence from the states we find the following limit lim Σi (τ, k) = −hk| Ĥint Ĥint |ki,  τ →0  (2.103)  where we used Û (0, 0) = 1. As a result, ih i ih h ih 1 X g(k − q, q)g(k, −q) h0|ck c†k ck−q bq c†k−q ck b†q c†k |0i N q 1 X =− g(k − q, q)g(k, −q). (2.104) N q  Σi (τ = 0, k) = −  In the thermodynamic limit where N → ∞ we can replace the sum by an integral Σi (τ = 0, k) = −  1 2π  Z  0  2π  dq g(k − q, q)g(k, −q).  (2.105)  66  2.3. Self-Energy Diagrammatic Monte Carlo  2.3.4  Updates  The minimal set of updates is fairly similar to G-DMC. It must include an update to insert a phonon, an update to delete a phonon and an update to change the total length of the selfenergy diagram. We must now start with at least one phonon and we cannot allow this last phonon to be removed. This means that an extra update to change the momentum of a phonon also needs to be added. The absence of the bare propagator before and after the proper part means that the insertion of a phonon must now allow for a change of the diagram length by considering also phonons inserted past the current length of the diagram. In this version we will actually replace the update to change the length of the diagram by a more general update that will shift a vertex position. Once again we will balance this update with itself since the inverse of shifting a vertex is to shift it back to its original position. We first select one of the vertices at random, except for the first one which we will keep fixed at τ = 0, and then select a new position for this vertex anywhere between the previous vertex and the next one. The last vertex is a special case for which we will select a new position between the previous one and τmax . The position can be chosen according to some suitable probability distribution to more closely match the weight ratio, but in the simplest version where the position is uniformly distributed, the acceptance ratio is Rshift =  Wν ′ . Wν  (2.106)  For the insertion we follow the same basic rules as for G-DMC, but instead of sampling the position of the second vertex on [τ1 , τ ] we instead choose it on [τ1 , τmax ] and leave it to the exponential distribution to prevent too many update suggestions that would result in very long and costly diagrams. We also prevent the removal the first phonon of the diagram. The acceptance ratio is thus almost the same as before Rinsert = F(ν ′ ) ·  Wν ′ 2πτ 1 − e−ω(q)(τmax −τ1 ) , · Wν nph ω(q)e−ω(q)(τ2 −τ1 )  (2.107)  where we need to add an extra factor given by F(ν ′ ) = 1, 0,  if ν ′ is a proper self-energy diagram otherwise,  (2.108)  to prevent improper self-energy diagrams from being generated.  67  2.4. Bold Diagrammatic Monte Carlo The removal of a phonon is found in a similar way and the acceptance ratio is Rdelete = F(ν ′ ) ·  Wν ′ (nph − 1) ω(q)e−ω(q)(τ2 −τ1 ) · . Wν 2πτ 1 − e−ω(q)(τmax −τ1 )  (2.109)  Finally the update to change the momentum of a phonon simply selects one of the phonons and changes the momentum from its previous value to any other value with equal probability. Balancing this equation with itself gives the acceptance ratio Rchange momentum =  Wν ′ . Wν  (2.110)  We should also mention that in the special case of a Hamiltonian with more than one branch, an update similar to the previous one needs to be included. For G-DMC, such an update was not necessary since a phonon from one branch could be removed, and one from a different branch can then be inserted in its place. The acceptance ratio for insertion and deletion would then of course need a factor of nbranch and 1/nbranch respectively. For Σ-DMC, however, we are not allowing the first phonon to be removed, and the branch and the momentum of this first phonon need to be sampled with a specific update.  2.4  Bold Diagrammatic Monte Carlo  The Bold Diagrammatic Monte Carlo (BDMC) algorithm [77, 81, 82] is an even faster DMC sampling method based on a self-consistent version of the Σ-DMC algorithm. The main idea is to use a more complete drawing element than the bare propagator such that each diagram drawn will account for many diagrams of the self-energy expansion at once. If this drawing element depends on the self-energy diagrams already summed, the resulting number of Σ diagrams will grow much faster with simulation time than the approximate linear increase achieved by ΣDMC. Even better, if the drawing element depends on the Green’s function diagrams generated so far through the Dyson equation, a single bold Σ diagram now accounts for an infinite number of diagrams, and we obtain an infinite set of bare Σ diagrams which grows even faster.  2.4.1  Bold line and double-counting  We first show how including two diagrams in our basic drawing element can generate a much larger set of diagrams. Once again we assume that the diagrams are in the frequency domain, where the length of the propagators do not matter, to simplify the figures. The actual implementation will of course use the imaginary time representation. Figure 2.10 first defines a 68  2.4. Bold Diagrammatic Monte Carlo bold propagator as a dashed thick line. In this simple example, the thick line only includes two diagrams, the bare propagator and the first order diagram for the Green’s function. The first-order diagrams for the self-energy generated with this thick line give two diagrams. The two second-order diagrams drawn with thick lines account for 15 types of diagrams. We thus include many more diagrams at once with this very simple propagator. We have, however, double-counted 2 types of diagrams up to this order. To properly evaluate the self-energy we will need to forbid double-counting. The crossed phonon lines of the second self-energy diagram do not result in any double counting by themselves, or with the first order self-energy diagrams. Crossing diagrams are therefore worth studying and might provide a hint at the solution to avoid double counting.  2.4.2  Bold algorithm  The more complete the thick line used, the faster the diagrams are accounted for. Instead of fixing the thick line to a subset of diagrams it would be advantageous if it could grow in complexity as the simulation progresses. Fortunately, for Σ-DMC we have a simple relationship between self-energy diagrams and Green’s function diagrams, in the form of Dyson’s equation. The geometric series nature of Dyson’s equation means that even first order diagrams for Σ translate to an infinite set of diagrams for G. We will therefore define the bold line or propagator as the sum of Green’s functions diagrams obtained from our best approximation to the self-energy that we have available. The basic algorithm will thus go as follows  69  2.4. Bold Diagrammatic Monte Carlo BDMC 1. Initialize Giapprox (τ, k, µ) = Gi0 (τ, k, µ) for a set of momenta {k}. 2. Draw a first diagram for each k (first order for example). 3. Generate diagrams for Σi (τ, k, µ) for each k with the usual update mechanism with appropriate restrictions to avoid double counting. Diagrams are drawn using Giapprox for electron propagators. 4. Repeat step 3 a predefined number of times. 5. Fourier transform the current approximate result for Σi (τ, k, µ) to obtain Σi (ξ, k, µ). 6. Use Dyson equation for each k to get an approximate value for Giapprox (ξ, k, µ). 7. Fourier transform back to get a new Giapprox (τ, k, µ). 8. Go back to step 3. 9. After enough iterations, or bold loops, Giapprox (τ, k, µ) ≈ Gi (τ, k, µ) We note the need to solve for all momenta at once because the momentum of a specific electron propagator can have any value after a phonon is created. The simplest way of obtaining the self-energy and thus the Green’s function for all momenta is to define a grid in momentum space, and have a diagram for each of these values updated separately. The Green’s function for other momenta, when needed, can be approximated by interpolation. The chemical potential should also be made momentum-dependent, to ensure that each value of momentum calculated samples the large τ behaviour. When calculating the bold propagator of momentum k′ inside a diagram of momentum k, we will need to match the chemical    potential by adjusting Giapprox τ, k′ , µ(k′ ) to Giapprox τ, k′ , µ(k′ ) · exp [µ(k) − µ(k′ )]τ to calculate the value of the diagram.  The set of updates can be exactly the same as for Σ-DMC with the one modification to F(ν ′ ) which should now return 1 only if the diagram is not going to cause any double counting, and 0 otherwise. This condition will be made explicit in the following section.  2.4.3  Restrictions to avoid double counting  Since we are now using a bold line, we need to avoid drawing a diagram which could be obtained by expanding the bold line. In other words, any phonon or group of self-contained phonons 70  2.4. Bold Diagrammatic Monte Carlo lines that can be absorbed in the bold line without changing the topology and the momentum of the rest of the diagram should not be allowed. By a self-contained group of phonons, we mean that the group only has one incoming and one outgoing electron line of the same momentum, and no other phonon line. Figure 2.11 gives a few examples of forbidden and allowed diagrams. Figure 2.12 shows a simple algorithm to check if a diagram is forbidden or allowed by assigning each phonon a unique number and creating lists of phonons covering each electron propagator. Each propagator needs to have a distinct phonon list to be allowed. Allowed diagrams are referred to as fully crossed diagrams.  2.4.4  Fast Fourier Transform  The need to go to frequency space to use Dyson’s equation implies that we will need to evaluate a numerical Fourier transform. The continuous Fourier transform could be obtained by calculating the continuous integral numerically and imposing τmax as an upper limit to the integral, but a much more efficient technique consists in using a Discrete Fourier Transform (DFT) instead, for which efficient algorithms called Fast Fourier Transform (FFT) exist. The use of DFTs to approximate continuous signals and analyze them numerically is an important topic of digital signal processing, and we shall only scratch the surface here. Figure 2.13 summarizes the relations between the Fourier transform of a periodic and aperiodic signal and the Fourier transform of a periodic and aperiodic discrete sequence. Sampling in time (frequency) is easily done and corresponds to a folding in frequency (time). Folding means more than just periodizing the function in question. It is made periodic by repeating an infinite number of times the aperiodic signal, each time shifted by one period. If the original signal is zero outside of the period of the repeated signal we can recover exactly the aperiodic signal from the periodic one. If the function is not zero outside of this range, each value of the periodic function is the sum of many non-zero values of the original function. This phenomenon is called aliasing. The bandwidth (duration) of the function to fold thus needs to be smaller than the sampling frequency (period) for the inverse operations of unfolding and reconstruction. This is known as the sampling theorem, and states that the sampling frequency needs to be larger than the bandwidth of the signal for the sampling to be lossless. We define the bandwidth (duration), as the full range of frequency (time) where the function is non-zero (as opposed to the usual signal processing definition which only considers positive frequencies). Reconstructing a function from its samples can be done in many ways. Figure 2.13 indicates the ideal interpolation formula for each conversion for completeness. For our specific needs, we assume that we start from a continuous time function Σi (τ ) which we sample at some sampling frequency Fs = 1/T , with T being the distance, or time, 71  2.4. Bold Diagrammatic Monte Carlo between two samples. Of course Σi (τ ) is already stored in some discrete form, but for the current purpose, the orthonormal representation described above can be considered a continuous representation which we need to sample. Since we only keep this function up to some τmax , we effectively set Σi (τ > τmax ) = 0. This is only an approximation, and we need to remember that the DFT will not be an exact representation of the Fourier transform. This does mean, however, that we need not worry about aliasing at this point. Next we can use a FFT procedure to obtain the DFT, which we denote Gi (k). From there, we need to get back to Σi (ξ) to apply Dyson’s equation. We do not need to reconstruct this quantity, only its samples. The sample of the Fourier transform of the discrete and aperiodic function (with frequency variable w) is related to the DFT coefficients (labelled by k) by Σ  i    2πk w= N    = Σi (k),  (2.111)  where N is the number of samples used for the DFT. Σi (w) is in turn related to the Fourier transform by i  Σ (ξ) = T Σ such that  i     ξ w= , Fs  (2.112)    2πFs k Σi ξ = = T Σi (k). N  (2.113)  The left-hand side is the required sample. If we perform the same operation for Gi0 we simply get the Green’s function with Dyson’s equation i  G    2πFs k ξ= N    =  Gi0    ξ=  2πFs k N  or in terms of the DFT Gi (k) =  [Gi0 (k)]−1  1 −1  −  Σi  1 . − T 2 Σi (k)    ξ=  2πFs k N  ,  (2.114)  (2.115)  Applying the inverse DFT of the last quantity gives us the discrete and periodic form of Gi (τ ). We define an aperiodic Green’s function simply by setting the first N values to the result of the inverse DFT and the rest to zero. Finally, we reconstruct the continuous Gi (τ ). Figure 2.13 suggest an ideal interpolation scheme, but for our purpose we need to use a rather large N to get an acceptable accuracy. With a high enough sampling frequency, we can simply use a linear interpolation scheme. One more technical detail needs to be taken care of. The Fourier transform is defined 72  2.4. Bold Diagrammatic Monte Carlo for continuous and smooth functions. When a function has a jump discontinuity, the synthesis equation of the Fourier transform (i.e. the inverse Fourier transform) will reproduce the original function everywhere, except at discontinuities, where it will converge to the average of the left and right limit of the original function. This means that when we sample Σi (τ ), the first point should be set to the average between Σi (0) and Σi (τmax ). If the function is padded with zeros before the DFT, the first value is simply Σi (0)/2 and the last non zero point is Σi (τmax )/2. When obtaining the Green’s function we also need to double the first value assuming some padding was used. Padding in general does not improve the accuracy of the DFT used as an approximation for the Fourier transform. It increases the resolution in frequency and thus helps to compare the DFT to the exact Fourier transform if it is known, but it will not improve the results for Gi (τ ) in the end. Having at least one zero will however simplify obtaining Gi (τ = 0).  2.4.5  Error analysis  We have seen how the sample standard error can be calculated or estimated using the blocking method. This technique is still valid when applied between two bold loops, but each update of Giapprox redefines the set of updates, and the overall probability distribution of the algorithm. We could still approximate the error by starting the error analysis when the bold code has reached an equilibrium where Giapprox does not change much. It is however costly in memory, as the error analysis described above has to be carried out for each MC estimator, namely for each bin j of the self-energy Σi (τj , k), or for each of the coefficients of the orthonormal functions used to describe the self-energy. The actual errors on the energies or the quasiparticle weights then need to be calculated according to their dependence on each MC estimator. A more practical way is to simply observe the polaron energy itself over a period of many bold loops, once Giapprox does not change much, and use the sample standard deviation of this sample. Even more simply, we can use the maximum and minimum values observed for each quantity over a reasonably long time, and subtract them to get an estimate of the error. Since we typically calculate each quantity for 40 to 50 different values of the momentum at the same time, and since the error does not typically depend strongly on the momentum, an average over these estimated errors provides an acceptable measure of the error. Another option, and one that we often use, is to run a number of simulations in parallel, to obtain a sample of MC results that can be used to estimate the error. It is also the simplest way of harnessing readily available multicore processors to improve the precision of our results.  73  2.4. Bold Diagrammatic Monte Carlo  =  =  +  + double counting  =  =  double counting  +  + 2  +  +  +  +  +  +  +  +  +  +  +  Figure 2.10: Drawing first and second order self-energy diagrams with a more complicated propagator composed of the bare propagator and first order Green’s function diagrams.  74  2.4. Bold Diagrammatic Monte Carlo  =  +  +  +  +  +  +  +  +  +  +  +  +  forbidden groups of phonons that can be absorbed in the bold line  +…  Figure 2.11: Forbidden and allowed self-energy bold diagrams up to third order  0 1  0 2  3  0 0 0 0 0 0 3 1 1 2 2 3 2 3 all phonon lists are different, diagram allowed  1  2  0 0 0 0 1 1 2 2  3 0 0 3 3  repeated phonon lists, diagram forbidden  Figure 2.12: Algorithm to check if a self-energy diagram is allowed in BDMC. Each phonon is assigned a unique number and each propagator has a list of phonons covering it. Diagrams with each list being unique are allowed, and diagrams with repeated lists are forbidden.  75  2.4. Bold Diagrammatic Monte Carlo  fa(t)  Sampling in ω or folding in t  Continuous and aperiodic function  fp(s)  1 fa (κΩ0 ) Tp ∞  fp (s) = fa (s − lTp )  Continuous and periodic function  fp (κ) =  fa (t) =    t ∞  dω −iωt e fa (ω) 2π  −∞  ∞  fa (ω) =  Ω0 =  Tp  l=−∞  fp (s) =  fa (t) = fp (t) for t < Tp  dt eiωt fa (t)  −∞  ![fa(ω)]  fa (ω) =  ∞   0  ∞   fp (κ) =  otherwise  s  e−iκΩ0 s fp (κ)  κ=−∞  Tp  1 Tp  2π Tp  ds eiκΩ0 s fp (s)  0  fp (κ)R(ω − κΩ0 )  ![fp(κ)]  κ=−∞  1 − e−iW Tp R(W ) = iW  Only exact if fp(t) has duration ≤ Tp  ω  ga (w) = Fs  !  "  f a w − v Fs  #  v=−∞  Sampling frequency ∞   fa (t) =  ga (m)  m=−∞  fa (ω) = T ga    ω Fs  Fs =  1 T  sin[π(t − mT )/T ] π(t − mT )/T     for  ω  ≤ Fs 2π 2  Only exact if bandwidth ≤ Fs ga(m)  Discrete and aperiodic function  Sampling in s or folding in κ  ∞   Sampling in w or folding in m gp (n) =  T    m  gp (k) = ga  ![ga(w)]  2π  2πk N    nTp N  fp (k − lN )  l=−∞  fp (κ) = gp (κ) for κ < N 0 otherwise ! N −1  2π n fp (s) = gp (n)Q Ω0 s − N n=0 sin(KN/2) −iK(N −1) 2 Q(K) = e N sin(K/2)  Only exact if bandwidth ≤ N gp(n)  Discrete and periodic function (DFT) N  n  N −1 1  −i2πnk/N e gp (k) gp (n) = N  k=0 ga (m) = gp (m) for m < N N −1  0 otherwise gp (k) = ei2πnk/N gp (n)  N −1  2π n=0 gp (k)P w− k ga (w) = N k=0 sin(W N/2) iW (N −1) ![gp(k)] P (W ) = e 2 N sin(W/2)  m=−∞  -2π  ∞   gp (k) = N  ga (n − lN )  l=−∞   dw −iwm e ga (w) ga (m) = −π 2π ∞  ga (w) = eiwm ga (m) π  ∞   gp (n) = fp (nT ) = fp  Reconstruction and unfolding  ga (m) = fa (mT )  [fp(κ)]  Reconstruction and unfolding Reconstruction and unfolding  Sampling in t or folding in ω  [fa(ω)]  κ  Ω0  w [ga(w)]  Only exact if ga(m) has duration ≤ N  Reconstruction and unfolding  k  N [gp(k)]  Figure 2.13: Relation between the Fourier transform of periodic and aperiodic continuous and discrete signals. The function represented in this graph is the decreasing exponential Θ(t)e−t . 76  2.5. Summary  2.5  Summary  1. The Monte Carlo technique (section 2.1) is a stochastic algorithm that evaluates quantities expressed as the ratio of two N -dimensional sums or integrals. A configuration index ν is defined as a collection of all the summation indices or integral variables. A quantity to be calculated is defined as a weighted average over the configurations {ν}, with each  configuration having a value Aν and a weight Wν . The configuration space is often found  to be unmanageably large, and we replace the sum over the full space as a sum over a random subset of configurations. Terms in the subspace need to be selected with a probability proportional to their weight. This subset is created as a random walk in configuration space, such that visited configurations are added to the average. From the current configuration, a number of types of moves, or updates, are defined to move to another configuration. 2. Our favourite description of this method is based on the exploration of a graph (section 2.1.4), where nodes are configurations, and arrows have a probability to be chosen. The probabilities need to be normalized, or balanced. This is ensured by the balance equation (section 2.1.5). The construction of the graph as a whole is a problem even more difficult than the original calculation. It can, however, be greatly simplified by devising it in such a way that only local information about the current node, and possibly some limited information about its neighbours, is needed. The first simplifying step is to require that updates be balanced in pairs. The second is to divide the update process into a suggestion step and an acceptance step. The suggestion step can be built to require no knowledge of neighbouring configurations, while the acceptance step will only depend on the ratio of the configuration weights of the pair of nodes whose probabilities are being balanced, and the suggestion probabilities of the associated pair of updates. The resulting algorithm is called the Metropolis algorithm (section 2.1.6). 3. A sign problem (section 2.1.8) refers to the case of a configuration space where the configurations weights are not positive definite. This hinders greatly our ability to solve problems where they occur. In the best case, convergence is simply slowed. If the sign problem is strong, meaning that the average of the configuration weights goes to zero, the Monte Carlo method will fail completely. 4. The error in a Monte Carlo calculation (section 2.1.9) is almost exclusively statistical in nature. It can be estimated using the standard error of the sample, defined in (2.26). Errors should decrease as the square root of the simulation time. 77  2.5. Summary 5. The DMC technique (section 2.2) is a sampling of a Feynman diagrammatic expansion. We consider lattice polaron models with any type of coupling depending on both the electron momentum and the phonon momentum (section 2.2.1 and equation (2.33)). The polaron Green’s function (section 2.2.2 and equation (2.35)) is the quantity usually calculated and we refer to the method as G-DMC. The Green’s function is the probability amplitude for putting the electron in the system at some initial time, and finding it back in the same state at some later time. First and second order diagrams for the Green’s function are shown in Figure 2.5 and 2.6. The Green’s function is calculated in imaginary time τ = it (section 2.2.3) since the imaginary time Green’s function (2.54) is positive definite instead of being complex-valued like the real time version or the frequency (real or imaginary) versions. The ground state energy and the quasiparticle weight can be obtained from the large time limit of the imaginary-time Green’s function (section 2.2.5). Normalization simply follows from Gi (τ → 0) = 1 (section 2.2.6), and the set of updates  necessary include phonon insertion, phonon deletion, and diagram length change.  6. Σ-DMC is very similar to G-DMC but calculates a much smaller set of diagrams (section 2.3.1). Figures 2.8 and 2.9 show diagrammatically how the Green’s function can be expressed as a function of Σ using Dyson’s equation, and which diagrams contribute to this new quantity. Even a single self-energy diagram contributes an infinite number of Green’s function diagrams, such that convergence is greatly improved. The polaron ground state energy and quasiparticle weight can be obtained directly from Σ (section 2.3.2), and the set of updates is similar to G-DMC. We must, however, start with at least one phonon and include an update to change its momentum. Proper care needs to be taken to avoid generating an improper diagram (section 2.3.4). 7. BDMC is a further improvement to Σ-DMC with a faster sampling (section 2.4). It uses a more complicated propagator (bold line) than the bare electron propagator (normal line), such that each diagram drawn accounts for many diagrams of the Σ expansion. Instead of fixing the bold line to some subset, we use a self-consistent definition depending on the set of all Green’s function diagrams that can be drawn from the set of self-energy diagrams summed so far (section 2.4.1). This technique will require the use of Dyson’s equation to relate the two quantities. Since this equation is only valid in frequency, we use Fast Fourier Transformations to convert between frequency and time (section 2.4.4). Restrictions must be enforced to ensure that a diagram is not counted twice (section 2.4.3). Diagrams allowed are fully crossed diagrams (Figures 2.11 and 2.12). We also need to solve for all momenta at once due to the self-consistency properties of the method. 78  Chapter 3  Momentum Average Approximation for Models with Dispersive Phonons The Momentum Average (MA) approximation is an analytical approximation proposed and developed by Berciu’s group at the University of British Columbia. It consists of summing all Feynman diagrams in the self-energy, but with each diagram simplified somewhat so that the full sum can be performed analytically. The technique was introduced by Berciu [7] and showcased more extensively by Goodvin, Berciu and Sawatzky [37] for the Holstein model. The technique was shown to be accurate over the entire coupling range for calculating the properties of the lowest polaron eigenstate, but it had a few shortcomings. In the case of the Holstein model, for example, it did not recover the proper position of the polaron plus one phonon continuum, it produced a momentum-independent self-energy and the accuracy was slightly worse for lower-dimensional systems. These issues were addressed by Goodvin and Berciu [9] shortly afterwards with a way to systematically improve MA. There has been an important number of extensions of MA to address polaron physics beyond the simple Holstein model. Most directly relevant for this thesis are extensions developed to study models with multiple phonon branches by Covaci and Berciu [17], models with an electron-phonon coupling g(q) depending only on the phonon momentum such as the breathing mode Hamiltonian [36], and models with an electron-phonon coupling g(k, q) depending on both the phonon and the electron momentum, such as the Edwards fermion-boson model [8], and the SSH model. Others include polarons near surfaces [38], polarons in the presence of disorder [6], polaron in systems with Rashba spin-orbit coupling [19] and quasiparticles near a Dirac point in rippled graphene [18]. This long list is a testament to the wide-ranging applicability of the MA technique. BDMC results are essentially exact within the limits of its error bars. However, reaching proper convergence can prove very time consuming. There is therefore a need for a (quasi-) analytical method to quickly explore the parameter space and to help fine-tune the BDMC code. Because of the obvious limitations of perturbation theory to the small coupling regime, and its shortcomings when applied to polaron models at larger momentum, MA was the obvious choice to complement the BDMC technique in our work. The recent extension to g(k, q) models paved 79  3.1. Review of previous MA techniques the way to study the SSH coupling with MA alongside the BDMC method, as presented in Chapter 4. The MA results for that chapter were produced by Berciu while the BDMC results were produced by the author, and we find an excellent agreement between the two techniques. At the same time, DMC results are often useful to gauge the accuracy of new approximations and this is exactly how we tested the new MA methods proposed in this work. One of the objectives of this project was to study the formation of polarons in the presence of acoustic phonons. Unfortunately, all the MA techniques cited above only apply to Einstein phonons. In fact, we are unaware of any numerical technique or analytical approximation valid for the acoustic phonons, other than variational techniques or semi-classical work (see references mentioned in the last section of [34] pertaining to acoustic phonons). This left only perturbation theory, with its known limitations, and of course BDMC. To fill this void we hereby introduce one exact technique for finite size systems inspired from MA, and two related MA approximations for the thermodynamic limit. All three admit dispersive phonons as well as momentum-dependent g(k, q) couplings. We denote the first approximation MAωq , and its multigrid extension mgMAωq . The three techniques can be extended to multiple phonon branches and can, in theory, work in any dimension. Practical computer memory limitations make them better suited for one-dimensional systems, however. These methods were developed as a side project at the very time this thesis was being written. We believe this late inclusion proves very beneficial as the agreement between BDMC and mgMAωq is excellent, providing a useful corroboration for our treatment of models with acoustical phonons. Unfortunately, space and time constraints prevent us from presenting an in-depth analysis and thorough comparison of the results from MAωq and mgMAωq with other techniques. We limit ourselves in presenting these methods here, and will let the results in the next chapter speak for their success.  3.1  Review of previous MA techniques  The MA approximation proposes a systematic way to sum all diagrams for the self-energy, with the concession that we will forgo an exact answer when carrying out the nph momentum integrals of a diagram. This is a very ambitious program considering the number of diagram topologies for each order. Designing a procedure that would explicitly draw all possible diagrams, making sure all diagrams are accounted for without double counting, and then performing the multidimensional momentum integration, is of staggering complexity for any order much larger than 3. BDMC dealt with this complexity by selecting diagrams stochastically and applying simple updates to the current configuration, to eventually generate all configurations. The random walk through configuration space took care all at once of the ergodicity condition, 80  3.1. Review of previous MA techniques avoiding double-counting and carrying out multidimensional integrals. Another approach to deal with this type of complexity is to use recursive or self-consistent equations. Expressing the Green’s function using the self-energy is an example, even though the self-energy is similarly challenging to calculate in itself. Therefore we seek a recursive technique which will take care of generating all possible diagrams, and which will allow us to carry a single momentum integral at a time. Dyson’s identity is a natural starting point with its self-consistent expression for the Green’s function. We only review here the basic MA techniques needed to introduce the new approximations. We start with the case of the Holstein polaron, then briefly cover the modifications to consider a momentum-dependent coupling, and to consider multiple phonon branches.  3.1.1  MA for Holstein  The material in this section is covered more thoroughly in [7, 9, 37]. Recall the one-dimensional Holstein Hamiltonian with one branch of Einstein phonons of energy ω0 and electron-phonon coupling g: Ĥ =  X  ǫ(k)c†k ck + ω0  q  k  |  X  {z  Ĥ0    g X † b†q bq + √ ck+q ck b†−q + bq , N k, q } | {z }  (3.1)  Ĥel-ph  with an electron dispersion given by ǫ(k) = −2t cos (k), and hopping integral t. We solve for the Green’s function  G(k, ω) = h0|ck Ĝ(ω)c†k |0i,  (3.2)  where Ĝ(ω) = [ω−Ĥ+iη]−1 . We also define a bare Green’s function with Ĝ0 (ω) = [ω−Ĥ0 +iη]−1  such that  G0 (k, ω) = h0|ck Ĝ0 (ω)c†k |0i = By applying Dyson’s identity,  1 . ω − ǫk + iη  Ĝ(ω) = Ĝ0 (ω) + Ĝ(ω)Ĥe-ph Ĝ0 (ω),  (3.3)  (3.4)  81  3.1. Review of previous MA techniques inserting Ĥe-ph , and using bq1 |0i = 0 we find   g X † † √ G(k, ω) = G0 (k, ω) 1 + h0|ck Ĝ(ω) c bq |0i , N q1 k−q1 1   g X F1 (k, q1 , ω) , = G0 (k, ω) 1 + √ N q1  (3.5) (3.6)  where we defined a new generalized Green’s function F1 (k, q1 , ω) = h0|ck Ĝ(ω)c†k−q1 b†q1 |0i.  (3.7)  This propagator is a sum over all the possible processes with an initial state comprised of one electron and one phonon and a final state with one electron and the phonon vacuum. We effectively pick the first string of the ball of yarn that is the self-energy, thus pulling out a single momentum integral and define an object representing the rest. To obtain the wanted recursive relation, we then pull one more phonon line out of F1 (k, q1 , ω), i.e. we apply Dyson’s identity to (3.9) and use h0|b†q1 |0i = 0 to find F1 (k, q1 , ω) = h0|ck Ĝ(ω)Ĥe-ph Ĝ0 (ω − ω0 )c†k−q1 b†q1 |0i  g X † = G0 (k − q1 , ω − ω0 )h0|ck Ĝ(ω) √ ck−q1−q2 ck−q1 b†q2 + b−q2 c†k−q1 b†q1 |0i N q2   g g X † ck−q1 −q2 b†q2 b†q1 |0i , = G0 (k − q1 , ω − ω0 ) √ G(k, ω) + h0|ck Ĝ(ω) √ N N q2   X g = G0 (k − q1 , ω − ω0 ) √ F2 (k, q1 , q2 , ω) . (3.8) G(k, ω) + N q2 The last line defines another Green’s function, F2 (k, q1 , q2 , ω) = h0|ck Ĝ(ω)c†k−q1 −q2 b†q2 b†q1 |0i.  (3.9)  With the identification G(k, ω) ≡ F0 (k, ω), we see that F1 can be related to F0 and F2 . This  can be generalized easily for Fn , with  Fn (k, q1 . . . qn , ω) = h0|ck Ĝ(ω)c†k−qt b†qn . . . b†q1 |0i,  (3.10)  82  3.1. Review of previous MA techniques and  n g X Fn−1 (k, q1 . . . qi−1 , qi+1 . . . qn , ω) Fn (k, q1 . . . qn , ω) = G0 (k − qt , ω − nω0 ) √ N i=1  X Fn+1 (k, q1 . . . qn+1 , ω) , (3.11) + qn+1  where qt =  n X  qi .  (3.12)  i=1  To solve this set of recursive equations relating the generalized Green’s functions Fn , it is convenient to define a related set of momentum-averaged Green’s function fn defined as fn (k, ω) =  1 X Fn (k, q1 . . . qn , ω). N n q ...q 1  (3.13)  n  Summing over momenta q1 . . . qn on both side of (3.11) and using (3.13), " g 1 X G0 (k − qt , ω − nω0 ) √ · fn (k, ω) = n N q ...q N n 1 # X n X Fn+1 (k, q1 . . . qn+1 , ω) . Fn−1 (k, q1 . . . qi−1 qi+1 . . . qn , ω) + · |i=1  {z A  }  (3.14)  qn+1  |  {z B  }  Let us treat each of the term labelled “A” and “B” in (3.14) separately and denote their contribution to fn as fnA and fnB . The first can be expressed as a function of fn−1 exactly. We take qt′ = qt − qi and write fnA (k, ω) =  (3.15)  n X g X 1 1 X √ G0 (k − qt′ − qi , ω − nω0 ). Fn−1 (k, q1 . . . qi−1 qi+1 . . . qn , ω) n−1 N q N i=1 N {q } i j j6=i  Since G0 only depends on k − qt′ − qi through a cosine function, the sum or the integral over  the full Brillouin zone, with qi as the integration variable, does not depend on either k or qt′ , as they can be eliminated through a change of variable. The integral over the other momenta  83  3.1. Review of previous MA techniques can then be carried out for Fn−1 and replaced by fn−1 , which yields g fnA (k, ω) = √ nfn−1 (k, ω)ḡ0 (ω − nω0 ), N  (3.16)  where we used the momentum-averaged Green’s function 1 X G0 (q, ω). N q  ḡ0 (ω) =  (3.17)  For fnB however, we cannot separate the variables in the same way since there is no free momentum variable. Fn+1 depends on all n + 1 momenta. This is where an approximation is needed. We replace the Green’s function by its momentum average and substitute the sum over Fn+1 by fn+1 . We obtain √ fnB (k, ω) ≈ g Nfn+1 (k, ω)ḡ0 (ω − nω0 ).  (3.18)  All diagrams are accounted for, but we have essentially replaced all electron propagators by a momentum-averaged propagator. After this simplification, the sums are gone and we can solve the set of recursive equations simply. We define √ ng αn = √ ḡ0 (ω − nω0 ) and βn = g N ḡ0 (ω − nω0 ), N  (3.19)  such that (3.14) simplifies to fn (k, ω) = αn (ω)fn−1 (k, ω) + βn (ω)fn+1 (k, ω).  (3.20)  For f1 and solving recursively we obtain the continued fraction f1 (k, ω) =  1−  α1 (k, ω) α2 (k, ω)β1 (k, ω)  f0 (k, ω),  (3.21)  α (k, ω)β (k, ω) 1− 3 1−...2  and using f0 = G and (3.6) we find G(k, ω) =  1−  G0 (k, ω) G0 (k, ω)g 2 ḡ0 (ω−ω0 )  .  (3.22)  2g 2 ḡ0 (ω−ω0 )ḡ0 (ω−2ω0 ) 1− 1−...  84  3.1. Review of previous MA techniques The self-energy follows, with Σ(k, ω) =  1−  g2 ḡ0 (ω − ω0 )  2g 2 ḡ0 (ω−ω0 )ḡ0 (ω−2ω0 )  .  (3.23)  3g 2 ḡ0 (ω−2ω0 )ḡ0 (ω−3ω0 ) 1− 1−...  On physical grounds, fn is expected to become smaller with increasing n as it describes processes that are increasingly less likely. The denominator of G0 contains the term −nω0 such  that for a fixed ω we can safely cut the continued fraction at some sufficiently large nmax and take fnmax = αnmax fnmax −1 .  (3.24)  With this starting point, we can work our way up the continued fraction and obtain G and Σ. nmax should be increased until convergence is reached. We note that this approximation is of course exact in the zero-coupling limit g = 0, but also exact in the strong coupling limit t = 0. At intermediary coupling, the momentum average allows us to bridge the gap between these two limits quite accurately. This approximation can be systematically improved. The above approximation is referred to as MA(0) . Instead of replacing all bare electron propagators by their momentum average, we can keep all propagators with one phonon G0 (k − q, ω − ω0 ) exactly, and only replace all the G0 (k−qt , ω −nω0 ) with n > 1 by their momentum average, giving us the MA(1) approximation.  Higher order approximations are similarly defined.  3.1.2  MA for models with phonon-momentum-dependent coupling g(q)  We only briefly mention how a model with a phonon-momentum-dependent coupling g(q) is treated. The problem is somewhat simplified by associating the value of |g(q)|2 with the  phonon creation vertex and a value of 1/N with the annihilation vertex. This is possible because diagrams with dangling phonon propagators do not contribute, and vertices come in pairs. Equation (3.14) then takes the form " 1 X fn (k, ω) = n G0 (k − qt , ω − nω0 )· (3.25) N q ...q n 1 # X n X 1 Fn+1 (k, q1 . . . qn+1 , ω) . |g(qi )|2 |Fn−1 (k, q1 . . . qi−1 qi+1 . . . qn , ω) + · Nq i=1 n+1 {z } | | {z } A  B  85  3.1. Review of previous MA techniques A number of similar approximations can then be used to get a simplified and solvable set of recursive equations. Goodvin and Berciu [36] suggest replacing again all electron propagators with their momentum average ḡ0 (ω − nω0 ), but replacing the electron-phonon coupling by its  momentum average defined as  ḡ 2 =  X q  |g(q)|2 .  (3.26)  They, however, treat the case of n = 1 differently since no approximation is necessary to relate f1 to f0 . For this special case, another momentum average is defined: ḡ¯0 (k, ω) =  1 X |g(q)|2 G0 (k − q, ω). N q  (3.27)  The continued fraction for the self-energy is then Σ(k, ω) =  3.1.3  ḡ¯0 (ω − ω0 )  2ḡ 2 ḡ  0 (ω−ω0 )ḡ0 (ω−2ω0 ) 3ḡ 2 ḡ0 (ω−2ω0 )ḡ0 (ω−3ω0 ) 1− 1−...  1−  .  (3.28)  Variational interpretation of MA  The more recent work on MA, including the extensions allowing for a coupling g(k, q) depending on both the phonon momentum and the electron momentum [8], rely on the real-space variational interpretation of MA. Since a polaron is a particle dressed by a cloud of bosons, it is advantageous to be able to select directly which bosonic states should be kept in the variational space to describe the cloud. MA allows for this selection of states which can then be summed efficiently. To get access to the real-space description of the cloud, we work in a mixed representation where the electron is described in momentum space, but the phonons are described in real space. Generalized Green’s functions would then take the following general form Fn (k, q1 . . . qn , ω) =  1 N n/2  X  ei  j1 ...jn  Pn  i=1 qji Rji  h0|ck Ĝ(ω)c†k−qt b†jn . . . b†j1 |0i,  (3.29)  where Rj is the position of the j th atom. We can then define more specific generalized Green’s functions which physical intuition suggests might be more important. We can, for example, define one for the case where bosons are all on the same site Fn(1) (k, qt , ω) =  1 N n/2  X j   n eiqt Rj h0|ck Ĝ(ω)c†k−qt b†j |0i.  (3.30)  86  3.1. Review of previous MA techniques Similarly, a two-site cloud Green’s function (2) Fn,m (k, qt , ω) =  1 N n/2  X j  n−m  m  |0i, b†j+1 eiqt Rj h0|ck Ĝ(ω)c†k−qt b†j  (3.31)  with n phonons and m of them on the first site, can be introduced. Restrictions need to be applied to ensure these are distinct from F (1) , so we require n ≥ 2 and 1 ≤ m ≤ n−1. This type  of description lets us decide what type of states are kept for the cloud. The set of recursive equations relating these quantities becomes more complicated, as each generalized Green’s function might now depend on a large subset of the others, but they can nevertheless be solved relatively easily. The added work comes however with the benefit of a better understanding of the relative importance of various bosonic states to the polaron state. In the case of the Holstein model, MA(0) involves only the one-site cloud Green’s functions (1)  (3.30). Averaging Fn (k, qt , ω) over qt would give us a Green’s function directly proportional to fn (k, ω) in (3.14), with a factor depending on N and n due to the different choice of scaling. Typically, a description that goes up to a three-adjacent-sites cloud is sufficient for g(k, q) models. This is what was used by Berciu to study the SSH model in Chapter 4, and for the description of the polaron in the Edwards fermion-boson model [8]. We note that the number of phonons on each of the adjacent sites can be arbitrarily large, while the electron can be anywhere, in the cloud or away from it. This works really well for describing the polaron eigenstates under the continuum, but as this remains an MA(0) type approximation, the continuum itself might require more work. An MA(n) approximation, by keeping exact propagators for one electron and up to n phonons, would do better by allowing up to n phonons to be arbitrarily far from the cloud. These are indeed the states that contribute to the continuum.  3.1.4  MA for Holstein with two phonon branches  Once again we limit ourselves to a brief overview of the technique to set the foundation for the new ones introduced in the next section. We follow [17] and restrict ourselves to the case of two dispersionless phonon branches with Holstein coupling. The Hamiltonian is generalized as follows Ĥ =  X k    1 X bq † bq + √ g1 c†k+q ck b−q † + bq N k, q q   X 1 X † † † √ Bq Bq + + ω2 g2 ck+q ck B−q + Bq , N k, q q  ǫ(k)c†k ck + ω1  X  (3.32)  87  3.1. Review of previous MA techniques where the subscript of ω and g numbers the phonon branch, and bq (Bq ) is the phonon annihilation operator for the first (second) branch. Conceptually, the calculation is very similar to the single-branch Holstein. We can apply Dyson’s identity to the Green’s function G(k, ω) to pull one phonon line out, and define two generalized Green’s functions: one for an electron plus one phonon of type 1 and one for an electron and one phonon of type 2. We denote momenta and operators for the first branch with lowercase letters and capital letters for the second branch. Let us define the generalized Green’s function for n phonons of the first type and m phonons of the second type as † † Fnm (k; q1 . . . qn ; Q1 . . . Qm ; ω) = h0|ck Ĝ(ω)c†kt b†q1 . . . b†qn BQ BQ |0i, m 1  with kt = k − qt − Qt ,  with qt =  n X  qi  and Qt =  i=1  m X  Qj .  (3.33)  (3.34)  j=1  This function can in turn be related to similar functions with one extra phonon or one phonon less of either kind by the exact equation  Fnm (k; q1 . . . qn ; Q1 . . . Qm ; ω) = G0 (kt , ω − nω1 − mω2 )  (3.35)  n g X √1 Fn−1,m (k; q1 . . . qi−1 qi+1 . . . qn ; Q1 . . . Qm ; ω) N i=1  m g2 X Fn,m−1 (k; q1 . . . qn ; Q1 . . . Qj−1 Qj+1 . . . Qm ; ω) +√ N j=1 g1 X +√ Fn+1,m (k; q1 . . . qn+1 ; Q1 . . . Qm ; ω) N qn+1  g2 X Fn,m+1 (k; q1 . . . qn ; Q1 . . . Qm+1 ; ω) . +√ N Qm+1  We replace all electron propagators by their momentum averages to obtain the MA(0) approximation. This time, we define a momentum-averaged and rescaled Green’s function fnm (ω) =  1 N  n+m 2  X  q1 ...qn , Q1 ...Qm  Fnm (k; q1 . . . qn ; Q1 . . . Qm ; ω) . G(k, ω)  (3.36)  The set of recursive equations resulting from expressing (3.35) as a function of (3.36) can be solved in matrix form. We define a vector of all the generalized Green’s functions that contain 88  3.1. Review of previous MA techniques n phonons of any kind, and omit the frequency dependence to find Vn = [fn,0 fn−1,1 . . . f1,n−1 f0,n ]T .  (3.37)  The relation to Vn−1 and Vn+1 is simply Vn = An Vn−1 + Bn Vn+1 .  (3.38)  The nonzero elements of An and Bn are easily derived from (3.35) and we only state the answer here. An is a matrix of size (n + 1) × n with nonzero elements along the diagonal  (An )i,i = (n − i)g1 ḡ0 ω − (n − i)ω1 − iω2 ,  (3.39)   (An )i+1,i = (i + 1)g2 ḡ0 ω − (n − 1 − i)ω1 − (i + 1)ω2 ,  (3.40)  for i ∈ [0, n − 1]. Bn is a matrix of size (n + 1) × (n + 2) with nonzero elements along  (Bn )i,i = g1 ḡ0 ω − (n − i)ω1 − iω2 ,  (Bn )i,i+1 = g2 ḡ0 ω − (n − i)ω1 − iω2 ,  (3.42)  Vn = Mn Vn−1 ,  (3.43)  (3.41)  for i ∈ [0, n]. By assuming that Vnmax +1 = 0 and solving recursively we get that  with Mn =  1 1 − Bn 1−B  1  1 n+1 1−... An+2  An+1  An .  (3.44)  The continued fraction goes down all the way to the denominator 1 − Bnmax −1 Anmax . We solve for the self-energy using V0 = (1) and  Σ(ω) = (g1 g2 ) · V1 = (g1 g2 ) · M1 .  (3.45)  The technique can be generalized to a larger number of phonon branches and to include momentum-dependent couplings g(q). To some extent, the needed modification for those cases will be presented below as we develop an approximation that will admit dispersive phonons.  89  3.2. Exact solution for finite systems  3.2  Exact solution for finite systems  In this section we apply the formalism presented above in an unexpected way, to consider a finite-size system with a dispersive phonon branch. The solution does not require any momentum average approximation and is thus exact. We present it first as it incorporates a few of the core elements of the full MA approximation for dispersive phonons and infinite systems presented in the next section. Due to the finite size of the lattice, there is a discrete set of momentum values allowed qj = 2πj/N with j = 0, 1, . . . , N − 1. The first insight that will lead to our exact solution is  to identify the momentum as a phonon sub-branch index. We effectively divide the Brillouin  zone into a number of regions such that the phonon branch is divided into a number of subbranches, one per momentum region. This conceptual identification of the original branch as different sub-branches is purely a matter of notation and has no further physical meaning. Essentially, we consider N sub-branches, each with a single momentum value qj , a constant coupling gj = g(qj ) and a phonon energy ωj = ω(qj ). By considering exactly N sub-branches, we can replace each momentum summation by an equivalent sum over a sub-branch index. Ĥ =  X i  ǫ(ki )c†ki cki +  X j    1 X † ωj b†qj bqj + √ gj cki +qj cki b†−qj + bqj , N i, j  (3.46)  where i and j now index the electron and phonon momentum and are not position indices. We start as before, solving for the Green’s function and applying Dyson’s identity repeatedly to pull phonon lines out of the cloud and to define a generalized Green’s function with a specific number of phonon nj in each branch for j ∈ [0, N − 1]. We start with   1 X † † G(ki , ω) = G0 (ki , ω) 1 + h0|cki Ĝ(ω) √ g−j1 cki −qj bqj |0i , 1 1 N j1    1 X g−j1 F ki ; {nj = 0}j=0...N −1, j6=j1 , nj1 = 1; ω , = G0 (ki , ω) 1 + √ N j1  (3.47) (3.48)  where we used the one-phonon case of  nN−1  n0 |0i, . . . b†q0 F (ki ; n0 , . . . nN −1 ; ω) = h0|cki Ĝ(ω)c†kt b†q0 with kt = ki −  N −1 X j=0  qj nj = ki −  N −1 X j=0  2πj nj . N  (3.49)  (3.50)  90  3.2. Exact solution for finite systems Similarly we define ωt = ω −  N −1 X  ωj nj ,  (3.51)  j=0  which will be useful shortly. The total number of phonons of the generalized Green’s function is defined as n=  N −1 X  nj .  (3.52)  j=0  There is no need to define a momentum-averaged Green’s function, but we introduce a rescaled Green’s function as before: f (ki ; n1 , . . . nN −1 ; ω) =  1 F (ki ; n0 , . . . nN −1 ; ω) . n G(ki , ω) N2  (3.53)  A function fn , with a total number of phonons n, can be related to n functions fn−1 , with one phonon removed, and N functions fn+1 , with one phonon added, through  NX −1 nj gj f (ki ; n0 , . . . , nj − 1, . . . , nN −1 ; ω) f (ki ; n1 , . . . , nN −1 ; ω) = G0 (kt , ωt ) N j=0  +  N −1 X j=0   g−j f (ki ; n0 , . . . , nj + 1, . . . , nN −1 ; ω) .  (3.54)  Equivalent recursive expressions can group gj g−j together, with either of the sums yielding the same exact result. Next we define a column vector Vn , analogous to (3.37), where each element of the vector is a member of the set of all the generalized Green’s functions defined by (3.49) for which the total number of phonons equal to n. The exact ordering of these elements is arbitrary so we do not specify it here. The number of elements of this vector is simply (N − 1 + n)! = (N − 1)! n!     N −1+n , N −1  (3.55)  with one element for each possible way to distribute n phonons in N distinct sub-branches. The recursive set of equations (3.54) now takes a matrix form Vn = An Vn−1 + Bn Vn+1 .  (3.56)  The position of the nonzero elements of An and Bn depends on the exact ordering cho91  3.2. Exact solution for finite systems sen for the elements of Vn and Vn±1 . If f (ki ; n0 , . . . , nN −1 ; ω) is the αth entry of Vn , and f (ki ; m0 , . . . , mN −1 ; ω) is the β th entry of Vn−1 with n = m + 1, then An    α,β  =  N −1 X  δn0 , m0 . . . δnj , mj +1 . . . δnN−1 , mN−1  j=0  !  nj gj G0 (kt , ωt ). N  (3.57)  Similarly, if f (ki ; n0 , . . . , nN −1 ; ω) is the αth entry of Vn , and f (ki ; m0 , . . . , mN −1 ; ω) is the β th entry of Vn+1 with n + 1 = m, then Bn    α,β  =  N −1 X  !  δn0 , m0 . . . δnj +1, mj . . . δnN−1 , mN−1 g−j G0 (kt , ωt ).  j=0  (3.58)  Those two matrices are rather sparse. An has dimensions of (N − 1 + n)! (N − 1 + n − 1)! × , (N − 1)! n! (N − 1)! (n − 1)!  (3.59)  but has only N  (N − 1 + n − 1)! , (N − 1)! (n − 1)!  (3.60)  nonzero elements. Bn has dimensions of (N − 1 + n)! (N − 1 + n + 1)! × , (N − 1)! n! (N − 1)! (n + 1)!  (3.61)  but has only N  (N − 1 + n)! , (N − 1)! n!  (3.62)  nonzero elements. Solving for the set of recursive equations (3.56) is done in exactly the same way as for the two-phonon-branch Holstein model. We assume Vnmax +1 = 0, or equivalently Bnmax = 0, such that Vnmax ≈ Anmax Vnmax −1 . This means that Vn can be expressed as a function of Vn−1 as in  (3.43) using the matrix Mn as defined by (3.44). This matrix has the same dimensions as An . Since V0 = (0) we find that M1 =  1 1 − B1 1−B  1  1 2 1−... A3  A2  A1 ,  (3.63)  92  3.3. MA with dispersive phonons (MAωq ) and  N −1 X   g−j M1 j . Σ(ki , ω) = g0 . . . g−(N −1) · M1 =  (3.64)  j=0  The Green’s function is calculated with G(ki , ω) =  1 . ω − ǫ(ki ) − Σ(ki , ω) + iη  (3.65)  We thus have a solution for the self-energy and the Green’s function where no approximation other than Vnmax +1 = 0 has been made. This is for a system of finite size and finite size scaling needs to be done to study the infinite system. Preliminary tests show that the ground state energy is E0 (N ) ≈ limM →∞ E0 (M ) − C/N where C is a constant. Memory limitations prevent solving for a large number of phonons and a large system size. A full performance analysis  is left for future work, but we give a few pointers here. With N = 8 and nmax = 8 we can calculate one point in (k, ω) in about 12 minutes using under 2 gigabytes of memory on an off-the-shelf intel core i7 desktop computer. For N = 9 and nmax = 8, we need approximately 100 minutes and 5 gigabytes of memory for one value. Limiting one of those two parameters allows to reach a much larger value for the second one. With N = 2 for example, we can solve for more than 100 phonons in less than one second, while with nmax = 2 we can reach up to N = 100 in approximately 40 minutes. Both speed and memory requirements scale similarly for the other techniques presented below. The above results are easily generalized for couplings g(k, q) that depend on the electron momentum as well. This is why we have chosen a symmetric representation in (3.54) where the couplings are not grouped together as |gj |2 . We define a k-dependent coupling gj (k) = g(k, qj ),  (3.66)  and replace gj by gj (kt ) in the first term of (3.54) and in (3.57), while replacing g−j by g−j (kt ) in the second term of (3.54) and in (3.58).  3.3  MA with dispersive phonons (MAωq )  The solution presented for a finite system cannot be blindly extended to an infinite system which would have an infinite number of sub-branches. The memory requirements become prohibitive even for relatively small N . Instead of removing the momentum sum altogether and replacing it by a sub-branch index sum, we suggest going halfway and defining a number of sub-branches  93  3.3. MA with dispersive phonons (MAωq ) with an associated finite set of momentum values qj , and introducing a momentum average approximation around those points. We will explain this in more detail shortly, but let us first consider the case of a single momentum-restricted phonon sub-branch. This will let us introduce the remaining core elements of MAωq without the added complication of many sub-branches and the matrix equations.  3.3.1  MA for a single momentum-restricted sub-branch  Once again we consider a coupling g(q) depending only on the phonon momentum to simplify the notation, and will briefly comment afterward on how to extend this to g(k, q). We consider the case of a phonon branch with momentum restricted to some region of width ∆q centred at q̄. In the language introduced in the previous section, this is a sub-branch of the full physical branch. Obviously, by considering the case of single sub-branch, we are studying a very different Hamiltonian than the unrestricted original one. Instead of restricting momentum sums to this range directly, we define the function χ(q) = Θ(q − q̄ + ∆q/2)Θ(q̄ + ∆q/2 − q),  (3.67)  which is 1 on q ∈ [q̄ − ∆q/2, q̄ + ∆q/2], and 0 otherwise. The model under consideration is given by Ĥ =  X  ǫ(k)c†k ck +  k  X q    1 X χ(q)ω(q)b†q bq + √ g(q)c†k+q ck χ(−q)b†−q + χ(q)bq . N k, q  (3.68)  By now the procedure has been covered a few times and should be familiar. The exact recursive relation now reads:  n X 1 √ χ(qi )g(qi )Fn−1 (k; q1 . . . qi−1 , qi+1 . . . qn ; ω) Fn (k; q1 . . . qn ; ω) = G0 (kt , ωt ) N i=1  X χ(qn+1 )g(−qn+1 )Fn+1 (k; q1 . . . qn+1 ; ω) , (3.69) +G0 (kt , ωt ) qn+1  where we have used kt = k − qt = k −  n X j=1  qj ,  and ωt = ω −  n X  ω(qj ).  (3.70)  j=1  94  3.3. MA with dispersive phonons (MAωq ) The factor χ(qn+1 ) can be dropped without changing the final answer. We can see this by looking at the value of each pair of vertices together and rearranging them to attach g(q)χ2 (q) to the annihilation vertex and g(−q) to the creation vertex, and use χ2 (q) = χ(q). The momentum-averaged Green’s function is defined as fn (k, ω) =  1 X Fn (k; q1 , . . . , qn ; ω) , n G(k, ω) N 2 q1 ...qn  (3.71)  and is averaged over the entire Brillouin zone. Thus (3.69) becomes n X X Fn−1 (k, q1 . . . qi−1 , qi+1 . . . qn , ω) 1 1 χ(qi )g(qi )G0 (kt , ωt ) fn (k, ω) = n−1 N N 2 q ...q G(k, ω) n i=1 1 | {z }    fnA  +  1  N |  n+1 2   Fn+1 (k, q1 . . . qn+1 , ω) g(−qn+1 )G0 (kt , ωt ) . G(k, ω) q1 ...qn+1 {z } X  (3.72)  fnB  Let us first focus on the second term and consider the case of a constant coupling. In the case of the unrestricted Holstein model, we simply replaced G0 (k − qt , ωt ) by an average over  the Brillouin zone to get rid of the qt dependence and pull it out of the sum. This is sensible because qt takes any value in the Brillouin zone with equal probability. In the present case, although we are still summing over the full unrestricted range of momentum, Fn+1 is now zero for most of this domain and replacing G0 by its average over the whole Brillouin zone is not justified. To get a better answer we want to use an average which is taken over the nonzero contribution. In this case qt + qi is no longer uniformly distributed. The (n + 1)-phonon distribution of qt + qi will be centred at (n + 1)q̄, but can extend from (n + 1)q̄ − (n + 1)∆q/2  to (n + 1)q̄ + (n + 1)∆q/2. Each phonon momentum deviation from q̄ is uniformly distributed on [−∆q/2, ∆q/2] and is an independent variable. The probability distribution of the sum of n independent random variables is given by the n-fold convolution of their probability distributions. Here the probability distributions are uniform, giving us a triangular distribution for n = 2 and slowly widening and smoothing out to resemble a gaussian for large n, due to the central limit theorem. Figure 3.1 shows what this distribution looks like as a function of n. The formula for the n-fold convolution of n uniformly distributed random variables on [0, 1]  95  3.3. MA with dispersive phonons (MAωq )  Figure 3.1: n-fold convolution of uniform probability densities on [−1/2, 1/2] for n = 2, 3, 4, 5 and associated color: red, blue, green and cyan.  is given by [94] for example, and takes the form ρ̃n (x) =  (  1 (n−1)!  0  P  j n 0≤j≤x (−1) j   (x − j)n−1 if 0 < x < n, otherwise.  We also note that ρ̃n+1 (x) = (ρ̃n ∗ ρ̃1 )(x) =  X y  ρ̃n (x − y)ρ̃1 (y).  (3.73)  (3.74)  This means that qt , which we know can take values between nq̄ − n∆q and nq̄ + n∆q, can be  described by an n-fold convolution directly, instead of the (n + 1)-fold convolution given by fn+1 . If we define qt = nq̄ + δqn , the probability distribution of δqn is given by ρn (δqn ) = ρ̃n     δqn n + , ∆q 2  (3.75)  which is nonzero between −n∆q and n∆q. So for a constant coupling that can be factored out, and a dispersionless phonon branch,  96  3.3. MA with dispersive phonons (MAωq ) we would define the averaged bare Green’s function n∆q/2  ḡ0 (k, ω, n) =  X  δqn =−n∆q/2  ρn (δqn )G0 (k − nq̄ − δqn , ω − nω0 ),  (3.76)  and substitute it for the bare Green’s function in fnB , allowing us to replace Fn+1 by fn+1 . For a momentum-dependent coupling, we could average it separately, as is suggested for g(q) coupling in our review section above, but we are still left with the problem of a dispersive phonon branch. Furthermore, a coupling g(k, q) cannot be averaged separately as it depends on all other momenta. We can however treat the momentum dependence of the coupling, and of the dispersive phonons, by introducing one further approximation. We use qj ≈ q̄ +  δqn , n  (3.77)  wherever a dependence on one of the phonon momenta appears other than through δqn . For a given δqn , this is the average value of any specific momentum qj . We will treat the dispersion of the phonons in the same way. We define an averaged quantity that will replace both the Green’s function and the coupling of the creation vertex, and perform the average using the (n + 1)-fold convolution: (n+1)∆q/2  gg + 0 (k,   δqn+1 · ω, n) = ρn+1 (δqn+1 )g − q̄ − n+1 δqn+1 =−(n+1)∆q/2    δqn  · G0 k − nq̄ − δqn , ω − nω q̄ + , n   X  (3.78)  where we have used (3.77) to eliminate individual momenta qj . Using (3.74) we rewrite ∆q/2  gg + 0 (k,  n∆q/2   δqn+1 · (3.79) ω, n) = ρ1 (δqi ) ρn (δqn )g − q̄ − n+1 δqi =−∆q/2 δqn =−n∆q/2    δqn  · G0 k − nq̄ − δqn , ω − nω q̄ + . n X  X    We can eliminate δqn+1 by replacing δqi by δqn+1 /(n + 1) in δqn+1 = δqn + δqi such that δqn+1 δqn ≈ . n+1 n  (3.80)  97  3.3. MA with dispersive phonons (MAωq ) We note here that we could have kept the δqi dependence explicitly without introducing a further approximation. This would have entailed carrying out a double integral to obtain gg + 0, which is more computationally intensive for little added benefit. Once δqn+1 is removed, there is no remaining dependence on δqi and we can replace the sum over ρ1 (δqi ) by a unit factor. Back to (3.72) we find fnB (k, ω) = gg + 0 (k, ω, n)fn+1 (k, ω).  (3.81)  For the remaining term, we have fnA (k, ω) =  (3.82) n X  1 1 X χ(qi ) n−1 N q N 2 i=1 i |{z} | {z } | n ∆q/2π |  with ω, n) =  {qj } ∀j∈[1, n]\i  Fn−1 (k, q1 . . . qi−1 qi+1 . . . qn , ω) g(q )G0 (kt , ωt ), } | i {z G(k, ω) gg − 0 (k, ω,n)  {z  }  fn−1 (k, ω)  {z  }  same range of qt with nonzero contribution as fn  n∆q/2  gg − 0 (k,  X  X  δqn =−n∆q/2      δqn  δqn . (3.83) G0 k − nq̄ − δqn , ω − nω q̄ + ρn (δqn )g q̄ + n n   We replaced the Green’s function and the coupling of the annihilation vertex by a similar momentum average as the one used for fnB since the range of values of q1 . . . qn for which fn is P nonzero is the same as the range of values for which qi χ(qi )fn−1 is nonzero. The result is fnA (k, ω) = n  ∆q − gg (k, ω, n)fn−1 (k, ω). 2π 0  (3.84)  The Green’s function is given by G0 (k, ω)  G(k, ω) = 1−  − (∆q/2π)gg + 0 (k, ω,0)gg 0 (k, ω,1)  ,  (3.85)  − 2(∆q/2π)gg + 0 (k, ω,1)gg 0 (k, ω,2) 1− 1−...  98  3.3. MA with dispersive phonons (MAωq ) and the self-energy Σ(k, ω) =  − (∆q/2π)gg + 1 0 (k, ω, 0)gg 0 (k, ω, 1) . − G0 (k, ω) 1 − 2(∆q/2π)gg + 0 (k, ω,1)gg 0 (k, ω,2) 3(∆q/2π)gg + (k, ω,2)gg − (k, ω,3) 0  1−  1−...  (3.86)  0  We now look at the modifications needed for g(k, q). We redefine the two gg 0 and we find n∆q/2  gg + 0 (k,    (3.87)    (3.88)   δqn ω, n) = ρn (δqn )g k − nq̄ − δqn , −q̄ − · n δqn =−n∆q/2    δqn  , · G0 k − nq̄ − δqn , ω − nω q̄ + n X  and n∆q/2  gg − 0 (k,   δqn ρn (δqn )g k − nq̄ − δqn , q̄ + ω, n) = · n δqn =−n∆q/2    δqn  . · G0 k − nq̄ − δqn , ω − nω q̄ + n X  + B We then simply rewrite fnA (k, ω) using gg − 0 andfn (k, ω) using gg 0 .  3.3.2  Dividing one phonon branch into multiple sub-branches: MAωq  At this point all the needed approximations have been introduced. What is left is merely a generalization of the single restricted phonon branch inspired by the exact solution for finite systems. Since this is the technique we will be using later, we treat the general case of g(k, q). We rewrite the Hamiltonian one last time as Ĥ = and  X k  Nq −1  ǫ(k)c†k ck +  X X j=0  q  Nq −1   1 X X † † χj (q)ω(q)bjq bjq + √ g(k, q)c†k+q ck χj (−q)bj−q + χj (q)bjq , N j=0 k, q (3.89)      χj (q) = Θ q − (q̄j − ∆q/2) Θ (q̄j + ∆q/2) − q ,  (3.90)  with q̄j = 2πj/Nq = j q̄ and q̄ = 2π/Nq . We added the sub-branch index to the phonon operators for clarity, but we stress again that this index is artificial, and there is only one physical branch with phonon operators bq and bq † .  99  3.3. MA with dispersive phonons (MAωq ) We are defining Nq sub-branches such that all diagrams can be coarse-grained with a momentum average approximation around a set of momentum values qj . The resulting generalized Green’s function can then be used to write a set of recursive equations which are solved as in the finite system case. The momentum dependence of a phonon line is therefore divided into two parts: a coarse value qj and an offset δq ∈ [−∆q/2, ∆q/2].  We define the generalized Green’s functions exactly as for the Holstein model in (3.10), but  we use the somewhat redundant notation of identifying the sub-branch j of each phonon as a superscript of the momentum, as it simplifies greatly the expression for momentum restriction. We find †  †  Fn (k; q1j1 . . . qnjn ; ω) = h0|ck Ĝ(ω)c†k−qt bjqnn . . . bjq11 |0i.  (3.91)  The recursive relation for Fn takes the form Fn (k;  q1j1  . . . qnjn ;   n X 1 ji−1 ji+1 ω) = √ χji (qiji )g(kt , qiji )Fn−1 (k; q1j1 . . . qi−1 , qi+1 . . . qnjn ; ω) G0 (kt , ωt ) N i=1  X X jn+1 jn+1 j1 +G0 (kt , ωt ) g(kt , −qn+1 )Fn+1 (k; q1 . . . qn+1 ; ω) , (3.92) jn+1 q jn+1 n+1  and we define the momentum-averaged Green’s function f (k; n0 , . . . nNq −1 ; ω) =  where  1 n N2  Nq −1  n=  X i=0  ni ,  kt = k − qt = k −  X Fn (k, q j1 . . . qnjn , ω) 1 , G(k, ω) j j  (3.93)  q11 ...qnn  n X i=1  qiji ,  and  ωt = ω −  n X  ω(qiji ).  (3.94)  i=1  After the momentum average, the Green’s function simply has a number of phonons ni in the ith sub-branch, and similarly for each of the Nq sub-branches. A function fn , with a total number of phonons n, can be related to n functions fn−1 , with one phonon removed, and Nq  100  3.3. MA with dispersive phonons (MAωq ) functions fn+1 , with one phonon added, through f (k; n0 , . . . , nNq −1 ; ω) = Nq −1  X  i=0 Nq −1  +  X  ni  (3.95)  ∆q − gg (k; n0 , . . . , nNq −1 ; i; ω)f (k; n0 , . . . , ni − 1, . . . , nNq −1 ; ω) 2π 0  gg + 0 (k; n0 , . . . , nNq −1 ; i; ω)f (k; n0 , . . . , ni + 1, . . . , nN −1 ; ω).  i=0  The momentum average of the bare Green’s function is only slightly modified as n∆q/2  gg + 0 (k;  n0 , . . . , nNq −1 ; i; ω) =  X  Nq −1    ρn (δqn )g k −  δqn =−n∆q/2  Nq −1    · G0 k −  X j=0  X j=0   δqn · nj q̄j − δqn , −q̄i − n Nq −1  nj q̄j − δqn , ω −  X j=0    δqn  nj ω q̄j + , n  (3.96)  and n∆q/2  gg − 0 (k;  n0 , . . . , nNq −1 ; i; ω) =  X    ρn (δqn )g k −  δqn =−n∆q/2    Nq −1  · G0 k −  Nq −1  X j=0  where  X j=0   δqn nj q̄j − δqn , q̄i + · n  nj q̄j − δqn , ω −  Nq −1  X j=0    δqn  nj ω q̄j + , n  (3.97)  Nq −1  δqn = qt −  X  nj q̄j .  (3.98)  j=0  The rest of this derivation is identical to the exact solution for systems of finite size. We define a column vector Vn , whose elements are the members of the set of all f defined in (3.93) containing n phonons. Once again the ordering of the elements inside Vn is arbitrary and left unspecified. (3.56) is the recursive matrix equation to be solved. Dimensions of the vectors and matrices are the same as before, but with Nq sub-branches instead of N . If f (k; n0 , . . . , nNq −1 ; ω) is the αth element of Vn , and f (k; m0 , . . . , mNq −1 ; ω) is the β th element  101  3.3. MA with dispersive phonons (MAωq ) of Vn−1 , with n = m + 1, then we find An  !  Nq −1    α,β  =  X  δn0 , m0 . . . δni , mi +1 . . . δnNq −1 , mNq −1 ni  i=0  ∆q − gg (k; n0 , . . . , nNq −1 ; i; ω). 2π 0 (3.99)  Similarly, if f (k; n0 , . . . , nNq −1 ; ω) is the αth element of Vn , and f (k; m0 , . . . , mNq −1 ; ω) is the β th element of Vn+1 , with n + 1 = m, then Bn    !  Nq −1 α,β  =  X  δn0 , m0 . . . δni +1, mi . . . δnNq −1 , mNq −1 gg + 0 (k; n0 , . . . , nNq −1 ; i; ω).  i=0  (3.100)  We then solve for M1 , defined by (3.63), from which the self-energy is found to be Σ(k, ω) =  ḡ0+ (k)  ...    + (k) ḡN q −1  · M1 =  N −1 X j=0   ḡj+ (k) M1 j .  (3.101)  where the momentum-averaged coupling ḡj+ (k) over the allowed momentum range of the j th sub-branch is given by ḡj+ (k) =  X q  χj (q)g(k, −q).  (3.102)  The Green’s function is then calculated with (3.65).  3.3.3  Interpretation of MAωq and remarks  We have mentioned that MA for Holstein simply replaces all bare propagators G0 by their momentum average ḡ0 in all the self-energy diagrams. This average is over the whole Brillouin zone. Once this replacement has been done, we only need to sum over all diagram topologies. This also holds for models with momentum-dependent coupling g(q), but there we also need to replace each vertex by the square root of the momentum average of |g(q)|2 over the Brillouin  zone. The exact solution for finite systems is not a momentum average approximation; we simply used the same formalism as in MA for multiple branches. The identification of the momentum as a sub-branch index is purely conceptual and the diagrams are unaffected. The meaning of MAωq , on the other hand, might not be as directly obvious and it is worth spending some effort to understand what it does to the diagrams. We first observe that dividing the phonon branch into a set of sub-branches and restricting the phonon momenta using χj (q) does not change the Hamiltonian other than by suggesting we keep track of the momentum by keeping two values: a coarse-grained component and an 102  3.3. MA with dispersive phonons (MAωq ) offset. If we consider the case of a single sub-branch, we find that the limit of ∆q → 2π is equivalent to an unrestricted branch, despite the average taken using ρn . After folding ρn  back to the Brillouin zone, we simply have a constant distribution in this limit. We also should point out that when taking a momentum average inside a sub-branch, we are doing two different approximations. In the first place, we assume each phonon momentum is an independent variable such that the sum of all those contributions is an n-fold convolution of a constant distribution. We make this approximation wherever possible, i.e. where the momentum dependence appears as the sum qt of all phonon momenta. Where this fails, we revert to a less accurate assumption. This is the case for the coupling and the phonon energies that appear in the propagator, which depend on individual momentum variables. In this latter approximation, we consider that the sum of the momentum carried by phonons, qt , is evenly distributed between all phonons, which is only true on average. If ∆q is small enough, we can see that this approximation is not severe for ω(q). On a small enough momentum range, we can approximate ω(q̄ + δq) ≈ ω(q̄) +  dω(q) dq q=q̄ δq.  Summing over all phonons, we find that the  momentum dependence through the phonon energies can be expressed as a function of qt . This  means that, in this limit, we have effectively linearized the phonon dispersion relation around q̄. Since we do actually keep the full function ω(q) and not only its linear approximation, the second order correction cannot be expressed as a function of qt and the best we can do at this point is to divide qt equally between the n phonons. The other individual momentum dependence is through the coupling. Again, this cannot be expressed as a function of qt so we simply replace it by its average qt /n. A better approximation for this single momentum would involve summing over this variable explicitly, such that gg ± 0 would now require a double integral evaluation: one integral over this individual momentum qi and one integral over the rest qt − qi using ρn−1 .  In the full MAωq approximation, the approximations are more or less the same as for a  single restricted sub-branch. We replace all bare propagators and an adjacent coupling by a momentum average gg ± 0 that removes the offset from each momentum, but keeps the coarsegrained values. In carrying out the momentum average, we effectively linearize the phonon dispersion relation in a piecewise fashion around the coarse-grained momentum values. Higher order corrections to ω(q) are included, but averaged assuming the sum of phonon momentum offsets are evenly distributed between all phonons. The sum over the remaining coarse-grained momenta and over all diagram topologies is then carried out exactly. Since we associate one vertex and one propagator together, diagrams for the self-energy have a remaining unpaired vertex. This is why we introduced a momentum averaged coupling. In the case of a Green’s function diagram, we can pair the first vertex with the bare Green’s function 103  3.3. MA with dispersive phonons (MAωq ) that starts the diagram. This is however equivalent to averaging the coupling separately since we have + gg + 0 (k; n0 = 0, . . . , nNq −1 = 0; i; ω) = G0 (k, ω)ḡj (k).  (3.103)  The remaining unpaired bare propagator does not need to be averaged at all. Comparing the MAωq and MA for Holstein with the same maximum number of phonons allowed, we find that MAωq is more accurate because at least part of the phonon momentum is treated exactly, as opposed to being fully averaged out by MA. This advantage allows MAωq to obtain a better estimate for the energies of the lowest polaron eigenstates and the position of the polaron plus one phonon continuum, the latter being a weakness of MA(0) . Our preliminary comparison between MA(0) , MA(1) , MA(2) , with a large enough maximum number of phonons, and MAωq , limited to 10 phonons and Nq = 8, shows that MAωq does as well as MA(2) for the polarons eigenstates under the continuum and the position of the continuum, but the shape of the continuum is not well recovered. Again, a full comparison and analysis is outside the scope of the present document. We will however heed the implied warning and shall not attempt to describe the shape of the continuum for the models studied with MAωq in this work. A few more remarks on performances and memory requirements can be found in the next section. We end this section by writing explicitly the first and second order contributions to the self-energy under all of MAωq approximations. The first order diagram is given by Nq −1 1  Σ (k ω) =  X i=0  ḡi+ (k)gg − 0 (k; ni = 1, {nj = 0}j∈[0, Nq −1]\i ; i; ω).  (3.104)  + Inserting gg − 0 from (3.97) and ḡi from (3.102) yields  1  Σ (k, ω) =  Nq −1 "  X i=0  ·    ∆q/2  X  g k, −q̄i − δq  δq=−∆q/2 ∆q/2  X      ·  (3.105)     g k − q̄i − δq, q̄i + δq G0 k − q̄i − δq, ω − ω(q̄i + δq)  δq=−∆q/2  # ,  104  3.4. Multigrid extension of MA with dispersive phonons (mgMAωq ) The second order non-crossing diagrams are Nq −1  Σ  2 nc  X  (k, ω) =  i1 , i2 =0 i1 6=i2  ḡi+1 (k)gg + 0 (k; ni1 = 1, {nj = 0}j∈[0, Nq −1]\i1 ; i2 ; ω)·  (3.106)  · gg − 0 (k; ni1 = 1, ni2 = 1, {nj = 0}j∈[0, Nq −1]\{i1 , i2 } ; i2 ; ω)· Nq −1  X  +  · gg − 0 (k; ni1 = 1, {nj = 0}j∈[0, Nq −1]\i1 ; i1 ; ω)  ḡi+ (k)gg + 0 (k; ni = 1, {nj = 0}j∈[0, Nq −1]\i ; i; ω)·  i  · gg − 0 (k; ni = 2, {nj = 0}j∈[0, Nq −1]\i ; i; ω)· · gg − 0 (k; ni = 1, {nj = 0}j∈[0, Nq −1]\i ; i; ω).  The second order crossing diagrams are Nq −1  Σ2 c (k, ω) =  X  i1 , i2 =0 i1 6=i2  ḡi+1 (k)gg + 0 (k; ni1 = 1, {nj = 0}j∈[0, Nq −1]\i1 ; i2 ; ω)·  (3.107)  · gg − 0 (k; ni1 = 1, ni2 = 1, {nj = 0}j∈[0, Nq −1]\{i1 , i2 } ; i1 ; ω)·  Nq −1  +  X i  · gg − 0 (k; ni2 = 1, {nj = 0}j∈[0, Nq −1]\i2 ; i2 ; ω)  ḡi+ (k)gg + 0 (k; ni = 1, {nj = 0}j∈[0, Nq −1]\i ; i; ω)· · gg − 0 (k; ni = 2, {nj = 0}j∈[0, Nq −1]\i ; i; ω)·  · gg − 0 (k; ni = 1, {nj = 0}j∈[0, Nq −1]\i ; i; ω).  3.4  Multigrid extension of MA with dispersive phonons (mgMAωq )  The main limitation of MAωq is its memory requirements for storing An and Bn and solving the set of recursive equations. Even without actually inverting any matrix and using sparse matrix representation, computing time and memory requirements place stringent restrictions on the maximum number of phonons nmax , and the number of points Nq of the coarse-grained momentum grid, that can be considered. Using nmax = 8 and Nq = 8 allows for one value in (k, ω) to be calculated in about twenty minutes using just under two gigabytes of memory on an off-the-shelf intel core i7 desktop computer. This is a good compromise between speed and 105  3.4. Multigrid extension of MA with dispersive phonons (mgMAωq ) accuracy and is fast enough for exploring the parameter space of a model. Reducing Nq to a small value like 2, allows to reach more than 100 phonons in less than 6 seconds. To improve on MAωq , we seek a multigrid approximation that would allow us to change Nq as a function of n when solving the recursive matrix equations for Vn . Instead of taking Vnmax = 0, we look for a way to approximate Vnmax with the equivalent quantity, but calculated for Nq /2 points on the momentum grid. Computer algorithms that rely on this type of procedure are referred to as multigrid techniques, hence the acronym mgMAωq . There are a number of ways this can be achieved, and we only present one option which we tested and which produces more accurate results for the polaron states under the continuum and for the position of the continuum itself. A more thorough investigation and comparison to other techniques will be the subject of future work. (2)  For the sake of simplicity, let us define a Nq = 2 generalized Green’s function Fnm with two sub-branches and n (m) phonons in the first (second) sub-branch. We try to relate this to (4)  the Nq = 4 generalized Green’s function Fabcd . Phonons in the first sub-branch of F (2) include phonons from the two first sub-branches of F 4 while the two other sub-branches are included in the second sub-branch of F 2 . This means that for the Green’s function that should be related, we have n = a + b and m = c + d. We find for example (2)  (4)  (4)  F10 = F1000 + F0100 .  (3.108)  The reverse statements, however, 1 (2) (4) F1000 = F10 , 2  or  1 (2) (4) F0100 = F10 , 2  (3.109)  are only true on average. We will see shortly that both relations will be needed such that the possibility of treating a larger number of phonons comes at the expense of this further approximation. (2N ) (N ) We define a transfer matrix Ti which will map the vector Vi q to Vi q for a specific number of phonon i at which we want to double Nq . We have (2Nq )  Vi  (Nq )  = Ti Vi  ,  (3.110)  and the matrix that does the opposite transformation Ti+ is the pseudoinverse of Ti , also called the Moore-Penrose inverse. It is a generalization of the matrix inverse for non-square matrices  106  3.4. Multigrid extension of MA with dispersive phonons (mgMAωq ) and satisfies CC + C = C,  C + CC + = C + ,  (CC + )H = CC + ,  and (C + C)H = C + C,  (3.111)  where C H is the conjugate transpose of C. It follows that Nq  Vi  2Nq  = Ti+ Vi  .  (3.112)  The elements of Ti+ are either 1 or 0. Which elements are nonzero is easily seen from the generalization of (3.108). To obtain Ti we simply take the transpose of Ti+ and divide each column of the resulting matrix by the dot product of the said column with itself. We then chose to use the following set of equations around Vi : Vi−1 q =Ai−1 q Vi−2 q + Bi−1 q Vi  (2N )  (2N )  (2N )  (2N )  (2Nq )  (3.113)  (2N ) Vi q (N ) Vi+1q (N ) Vi+2q  (2N ) (2N ) (N ) (N ) =Ai q Vi−1 q + Ti Bi q Vi+1q (N ) (2N ) (N ) (N ) =Ai+1q Ti+ Vi q + Bi+1q Vi+2q (N ) (N ) (N ) (N ) =Ai+2q Vi+1q + Bi+2q Vi+3q  (3.114) (3.115) (3.116)  As an example of the modified continued fractions, the matrix M1 as defined in (3.63) takes the following form for a grid change at n = 2, M1 =  1 1 − B1 1−T  1  + 1 2 B2 1−... A3 T2  A2  A1 ,  (3.117)  where we have dropped the superscripts Nq and 2Nq . This change of momentum grid can be done more than once. We have found that using   2      4     8 Nq =  16      32     64  for n ∈ [50, 21] for n ∈ [20, 9] for n ∈ [8, 5]  for n ∈ [4, 3] for n = 2 for n = 1  yields better results than MAωq with Nq = 8 and nmax = 8, while taking only a few minutes more to calculate. Memory requirements are also acceptable with a peak use of about 4 gigabytes. 107  3.5. Summary  3.5  Summary  1. The Momentum Average approximation consists of summing all Feynman diagrams in the self-energy, but with each diagram having been somewhat simplified. It is first presented in its simplest form for the Holstein model (section 3.1.1). It relies on the definition of generalized Green’s functions (3.10) with one electron and n phonons. Each such Green’s function can be related to similar quantities having one phonon more or fewer. This allows us to write the recursive set of equations (3.11). To solve it, we define a momentum averaged Green’s function with (3.13). By replacing each propagator with its momentum average, we can rewrite the recursive relations as (3.20) and solve it to find the Green’s function as a continued fraction shown in (3.22), and the self-energy in another continued fraction as shown in (3.23). This simple analytical technique is exact in the zero-coupling and infinite-coupling limits, and it bridges the gap between them quite accurately. It fails, however, to recover the polaron plus one phonon continuum position. 2. This simple technique can be extended to g(q) couplings by introducing an averaged quantity for |g(q)|2 (section 3.1.2). It can also be extended to g(k, q) using a real-space  variational interpretation, where the phonon operators are Fourier transformed to real space (section 3.1.3). In this representation, one can select what type of phonon cloud states to keep. For g(k, q) models, it is usually sufficient to consider phonon clouds spread on up to three adjacent sites with an arbitrary number of phonons on each. This technique was used to produce MA results for the SSHo model (section 4.2). 3. MA can also be extended to solve for multiple optical phonon branches where, once again, each electron propagator is replaced by its momentum average. The two-branches Holstein model is presented in section 3.1.4. Generalized Green’s functions are defined for n phonons of branch 1, and m phonons of branch 2, as in equation (3.33). We also define a momentum-averaged version in (3.36), and we collect all such Green’s functions containing a total of n phonons of any type into a column vector Vn , defined in (3.37). When expressed as a function of Vn−1 and Vn+1 , the resulting set of recursive matrix equations is given by (3.38). The self-energy is then expressed as a continued fraction in (3.45). 4. So far, all MA approximations admit optical Einstein phonons only. In this work, we propose three new generalizations to study models with dispersive phonons (sections 3.2 and 3.3). All three allow for any g(k, q) coupling. They are based on the idea of dividing 108  3.5. Summary a dispersive phonon branch into a number of smaller sub-branches. This division is purely conceptual and does not change the Hamiltonian. 5. The first new technique is actually an exact solution that applies to finite size systems (section 3.2). No momentum average is required, but we use the same formalism as for MA with multiple phonon branches. For a system of N sites, we define N sub-branches, and replace the momentum sums by momentum index sums. The rest of the calculation is identical to MA with multiple phonon branches. Only a slight modification is needed to consider g(k, q) as explained at the end of section 3.2. Only a limited size N can be treated due to the important memory requirements. 6. For an infinite system, we need to introduce an approximation that we call MAωq (section 3.3). We again divide the branch into Nq sub-branches. We are simply replacing each momentum by two quantities: a coarse value and an offset. The coarse momenta are kept and treated exactly, as in the exact solution for finite systems, but the offsets need to be averaged out. The average is not taken over the entire Brillouin zone, but inside the sub-branches. Here we chose to average the bare Green’s function along with the coupling, which is necessary to consider g(k, q). It is also necessary to define a different average for the coupling associated with the creation vertex and the annihilation vertex. Two types of approximations are used to obtain the averaged resulting quantities (3.96) and (3.97). The first approximation is used for the dependency of G0 on the sum of the offset momenta. In this case we consider each offset as a random variable uniformly distributed. The distribution of a sum of such independent variables is given by an n-fold convolution (Figure 3.1 and equation (3.73)). The second approximation is less accurate, but needs to be used for single offset momentum dependencies coming from the phonon dispersion and the coupling. It consists in assuming that the momentum carried by the phonons is uniformly distributed between them, which is true only on average. 7. The last technique, the multigrid MAωq , is an improvement over the previous one (section 3.4), which allows a larger number of phonons to be considered. Due to the memory requirements of MAωq , both the maximum number of phonons nmax and the number of sub-branches Nq are very limited. If one of them is decreased, the other can be increased to a much greater number. We thus suggest to change the number of subbranches during the calculation, starting with very few, when the phonon number is large, and increasing it progressively as the phonon number decreases. This introduces an additional approximation, but allow us to reach more than 100 phonons.  109  Chapter 4  Phonon-Modulated Hopping We turn to the SSH model, made famous by Su, Schrieffer and Heeger [40, 91] in their treatment of the half-filled case of the one-dimensional polymer chain that is trans-polyacetylene. We use the single electron version of this Hamiltonian as a simple example of phonon-modulated hopping, and therefore as a model with coupling depending on both the phonon and the electron’s momentum. As a result, this model also has a sign problem which makes numerical Monte Carlo techniques converge more slowly. We consider the SSH coupling to both acoustic and optical phonons, first separately, and then together. We also look at a model with both an SSH coupling to optical phonons and a Holstein-type coupling to optical phonons. We find a sharp transition for the SSH coupling to optical phonons at a critical coupling for which the k = 0 effective mass of the polaron becomes infinite. Below this transition, the polaron is very similar to the usual lattice polaron with the expected polaronic behaviour. Above the transition, we find a surprisingly light polaron with a number of phonons which remains relatively small, a significant quasiparticle weight even at strong coupling, and a polaron ground state at finite momentum k 6= 0. This sharp transition seems robust, surviving the addition of a SSH coupling to acoustic phonons, or a diagonal coupling to optical phonons. The SSH model with only acoustic phonons does not seem to have such a transition at the couplings considered in this work.  4.1 4.1.1  Introduction Motivation  Our motivation for studying this model is twofold. On the one hand, phonon-modulated hopping is a wide ranging phenomenon which is often ignored for the sake of simplicity, but can have important consequences on transport properties, especially on quasi one-dimensional metals and conducting polymers [40]. Its absence in the discussion of the polaron problem is related to the Born-Oppenheimer approximation, which we introduced in section 1.3.1 of Chapter 1 in our derivation of the lattice polaron Hamiltonian. We recall that the Bloch state of the electron in the lattice was obtained under the assumption that the atoms are immobile 110  4.1. Introduction and at their equilibrium position. This means in turn that the tight-binding approximation for the state of the electron would produce a constant hopping integral independent of lattice distortions, and therefore, independent of phonons. This may well be justified in some systems, but the terms that are neglected can have non trivial effects on others. Examples include the dimerization of polyacetylene due to Peierls’s instability [70], the associated opening of a gap at the Fermi surface, and the presence of optical phonon branches in a monoatomic chain. These phenomena are observed despite a rather small change in the lattice constant (for polyacetylene, a change of 0.08Å along the chain axis with respect to the original lattice constant of 1.2Å ) [91]. At the very least, the effects of phonon-modulated hopping need to be studied to ascertain their importance and whether they can safely be ignored in specific cases. The second motivation for studying this model is the form of the SSH coupling in momentum space. As we will see, the SSH coupling g(k, q) depends both on the electron’s momentum and the phonon’s momentum. We mentioned in the introduction that the existence of a sharp transition in the polaronic ground state has been a central question in this field since its introduction. As discussed in section 1.5, Gerlach and Löwen [34] showed that for coupling g(q) depending exclusively on the phonon’s momentum, only a smooth crossover is possible. The SSH model, with its dependence on the electron momentum, is a prime candidate to look for a sharp transition. This comment applies more specifically to the SSH model with optical phonons, since all models with acoustic phonons are already outside the range of applicability of Gerlach and Löwen’s proof for the absence of a sharp transition due to the assumption (1.84), and this holds even for g(q) couplings. We emphasize that we will be investigating only the one-electron problem, as opposed to the very different half-filled case, which has traditionally been the focus of previous studies. Before our own recent contribution and that of our collaborators [60], studies in this regime have been either variational or perturbative. Due to the known failings of both of these techniques, a study relying on different methods was in order. We thus find ourselves in the very interesting position of having a model amenable to few techniques outside of those two categories, while having two completely independent and eligible methods at our immediate disposal.  4.1.2  Phonon-modulated hopping and non-diagonal coupling  The ground state of a half-filled 1D polyacetylene chain is well known to be a dimerized chain due to Peierls’s instability [70]. This phenomenon is a clear telltale sign of the failure of the usual Born-Oppenheimer approximation for this system. The dynamic Jahn-Teller effect is another example that comes to mind. It is another electronic effect, but leading in this case to the geometrical distortion of a non-linear molecule [44]. For molecules where this effect is 111  4.1. Introduction important, a description in the Born-Oppenheimer approximation, that is, carried out for the equilibrium configuration of the molecule in the absence of the electrons, would completely miss the distortion. We recall that in the tight-binding approximation, we first expand the Bloch wave function with a set of localized atomic orbitals centred at each atom. The amplitude of those states typically falls off quickly away from the atom. Different types of wave functions can be used for this calculation, but an exponential decay at long range is typical. Slater type orbitals [88] and Gaussian functions are two basis sets often used in similar calculations in chemistry, and they both have an exponential decay. The details of these techniques do not concern us here, but we simply observe that if the wave function fall off exponentially, the overlap of the wave functions of two neighbouring atoms should also decrease exponentially with the distance separating them. Tunnelling between two sites, and therefore the hopping integral, can be significantly altered with even slight deviations from the equilibrium position. In the case of trans-polyacetylene at half-filling, the naive way of correcting for this error is to assume a dimerized chain in the first place and calculate the Bloch states for this new equilibrium configuration. This does restore many of the previously missed features of this system, but is not as satisfying as a fully dynamic ab initio solution. As the electrons move, we can expect changes in the atom positions of a comparable order of magnitude as observed for the dimerized chain. To address this dynamical problem, Su, Schrieffer and Heeger [40, 91] considered a tight-binding model with a hopping constant that depends linearly on the actual distance between the two neighbouring sites. Since the position of the sites can be expressed using phonon operators, the resulting hopping term is said to be phonon-modulated. We should point out that Peierls’s instability was a well established phenomenon and others had studied models with inhomogeneous hopping before them, see the work by Rice [84] or by Pople and Walmsley [76] for examples on polyacetylene. To the best of our knowledge, Su, Schrieffer and Heeger were the first to consider the dynamical case and apply it to polyacetylene. The same model was, however, used ten years ealier by Barišić, Labbé and Friedel as an extension of the Hubbard Hamiltonian to describe superconductivity in transition metals [5]. It was pointed out shortly thereafter by Deegan [25], that this effect was not limited to the electron-phonon interaction of this model, and also that their results followed directly from previous work by Fröhlich and Mitra [29, 68]. The type of coupling described above is also referred to as non-diagonal in the site index, or non-local. It is defined as a nontrivial dependence of the hopping integral on the lattice coordinate [99]. It translates into a term in the Hamiltonian proportional to ci † cj where i 6=  j. Phonons can be created and annihilated as the electron moves, as opposed to a diagonal  coupling which only creates and absorbs phonons when an electron stays on the same site. 112  4.1. Introduction Non-diagonal coupling is expected to be especially relevant to the description of π-conjugated organic molecular crystals such as oligoacenes including naphthalene, anthracene, tetracene and pentacene [39, 90]. Unsurprisingly, we find in their structure of linearly fused benzene rings, the same alternation of single and double bonds as found in trans-polyacetylene. This type of coupling is also important for the description of solid state excimers such as α-perylene [98]. Several dimerized Mott magnetic semiconducting oxides such as ZnO, TiO2 , SrTiO3 and KTaO3 for example, also have non-diagonal coupling [69]. More specifically, the SSH coupling presented above is an example of antisymmetric nondiagonal coupling, where the tunnelling between a site and its neighbour on the right is enhanced, and tunnelling with its neighbour on the left is inhibited (or vice versa depending on the phase of the oscillation). Symmetric non-diagonal coupling describes a different physical system, where the tunnelling enhancement or inhibition with both neighbours are in phase. This work is only concerned with antisymmetric coupling. An example of a model with nondiagonal symmetric coupling is the Edwards model which has recently been studied with the same MA technique used in this chapter and developed by Berciu [8]. A few authors have looked at the SSH model in the single electron limit using perturbation theory. For example, Capone, Stephan and Grilli, worked on the SSH model with optical phonons [15], while Zoli [101], and more recently, Li, Chadler and Marsiglio [57], have considered the version with acoustic phonons. We shall delay a summary of their findings to the sections below corresponding to the specific model under consideration. We wish to point out immediately, however, that while perturbation theory does reasonably well in the weak coupling regime of the SSH Hamiltonian with optical phonons, the large momentum failings notwithstanding, it does much worse for the acoustic case, and one should tread very carefully. A number of other studies have looked at the effect of non-diagonal coupling using various variational ansätze [58, 90, 99]. More specifically, references [58, 99] rely on a phonon coherent state (PCS), as used in many polaron ansätze, to describe the dressing cloud (see also section 1.2.3 of the introduction). The particle itself is described in different ways depending on the method. The small polaron ansatz considers a basis of states with the particle spread over a single site, and the PCS is described relative to this site. The trial state is given by  i 1 X ikj † 1 Xh iqj ∗ −iqj † |ψsmall i = √ e cj exp − √ f (k, q)e bq − f (k, q)e bq |0i. N j N q  (4.1)  This approximation only makes sense if the hopping integral is relatively small. In the presence of a larger hopping integral, the particle is expected to be spread over a larger region. The Toyozawa ansatz simply allows for the delocalization of the particle around the polaron centre, 113  4.1. Introduction by allowing for a more complex phased-sum over states similar to the small polaron ansatz. The trial state then takes the form |ψToyozawa i = |pihp|pi−1/2 ,  (4.2)  with  i 1 X i(p−k)j 1 Xh † iqj ∗ −iqj † √ √ |pi = e h(p, q)ck exp − f (p, q)e bq − f (p, q)e bq |0i. N j N q  4.1.3  (4.3)  SSH Hamiltonian for a simple monoatomic chain  The Hamiltonian presented below is essentially the same as Su, Schrieffer and Heeger’s tightbinding model of trans-polyacetylene [40, 91], with the important difference that they considered the half-filled case, while we restrict ourselves to a single electron to study the effect of phonon-modulated hopping and polaron formation. The model can be generalized to any dimension, but we only look at the one-dimensional case here. We will further distinguish between a number of variations of this model, but we start here by deriving the SSH Hamiltonian in its most austere version, for the monoatomic chain. We will only be considering displacements parallel to the symmetry axis of the chain, with uj being the displacement of site j from its equilibrium position on a lattice of constant a. Since the displacement due to the presence of an electron is assumed small, we can expand the elastic energy of the lattice (which in polyacetylene is due to the bonding energy of σ bonds) to second order about the equilibrium state Ĥa (elastic) =  1X K(uj+1 − uj )2 , 2  (4.4)  j  with K the effective spring constant. The second term is the kinetic contribution of the lattice motion Ĥa (kinetic) =  1X M u̇2j . 2  (4.5)  j  The next contribution is the kinetic energy of the electron (π electron in pz orbitals for trans-polyacetylene), which, as advertised, is treated in the tight-binding approximation: Ĥe (kinetic) =  X j, s   tj+1, j c†j+1, s cj, s + c†j, s cj+1, s ,  (4.6)  with c†j, s the creation operator for an electron of spin s on site j. The spin index can be 114  4.1. Introduction dropped in this one-electron case. Again we assume that the uj are small, and expand the hopping integral up to linear contributions with tj+1, j = t0 − α(uj+1 − uj ),  (4.7)  where (uj+1 − uj ) is zero for the equilibrium state of the chain without any electron. The full Hamiltonian is thus Ĥ = −  X j  X p2j   1X K(uj+1 − uj )2 + . t0 − α[uj+1 − uj ] c†j+1 cj + c†j cj+1 + 2 2M j  (4.8)  j  The two last terms are the phonon contribution to the Hamiltonian. To diagonalize them, we Fourier transform using 1 X iqaj uj = √ e uq N q and find Ĥphonon = with ωq2 = and the phonon frequency  1 X −iqaj e pq , and pj = √ N q  MX 2 1 X pq p−q + ω uq u−q , 2M q 2 q q   aq   aq  4K = 4ω02 sin2 , sin2 M 2 2 ω0 =  r  K . M  (4.9)  (4.10)  (4.11)  (4.12)  After defining the second-quantized phonon creation and annihilation operators ! M ωq i bq = p−q , uq + 2h̄ M ωq ! r M ωq i † bq = pq , u−q − 2h̄ M ωq r  (4.13) (4.14)  we find the expected phonon part in momentum space: Ĥphonon =  X q   1 . h̄ωq b†q bq + 2  (4.15)  Note that only longitudinal phonons are considered since we assumed all displacements uj to 115  4.1. Introduction be along the chain axis. In reality, transverse phonons are also allowed, but the coupling is likely to be weaker since a small displacement perpendicular to the chain will not change the modulus of the distance between neighbouring atoms as much. Inserting tj+1, j from (4.7) in (4.8), we get a term proportional to t0 and diagonal in the site index. Once Fourier-transformed, it yields the usual kinetic energy of the electron Ĥelectron =  X  ǫk c†k ck ,  (4.16)  k  with the electron dispersion relation ǫk = −2t0 cos (ka).  (4.17)  The part proportional to α represents a non-diagonal electron-phonon coupling. Collecting all the terms affected by displacement uj we can write Ĥe-ph = α  X j    uj c†j cj−1 + c†j−1 cj − c†j+1 cj − c†j cj+1 ,  (4.18)  with the Fourier transform h i   1 X Ĥe-ph = √ 2iα sin a(k + q) − sin (ak) uq c†k+q ck . N k, q  (4.19)  Expressing uq as a function of b†q and bq and regrouping all the contributions yields Ĥ =  X k  ǫk c†k ck +  X q   1 X g(k, q)c†k+q ck b†−q + bq , h̄ωq b†q bq + √ N k,q  with coupling g(k, q) = 4iα sin   aq  2  h  q i cos a k + 2  s  h̄ , 2M ωq  (4.20)  (4.21)  where we simply used a trigonometric identity to replace the sum of sines into the product shown above. The resulting Hamiltonian is of the general form (2.33) which we can study with the BDMC technique. Since we will be interested only in the Green’s function and the self-energy, only diagrams with phonon lines attached at both ends to the electron line need to be considered. For such diagrams, the interaction vertices always come in pairs and the corresponding contribution  116  4.1. Introduction from the coupling is g(k, −q)g(k′ − q, q) = 16α2  h  h   aq  q i q i h̄ cos a k − cos a k′ − . sin2 2M ωq 2 2 2  (4.22)  We can therefore ignore the factor of i and define an effective coupling g(k, q) = 4α sin   aq   h  q i cos a k + 2  2  s  h̄ . 2M ωq  (4.23)  The corresponding electron-phonon interaction term remains hermitian thanks to the absolute value of the sine function. Let us again set the lattice constant to 1 and insert ω(q), to find ga (k, q) = 2α̃ with  r  sin  q   r  α̃ = α  2   q , cos k + 2  h̄ . M ω0  (4.24)  (4.25)  The subscript a is used to emphasize that we are coupling to acoustical phonons. It is important to note that the SSH coupling opens new channels for the delocalization of the electron. We can therefore look at the possibility of multisite hopping processes which give rise to next nearest neighbour hopping. Even when t0 = 0, the electron can hop to a neighbouring site by creating or absorbing a phonon on the destination or the departure site. The processes that leave a phonon behind are exponentially suppressed, but others would contribute to the electron kinetic energy while leaving no phonon behind. For example the particle can hop to the neighbouring site while creating a phonon on the destination site and then hop one site further destroying this phonon. In a perturbative approach, the amplitude for this process is t2 = −α2 /ω0 . The negative sign is due to the fact that the atom displacement  increases one bond length and decreases the next. This means that the dispersion will have a term like −2t2 cos(2k) with t2 negative and thus favouring a minimum at π/2.  4.1.4  Sign problem  Equation (4.22) clearly shows why a model depending on both the phonon’s and the electron’s momentum has a sign problem, whereas a more common model depending only on the phonon’s momentum does not. The coupling, after the Fourier transform to momentum space, is typically some combination of trigonometric functions depending on the momentum. Models with g(q) 117  4.1. Introduction couplings satisfy ω(q) = ω(−q) and  g(q) = g∗ (−q).  (4.26)  The first is not surprising as the energy of a phonon moving in one direction should not be different from a phonon moving in the opposite direction in a purely isotropic medium. The second simply follows from the hermiticity of the Hamiltonian. Using these, and looking at pairs of interaction vertices, we see that the phonon contribution |g(q)|2 is positive definite. The only way to obtain a negative sign from a pair of electron-phonon interaction vertices  is if they depend also on the electron momentum k. In this case, the electron momentum is not necessarily the same for both vertices, and (4.22) cannot be expressed as the square of a product of trigonometric functions. For g(k, q) couplings, we still have g(k, −q) = g ∗ (k − q, q),  (4.27)  such that pairs of vertices with k = k′ have a positive contribution of g(k, −q)g∗ (k − q, q) =  |g(k, −q)|2 . This is true for all first order diagrams, for example. A sign problem only appears starting from second order diagrams, and then, only for crossing diagrams. This also means  that diagrams kept in the self-consistent Born approximation (SCBA), i.e. all non-crossing diagrams, are positive definite. Other diagrams can, of course, take any sign as pairs of vertices can have any combination of k and k′ . In the weak coupling regime we expect the polaron to be composed of a small number of phonons and since we see that first order diagrams are positive, as are most of the second order diagrams, we conclude that this is a rather weak sign problem and the answer should converge without problem. At stronger coupling it is difficult to predict the sign of higher order diagrams. However, if the number of phonons is sufficiently large to consider that k and k′ of each pair of vertices are not correlated for crossed diagrams, we expect their contribution to average out to zero. Added to the fact that SCBA diagrams are positive definite, again this model does not appear to be as severely affected by the sign problem as a full many-body fermionic problem for example, where no such significant group of diagrams are positive definite or can be assumed to cancel.  4.1.5  Other variants of the SSH Hamiltonian  We shall henceforth refer to the SSH Hamiltonian with longitudinal acoustic phonons presented above as the SSHa Hamiltonian. A diatomic chain will of course have both acoustic and optical phonons branches. The optical branch is often approximated by a dispersionless Einstein phonon branch. We will therefore also look at two longitudinal phonon branches with  118  4.2. SSHo — SSH coupling to one dispersionless optical branch SSH coupling, one acoustic, and one dispersionless optical, and refer to this model as SSHoa. Although considering an optical phonon branch by itself is physically dubious, it nevertheless allows for comparison with a wealth of results for other types of coupling to dispersionless Einstein phonons. We will refer to a dispersionless longitudinal optical branch as SSHo. This model will be the first one to be presented below because of its relative simplicity. The SSH coupling can of course coexist with more usual types of electron-phonon coupling. In fact, all systems must have acoustic phonons with diagonal coupling. We will briefly look at the interplay of more than one phonon branch when we discuss the SSH model with diagonal and non-diagonal coupling to optical phonons. The diagonal electron-phonon coupling we consider is the simplest one: the local Holstein coupling, as the Holstein model encompasses most polaronic features we have become used to in the condensed matter literature. The techniques used, however, are trivially extended to any other coupling as well as to any number of phonon branches in any dimension, the added computational burden notwithstanding.  4.2  SSHo — SSH coupling to one dispersionless optical branch  Since models with acoustic phonons are more complicated to study (see section 4.3), we first turn to the SSH model with only one optical branch approximated by Einstein phonons: SSHo. This model should have a closer behaviour to the ubiquitous Holstein model, allowing us to isolate the effect of the phonon modulated hopping and to compare to a wealth of both analytical and numerical results. It also turns out to have the most interesting behaviour.  4.2.1  Hamiltonian, parameters and normalization  The Hamiltonian is the same as in (4.20), but we want to use Einstein phonons, so we set the phonon dispersion relation to a constant ω(q) = ω0 ,  (4.28)  such that the coupling in (4.24) becomes go (k, q) = 2β̃ sin  q 2   q , cos k + 2  (4.29)  119  4.2. SSHo — SSH coupling to one dispersionless optical branch where β̃ is the coupling strength to the optical phonons β̃ = β  r  2h̄ , M ′ ω0  (4.30)  analogous to α and α̃. An alternative form for the coupling is given by   go (k, q) = β̃ sin(k + q) − sin(k)  for  q ∈ [0, 2π].  (4.31)  Note that this last expression is only valid for q ∈ [0, 2π], since we dropped the absolute  value around sin(q/2). Outside of this range we need to multiply the above expression by sign(sin (q/2)) to make sure the coupling is properly defined outside of the first Brillouin zone. Next we define the two usual dimensionless parameters. The adiabaticity ratio ω0 /t0 has the usual definition. Since we will set t0 = 1 as our unit of energy, we can simply use ω0 as the adiabaticity ratio. The second parameter is the dimensionless electron-phonon coupling, defined as the ratio of the polaron energy in the strong coupling limit with t0 → 0, to the energy  of the bare electron with β̃ → 0. For the Holstein model, the strong coupling limit energy can  be obtained analytically using the Lang-Firsov canonical transformation (see section 1.4.2). It relies on the fact that in the strong coupling limit, the electron is localized. This is not the case anymore with the SSH model, as the non-diagonal coupling allows for electron hopping processes even when t0 = 0. We choose to use an analogous definition to the effective coupling of the Holstein model λo =  β̃ 2 . 2t0 ω0  (4.32)  Effectively it means that for lack of a better estimate, we ignore the momentum dependence of the coupling, and assume a constant Holstein-like coupling. Equivalently we could obtain an effective coupling by averaging over all possible phonon and electron momenta, which yields the same answer up to a multiplicative constant. In the strong coupling limit the number of phonons is expected to be large enough that such an average provides a good enough estimate. We also need to obtain the limit Σi (τ → 0, k) in order to properly normalize the diagrams  in a BDMC code:  8β̃ 2 Σ (τ → 0, k) = 2π i  Z  π 0  dq sin2 (q/2) cos2 (k − q/2) = λo ω0 t0 [2 − cos(2k)].  (4.33)  120  4.2. SSHo — SSH coupling to one dispersionless optical branch  4.2.2  Previous studies  As mentioned in the introduction, Capone, Stephan and Grilli [15] have studied this model with perturbation theory and obtained the following expression for the effective mass of the polaron at k = 0 " #  m0 ω0 2t + ω0 2ω0 p + −1 . = 1 + λo p 2 m∗ 2t ω0 + 4tω0 ω02 + 4tω0  (4.34)  In the adiabatic limit, (4.34) simplifies to  3 m0 = 1 + λo m∗ 2  r  ω0 , t  ω0 ≪ t,  (4.35)  and in the antiadiabatic limit  m0 t  2 + = 1 + λ , o m∗ ω0  ω0 ≫ t.  (4.36)  They do not look at momentum-dependent properties in this work. They also present results from Exact Diagonalization of four-site clusters with open boundary conditions, and find only a crossover between the weak and strong coupling regimes. Although they do find a sharp transition at yet larger couplings where the hopping integral vanishes, they rightfully call the state found at couplings above this point unphysical. This boundary is well beyond the regime where the SSH coupling applies, since the average deviation of the atoms from their equilibrium positions can no longer be considered small. More remarks on their results follow below. As for variational studies, we refer to the work of Liu, Zhao, Wang and Kato [58] which built on the work by Brown, Lindenberg, Zhao and colleagues [13, 85, 100] on the Global-Local ansatz. This ansatz is a further generalization of the Toyozawa ansatz that allows for a nontrivial spread of the particle in real space. It was proven superior to other variational ansätze presented above for Holstein [85]. In this paper, they consider both the SSH coupling and Holstein-type coupling to optical phonons simultaneously. They do find a sharp transition at a critical coupling and a ground state at finite momentum. However, non-physical discontinuities are found in the polaron dispersion in some regimes. A numerical comparison to variational results by Liu et al. needs to be delayed to section 4.2.9 since they provide numerical values only for a model with both non-zero non-diagonal coupling and non-zero Holstein coupling.  121  4.2. SSHo — SSH coupling to one dispersionless optical branch  4.2.3  Results and general comments  The results presented here have been partially published in [60]. This Letter also presents results produced with G-DMC by Mishchenko, and the Limited Phonon Basis Exact Diagonalization (LPBED) by De Filippis and Cataudella [22]. We will focus on results produced with BDMC and MA, but we will also compare to Rayleigh-Schrödinger (RS) and Wigner-Brillouin (WB) perturbation theory (PT). The BDMC results and the perturbation results were produced by the author. Except where otherwise stated, the MA results presented in this section have been produced by Berciu using an extension to the momentum average approximation described in [8]. In this approximation, the phonon cloud is allowed to extend to three contiguous sites with any number of phonons on those sites, while the particle can be arbitrarily far from the cloud. Further improvements to this approximation are possible, but this already yields a very good agreement with the BDMC results. These results have a variational interpretation and should give an upper bound for the dispersion relation. A number of reasons justify our choice of MA and BDMC to study the SSHo model. DMC belongs to a category of unbiased methods, where we make no assumption about the solution, or the type of diagrams that will contribute most to the final answer. All diagrams are summed, and the result is essentially exact within the statistical error bars. BDMC is essentially equivalent to other DMC techniques, but with a faster convergence. Such techniques are especially useful if we want to confirm the presence of a sharp transition. Perturbation theory is of course of limited applicability, since it applies to a very restricted range of coupling. Variational ansätze, on the other hand, often produce singular artifacts. While the MA approximation for g(k, q) couplings is also a variational method, and should therefore be checked against an unbiased method, we point out that after being applied to numerous systems and type of couplings, it has yet to produce any singular artifacts for any model. It should also be mentioned that the variational interpretation of MA lends itself to a systematic improvement, where the variational space can be enlarged in a controlled manner. This is unlike other ansätze which are much less flexible. MA also provide insight into the type of phonon cloud states that are most relevant. See also our remark below on the adiabatic regime. The other techniques presented in the Letter are not included in this thesis. The DMC results are essentially the same as BDMC, while LPBED is essentially an Exact Diagonalization (ED) technique based on the standard Lanczos algorithm and carried for a system where the phonon cloud has a similar structure as in the MA technique described above. In the Letter, a phonon cloud spread over five lattice sites was used with two additional phonons that can be  122  4.2. SSHo — SSH coupling to one dispersionless optical branch anywhere. This technique is not as fast as MA, however, and the number of phonons on each site of the cloud is more severely limited, unlike for MA. The agreement between BDMC, MA and LPBED is excellent and a presentation of the LPBED results here was deemed superfluous. An additional reason for choosing MA over LPBED in this work include the fact that LPBED could not easily be extended to treat dispersive phonons, something that will prove important in the next section on the SSHa model. The results included herein span the whole parameter range defined by the adiabaticity ratio and the dimensionless coupling. The adiabatic regime is studied for ω0 = 0.5 in Figures 4.1 and 4.2, while the non-adiabatic regime is studied for ω0 = 3.0, 4.0, 100 in Figures 4.3, 4.4, and 4.5. The effective coupling is varied from the small coupling regime to the strong coupling regime. The transition from one to the other happens at λo ≈ 1 for most adiabaticity ratios  considered. Error bars are omitted for BDMC when they are smaller than the actual size of the symbol used. Perturbative results and MA results are assumed to be exact within the limits of the approximation they rely on. The MA and BDMC results at ω0 = 3.0 for the energies and the quasiparticle weights are  in very good quantitative agreement. The ground state energy given by MA is found to be well within 1% of the BDMC results, while the quasiparticle weights of the ground state are also within 1%, up to intermediate couplings, and always within 6% for all coupling considered. The agreement is good enough that even for larger couplings and momenta, the MA energy remains within 2% of the BDMC prediction, and the quasiparticle weight is always within 6%. Such a good agreement shows that MA truly captures the key features of the structure of the phonon cloud. The phonon states ignored are confirmed to be much less important quantitatively. We should mention that the error bars of BDMC are considerably larger at stronger couplings and small adiabaticity ratios (e.g. ω0 = 0.5), such that, in this regime, a comparison between the two methods should not be construed as a direct estimate of the accuracy of MA. Nevertheless, at ω0 = 0.5 and intermediate couplings, where the convergence of BDMC is more severely affected, we still find the results agree to within 10%. All techniques presented here are more questionable in the strong adiabatic limit ω0 → 0,  because the number of phonons in the phonon cloud in this regime is much larger. The ability of MA to represent clouds with an arbitrary number of phonons, albeit spread over a limited number of sites, makes it the most reliable technique at stronger coupling in the adiabatic  regime. BDMC is more limited in this respect as convergence is drastically slowed when the number of phonons is too large. We should mention that whether MA or LPBED would be most accurate in the extreme adiabatic case is not clear. As the phonon energy decreases, the phonon cloud is expected to extend to more sites, but the number of phonons would also 123  4.2. SSHo — SSH coupling to one dispersionless optical branch increase significantly.  4.2.4  From weak coupling to strong coupling  We first observe that SSHo polaron in the weak coupling regime is very similar to the Holstein polaron (compare Figures 1.2 and 1.3 to Figures 4.2, 4.3, 4.4, and 4.5). Below λo = 1, we find a similar dispersion relation, with a bandwidth limited by the phonon energy ω0 (part (a) of Figures 4.2 to 4.5). In the adiabatic limit, the dispersion flattens at large momentum and looks more like the phonon dispersion (Figure 4.2). The polaron plus one phonon continuum is also found at E0 + ω0 (not shown here). The quasiparticle weight (part (b) of Figures 4.2 to 4.5) also decreases with increasing momentum, but remains larger than for the Holstein model. Overall, very little evidence of new polaron physics can be seen in the very weak coupling. The strong coupling limit, however, has a number of unusual features. The polaron ground state energy, usually non-degenerate and found at k = 0 as for the bare particle, is now degenerate and found at finite momentum for couplings λ > c λo (part (a) of Figures 4.2 to 4.5). The position of this minimum kgs is shown in part (d) of those same Figures, and is found to converge asymptotically toward π/2. This is accompanied by the appearance of a local maximum in the quasiparticle weight Z0 (k) close to π/2 at all ω0 , except in the extreme non-adiabatic limit where we observe a local minimum near π/2 instead (part (b) of Figures 4.2 to 4.5). The polaron in the strong coupling regime is a surprisingly light quasiparticle compared to the exponentially large effective mass of the small Holstein polaron, and the quasiparticle weight of the ground state Z0 (kgs ) approaches zero very slowly (part (e) and (f) of Figures 4.2 to 4.4). The minimum of the energy near k = π/2 was mentioned to be a consequence of the new channel for the delocalization of the electron opened by the SSH coupling (end of section 4.1.3), which contributes a term like −2t2 cos (2k) to the electron dispersion. In the adiabatic limit,  the continuum still plays an important role by pushing down the polaron state and reducing the polaron bandwidth when λo is increased. At larger adiabaticity ratios, on the other hand, the continuum is farther away and an increase in bandwidth is observed with increasing λo (Figure 4.5 (a)). The properties of the strong coupling SSHo polaron are therefore closely related to the phonon-modulated channel, whereas the weak coupling regime is dominated by the usual hopping constant with a more standard polaronic behaviour. The dispersion relation of the polaron E0 (k) obtained (part (a) of Figures 4.2 to 4.5) is qualitatively very similar to those obtained by Liu et al. with the Global-Local ansatz [58]. We do not however, observe any non-analyticity in the dispersion itself, confirming their suspicion of an ansatz or technique related problem in some regimes. Numerical comparison for the more 124  4.2. SSHo — SSH coupling to one dispersionless optical branch general case of section 4.2.9 also suggests a very good agreement to within a few percents.  0  1 0.8  -5 -10  0.4 0.2  -15 E0(k)  Z0(k)  0.6  0 -20  (b) 0  0.2  0.4  0.6  0.8  1  k/π 0.5  -25  0.4  -35 (a) -40  0  0.2  λ=0.25 λ=2 λ=5 0.4  0.6 k/π  λ=10 λ=50 λ=100 λ=500 0.8  kgs/π  0.3  -30  0.2 st  WB 1 order Σ−DMC  0.1 1  0  0  100 λο  Figure 4.1: Results for the SSHo model with ω0 = 0.5 from Σ-DMC limited to first order (one phonon) self-energy diagrams. Errors are too small to be represented here. (a) Lowest polaron eigenstate energy E0 (k) for an effective coupling spanning the weak to the very strong regime. (b) Quasiparticle weight of the lowest polaron eigenstate Z0 (k). The couplings and color code are the same as in (a). (c) Momentum of the ground state energy kgs of the lowest polaron state from the first order Σ-DMC results and comparison to first order Wigner-Brillouin perturbation theory.  125  4.2. SSHo — SSH coupling to one dispersionless optical branch  0.6 0.4 0.2  (a) 0  0.2  0.4 0.6 k/π  0.8  1  (c)  0  0.2  0.4 0.6 k/π  0.8  1  -2.5  WB 1 order  -3  WB 2 order st RS 1 order  (d)  0.4  st  0.3  nd  0.2  nd  RS 2 order MA BDMC  -3.5 0  1 λο  0.5  1  kgs/π  E0(kgs)  0  (b)  0.5  -2  0.1 1.5  0  2  0  0.5  1 (e)  0.8  1 λο  1.5  1.5 λο  2  2  (f)  0.75  0.6  *  Z0(kgs)  λ=0.25 λ=0.50 λ=1.0 λ=1.094 λ=1.21 λ=1.96  0.8  Z0(k)  0.5 0.4 0.3 0.2 0.1 0  m0/m  E0(k)  1  0.5 0.4 0.25  0.2 0  0  0.5  1  1.5 λο  2  2.5  3  0  0  0.5  1  2.5  3  Figure 4.2: Results for SSHo with ω0 = 0.5 from BDMC, MA, and first and second order RS and WB perturbation theory. Figures (a)-(b) present BDMC (circles) and MA (lines) results. They share the same couplings and color code, and show: (a) E0 (k) shifted such that E0 (0) = 0 and (b) Z0 (k) for the lowest polaron eigenstate. Figures (c)-(f) share the same color code, and show: (c) E0 (λo ), (d) kgs (λo ), (e) Z0 (λo ) and (f) the ratio m0 /m∗ (λo ), of the ground state of the lowest polaron eigenstate as a function of the effective coupling. 126  4.2. SSHo — SSH coupling to one dispersionless optical branch  3  1  1  0.6 0.4  0  0.2  (a)  -1 0  0.2  0.4 0.6 k/π  0.8  1  (c)  -3  0.2  0.4 0.6 k/π  0.8  1  MA BDMC st RS 1 order  -4 -5 -6  0.3 0.2  nd  RS 2 order st WB 1 order  -7  (d)  0.4  kgs/π  E0(kgs)  0  0.5  -2  0.1  nd  WB 2 order  -8 0  1  1  2 λο  3  4  0  0  1  2 λο  3  4  1  2 λο  3  4  1.5 (e)  0.8  (f) 1  0.6  m0/m  Z0(kgs)  0  (b)  *  E0(k)  2  λ=2.0 λ=2.5 λ=3.0 λ=4.0  0.8  Z0(k)  λ=0.25 λ=0.50 λ=1.0 λ=1.5  0.4  0.5  0.2 0  0  1  2 λο  3  4  0  0  Figure 4.3: Results for SSHo with ω0 = 3.0 from BDMC, MA, and first and second order RS and WB perturbation theory. Figures (a)-(b) present BDMC (circles) and MA (lines) results. They share the same couplings and color code, and show: (a) E0 (k) shifted such that E0 (0) = 0 and (b) Z0 (k) for the lowest polaron eigenstate. Figures (c)-(f) share the same color code, and show: (c) E0 (λo ), (d) kgs (λo ), (e) Z0 (λo ) and (f) the ratio m0 /m∗ (λo ), of the ground state of the lowest polaron eigenstate as a function of the effective coupling. 127  4.2. SSHo — SSH coupling to one dispersionless optical branch  4  1 λ=0.25 λ=0.50 λ=1.00 λ=1.50  0.6  1 0.4  0 -2  0.2  (a)  -1 0  0.2  0.4 0.6 k/π  0.8  1  0  0.2  0.4 0.6 k/π  0.8  1  0.5  -2 (c)  -3 E0(kgs)  0  (b)  -4  (d)  0.4 0.3  BDMC  kgs/π  E0(k)  2  0.8 Z0(k)  3  λ=2.00 λ=3.00  st  RS 1 order  -5  0.2  nd  RS 2 order st  WB 1 order  -6  0.1  nd  WB 2 order  0.5  1  1.5 λο  2  2.5  3  0  0  0.5  1  1.5 λο  2  2.5  3  1  1.5 λο  2  2.5  3  2 (e)  (f)  1.5  *  1 0.9 0.8 0.7 0.6 0.5 0.4 0.3  0  m0/m  Z0(kgs)  -7  1 0.5 0  0.5  1  1.5 λο  2  2.5  3  0  0  0.5  Figure 4.4: Results for SSHo with ω0 = 4.0 from BDMC and first and second order RS and WB perturbation theory. Figures (a)-(b) share the same couplings and color code, and show: (a) E0 (k) shifted such that E0 (0) = 0 and (b) Z0 (k) for the lowest polaron eigenstate. Figures (c)-(f) share the same color code, and show: (c) E0 (λo ), (d) kgs (λo ), (e) Z0 (λo ) and (f) the ratio m0 /m∗ (λo ), of the ground state of the lowest polaron eigenstate as a function of the effective coupling. 128  1  0.95 Z0(k)  4 3 2 1 0 -1 -2 -3 -4 -5  0.9 (a) 0  0.2  0.4  0.6  0.8  λ=0.50 λ=1.00 λ=1.50  (b) 1  0.85  0  0.2  k/π  0.6  0.8  1  k/π 0.5  -2 -4 E0(kgs)  0.4  λ=2.50 λ=3.00 λ=3.50  (c)  0.4  -6  0.3  -8  0.2  (d) st  WB 1 order nd WB 2 order  kgs/π  E0(k)  4.2. SSHo — SSH coupling to one dispersionless optical branch  st  0.1  -10 -12  RS 1 order nd RS 2 order  0  1  λο  2  3  0  0  1  2 λο  3  4  Figure 4.5: Results for SSHo with ω0 = 100 from BDMC and first and second order RS and WB perturbation theory. Figures (a)-(b) share the same couplings and color code, and show: (a) E0 (k) shifted such that E0 (0) = 0 and (b) Z0 (k) for the lowest polaron eigenstate. Figures (c)-(d) share the same color code, and show: (c) E0 (λo ), (d) kgs (λo ), of the ground state of the lowest polaron eigenstate as a function of the effective coupling.  129  4.2. SSHo — SSH coupling to one dispersionless optical branch  4.2.5  Sharp transition  Although the ground state energy E0 (kgs ) and the quasiparticle weight Z0 (kgs ) do clearly show two distinct regimes for small and large coupling, this is not unlike the Holstein model and other q-only dependent models. The most striking result to come out of this work, however, is the presence of a sharp transition in the ground state of the polaron, instead of the crossover as expected for g(q) couplings. The sharp transition might not be obvious when looking at E0 (k) and Z0 (k), where a smooth evolution from one regime to the other is observed with increasing coupling. Nevertheless, there is actually a discontinuity in the polaron ground state, as can be seen from the momentum of the ground state kgs (λo ), from the quasiparticle weight of the ground state Z0 |k=kgs (λ) and from the effective mass m∗ (kgs ) of the ground state, presented  above as m0 /m∗ (see part (c), (e) and (f) of Figures 4.2 to 4.4 and part (c) of Figure 4.5). At the singularity c λo , the effective mass diverges while the derivative with respect to λo for the two other quantities have a jump discontinuity. Other quantities also have singularities; for example our Letter [60] also shows the derivative of the ground state energy with respect to coupling β̃, therein referred to as 2α. All of these non-analyticities are direct consequences of the −2t2 cos(2k) term mentioned at  the end of section 4.1.3. The usual channel for the electron delocalization favours a minimum at k = 0, and the phonon-modulated channel favours a minimum at π/2. When we sum both contributions, an inflexion point in the dispersion relation occurs at the critical coupling. The effective coupling λo simply measures the relative importance of these two channels. The divergence of the effective mass at the critical coupling is very different from the diverging effective mass found in the infinite coupling limit of the Holstein polaron. What we have here is an inflexion point at k = 0. The bandwidth remains finite, the polaron is not localized, the quasiparticle weight does not vanish, and the number of phonons remains finite. Obviously, we cannot use the inverse bandwidth to evaluate the effective mass here. By effective mass, we really mean the inverse of the second order derivative of the dispersion relation with respect to momentum, taken at the momentum kgs of the ground state (see equation (1.93)). Figure 4.6 shows the phase diagram with the phase boundary separating the ground state with zero and non-zero momentum. This phase boundary goes to 0.5 for ω0 → ∞. This limit  is easily obtained by looking at the first order correction to the energy of the ground state. In  130  4.2. SSHo — SSH coupling to one dispersionless optical branch real frequency we find E0 (k) = − 2t0 cos k + = − 2t0 cos k +  1 X go (k, −q)go (k − q, q) + higher order terms, N q ω + 2t0 cos (k − q) − ω0  8t0 ω0 λo X sin2 (q/2) cos2 (k − q/2) + higher order terms, N ω + 2t0 cos (k − q) − ω0 q  (4.37)  and in the limit of large phonon frequency the propagator reduces to its static limit −1/ω0 and lim E0 (k) = −2t0 cos k −  ω0 →∞  8t0 ω0 λo X sin2 (q/2) cos2 (k − q/2) . N ω0 q  (4.38)  Higher order terms are neglected since they scale as (1/ω0 )nph −1 . In the thermodynamic limit we replace N by 2π and the sum by an integral, which yields the same form as in (4.33) and simplifies to   lim E0 (k) = −2t0 cos k − t0 λo 2 − cos (2k) .  ω0 →∞  (4.39)  The dispersion curvature in this limit is given by  i d2 h lim E (k) = −2t0 [1 − 2λo ], 0 dk2 ω0 →∞ k=0  (4.40)  and goes to zero for λo = 0.5. The effective mass thus diverges and the critical coupling is cλ o  = 0.5. We can also obtain an analytical expression for kgs in this limit by looking at where  the slope of the dispersion vanishes for coupling λo > 0.5. We have i dh lim E0 (k) =2t0 [sin (kgs ) − λo sin (2kgs )] = 0, dk ω0 →∞ k=kgs  =2t0 sin (kgs )[1 − 2λo cos (kgs )] = 0.  (4.41)  We want the non-trivial solution kgs 6= 0 which appears above c λo and is given by kgs   1 . = arccos 1 + 2(λ − c λo )   (4.42)  kgs approaches its asymptotic value of π/2 rather slowly in the extreme non-adiabatic limit. In fact, comparing various values of the adiabaticity ratio, the transition is much sharper at √ small ω0 . The momentum of the ground state near the critical coupling thus scales as c λo in the extreme non-adiabatic limit, and scales with a smaller power when the adiabaticity ratio is decreased. 131  4.2. SSHo — SSH coupling to one dispersionless optical branch  1.25 1.2 kgs > 0  1 1.1 0.75  kgs = 0  0.5 0.1  0.9  1  10  100  c  λo  1  0.8  kgs > 0  0.7  kgs = 0  0.6 0.5  0  0.5  1  1.5  2  ω0  2.5  3  3.5  4  Figure 4.6: SSHo phase diagram in the two-dimensional plane (ω0 , c λ0 ) from BDMC (circle) and MA (line). The sharp transition between kgs = 0 and kgs > 0 is shown. The inset shows the same results on a semilogarithmic scale and extends farther in the non-adiabatic regime. The asymptotic value as ω0 → ∞ of the critical coupling is c λ0 → 0.5 We want to stress the fact that we claim a sharp transition, or non-analyticity of the ground state of a single polaron, not a quantum phase transition. Although we work in the T = 0 limit, we only treat the single-electron case which at finite coupling is dressed by a relatively small number of phonons. Since any phase transition involves the cooperative behaviour of an infinite number of degrees of freedom, we limit ourselves to the statement that the ground state of the single polaron is non-analytic at the critical coupling c λo . However, this non-analyticity as a function of the coupling for a single polaron might have a signature in bulk properties, such as the polaron mobility, in the presence of a macroscopic number of polarons. See section 4.2.8 for more detailed comments. We should also mention that the Exact Diagonalization work of Capone et al. [15] leads  132  4.2. SSHo — SSH coupling to one dispersionless optical branch to a very different phase diagram. The MA results suggest that a three-site phonon cloud is necessary to obtain an accurate description of the polaron state. The phonon-modulated channel for the electron delocalization also requires a bare minimum of three sites. This casts serious doubts on the applicability of the results for four-site open-boundary clusters to the thermodynamic limit that we consider here. The unphysical region mentioned in their work, with a vanishing hopping integral and a tendency towards localization, has nothing to do with the sharp transition we find, with a finite bandwidth and an absence of localization.  4.2.6  Spectral function  We also include one example of the spectral function for ω0 = 3.0 and λ = 1.0 in Figure 4.7. The quasiparticle peak of the lowest polaron eigenstate is clearly visible and the continuum starts exactly at energy E0 (kgs ) + ω0 as for other models with Einstein phonons. We note also the presence of a first excited state below the continuum at kgs . This first excited state must have end points as the quasiparticle peak becomes a mere resonance inside the continuum away from kgs . This is similar to what is observed in reference [58]. This spectral function was calculated with mgMAωq . MA would have produced a more accurate result for the shape of the continuum, but this result was not available at the time of writing, whereas mgMAωq was readily available. As explained in chapter 3, this technique reproduces the position and height of the first few lowest lying states and the position of the continuum very accurately at moderate coupling and adiabaticity ratio and therefore serves our purpose equally well.  4.2.7  Perturbation theory  Figure 4.1 shows that the features discussed above appear even when restricting Σi (τ ) to first order diagrams. Perturbation theory will therefore be relevant to some extent. Figures 4.2 to 4.5 indeed show that below the critical coupling, perturbation theory provides a fair approximation for the ground state properties. As for other techniques we used, perturbation theory is more accurate in the non-adiabatic limit. We know of course that for Einstein phonons with ω0 < 4t0 , first order RayleighSchrödinger (RS) perturbation theory for the polaron energy E0 (k) has an unphysical maximum at some momentum kmax , and an unphysical diverging energy E0 → −∞ at larger momentum.  In spite of this unphysical feature, we can calculate the ground state energy and the effective st  1 . At this coupling the maximum in the ground state mass below the critical coupling c λRS o  completely disappears, and we get an unphysical diverging ground state energy E0 (π) → −∞. c λRS 1st o  can be used as a rough estimate of c λo . If we include the second order correction,  133  A(π, ω)  A(kgs, ω)  A(0,ω)  4.2. SSHo — SSH coupling to one dispersionless optical branch  -4  -3  -2  -1  0  1  ω/ t  Figure 4.7: Results for the SSHo spectral function for ω0 = 3 and λo = 1 from mgMAωq and with η = 0.01. The dotted line drawn at E0 (kgs ) + ω0 shows where the continuum starts. The first excited state is visible at kgs . With these parameters we see that this first excited state has end points.  we eliminate the unphysical maximum and the diverging negative ground state energy, but for ω0 < 4t0 , we find instead a positive diverging energy at larger momentum. The critical coupling c λRS 2nd is now a much better approximation of c λ , but the diverging energy at k = π leads o o nd ∗ c RS 2 . to an underestimated kgs and m above λo The Wigner-Brillouin perturbation theory results correspond to calculating Σi (τ ) directly. Figure 4.1 shows that the first order results from Σ-DMC match exactly the first order WB perturbation theory. Understandably, it works equally well as the RS perturbation theory in the strongly non-adiabatic regime because only one or two phonons are sufficient to recover the ground state. At smaller ω0 it does not yield a very accurate estimate of most quantities considered here. Its main advantage is to allow direct comparison with first order diagrams for 134  4.2. SSHo — SSH coupling to one dispersionless optical branch Σi (τ ) when testing the Σ-DMC implementation. It also can calculate the quasiparticle weight and the number of phonons at all momenta and ω0 , something RS cannot do. More details on the perturbative results calculations can be found in Appendix A on perturbation theory for polaron models. A comparison of the effective mass obtained by Capone, Stephan and Grilli, given by equation (4.34) is not presented explicitly here, but it shows that their expression only holds for very small coupling λ0 < 0.1. The effective mass that we obtain through the numerical derivative of the dispersion relation produced by perturbation theory agrees much better, and can be used up to c λo .  4.2.8  Relevance to real systems and experimental validation  We now briefly comment on the relevance of our calculation to experimental systems, and the work still needed to achieve a quantitative comparison. The first question is whether the SSHo model is physically sound, and if it contains all elements needed to describe single electron properties of real materials. A zero-filled monoatomic chain, for example, is a dubious candidate since the dimerization, and therefore the presence of optical phonons, does not occur for a single electron. Dimerization actually only occurs at half-filling. Other systems, of course, have a more complicated unit cell, and some of them seem to be well described by including only optical phonons. Rubrene, an oligoacene used for manufacturing single-crystal organic field effect transistors (OFET), seems a good candidate [93]. Nevertheless, a proper treatment of acoustic phonons is needed for the general case. Other missing elements include a correct treatment of diagonal couplings, and of interchain couplings, important for quasi one-dimensional systems. In this work, we will briefly cover two of these additions: a nondiagonal coupling to acoustic phonons and a diagonal coupling to optical phonons. It should be noted that in real systems, there can be a large number of modes of vibrations and phonon branches. Fortunately, the study of these systems can be simplified somewhat by combining similar modes together into an effective mode parameterized using a combination of ab initio calculations and comparison to experiments. An example of such a parameterization is given by Troisi in reference [93] for rubrene. In our Hamiltonian (4.20), we have also neglected the usual second nearest neighbour hopping (which should also be phonon-affected), higher order terms in the expansion of the hopping integral, and have assumed the vibrations of the atoms around their equilibrium position to be harmonic. The second nearest neighbour hopping is not necessarily negligible. For example, in polyacetylene it is found to be of the order of 10% of t0 [89]. This will push the critical coupling c λo to a larger value and should therefore be included. As for higher order terms in the hopping integral, and anharmonic phonons modes, 135  4.2. SSHo — SSH coupling to one dispersionless optical branch their effects remain an open question. The results presented here are for T = 0, and therefore will be relevant only at temperatures smaller than all other characteristic energy scales, including the phonon energy and the hopping integral. A DMC calculation at finite temperature could be done by calculating the Matsubara Green’s function [67]. This calculation would greatly improve our ability to compare to experiments, since the temperature dependence of various quantities are often reported. Also, a number of experiments on relevant systems are performed at temperatures close to room temperature, and there is a need to describe their properties in this regime. OFETs based on single crystals of oligoacene molecules are an example for which phonon-modulated hopping is known to be important, and for which room temperature properties have a very concrete technological impact. Another important consideration for comparison to experiments, is the need to consider finite electron densities. Understanding the properties of a single polaron is a necessary first step, but will likely prove insufficient except possibly in the very low density limit. Considering half-filled systems like trans-polyacetylene is out of the question without first understanding the two-particle model. Even then, it might be necessary to treat the full many body problem to reach a proper description of the finite density case. Lattice polarons have a tendency to form bipolarons due to the attractive interaction of their phonon clouds. At large distances, however, the phonon clouds have a much smaller overlap, and the repulsive long-range Coulomb interaction usually takes over. This is the case for the Fröhlich polaron, and similar phenomena are found for other models such as the Holstein-Hubbard model. Until we can turn our attention to the SSH bipolaron model, our results are most relevant to insulating systems where single (or a few) polaron(s) properties can be probed with ARPES or PES. Observing the SSHo transition in thermodynamic measurements would be futile in this limit. In transport measurements, the sharp transition might still have a signature in low electron densities due to the divergence of the effective mass. The mobility being a complicated average of a k-dependent quantity inversely proportional to the effective mass m∗ (k), we speculate that it might be suppressed to some extent at the critical coupling due to the k = 0 component, provided that polaronpolaron interactions are weak. A proper calculation at finite temperature is needed to confirm this. Experiments with a tuneable coupling are not easy to design, but the coupling in some dimerized Mott magnetic semiconducting oxides can be varied to some extent by pressure [69]. It is also important to identify which region of parameter space is most relevant for experiments. For example, the lattice constant is measured readily enough with x-ray scattering, and the phonon energy with neutron scattering. The average hopping integral t0 and the nondiagonal coupling β are a different story. These are parameters of an effective tight-binding 136  4.2. SSHo — SSH coupling to one dispersionless optical branch Hamiltonian, and are calculated using ab initio methods. One usually uses the result from such a calculation to predict the lattice constant, which is then compared to experiments. The calculation is optimized to match the experimental results, and the calculated hopping integral is obtained for the optimized parameters. Examples of such a calculation for trans-polyacetylene can be found in [32] and references therein. They report t0 = 2.1995 eV and β = 4.1552 eV Å−1 . More interesting to us is Troisi’s ab initio calculation for rubrene [93]. He uses a Molecular Dynamic (MD) simulation and quantum chemistry computation to obtain the distribution of hopping integrals. He then uses the average from this distribution as t0 and the standard deviation as β. He finds, at T = 300 K, an average transfer integral of t0 = 0.143 eV, β = 0.493 eV Å−1 , and the phonon energy of ω0nd = 6.2 × 10−3 eV, placing rubrene in the strong adiabatic  limit. Values for a diagonal Holstein coupling to optical phonons is also provided, with a strong coupling of g = 2.63 eV Å−1 , and a phonon energy of ω0d = 0.17 eV. A very rough estimate of the non-diagonal coupling can be obtained if we know the lattice constant and the average hopping integral. If we assume that the wave function used for the calculation of the hopping integral falls off exponentially at long distances according to   some characteristic length ξ, the overlap is then proportional to exp − (a + uj+1 − uj )/ξ ,  and therefore expanding around uj+1 − uj = 0 yields  h i (uj+1 − uj ) tj+1, j = t0 1 − + ... , ξ such that β=  t0 . ξ  (4.43)  (4.44)  Now, for the tight-binding Hamiltonian to be an accurate approximation, the wave function needs to be very small at distances corresponding to the lattice constant a. In other words, ξ should be much smaller than a. A lower bound for the non-diagonal coupling is then given by β>  t0 , a  (4.45)  where a and t0 are more readily available in the literature. For polyacetylene, this estimate is about half of the reported value for β. For rubrene, using an approximate lattice constant of 4 Å yields a value an order of magnitude smaller than the value calculated by Troisi, so this lower bond can be significantly less than the actual coupling.  137  4.2. SSHo — SSH coupling to one dispersionless optical branch  4.2.9  SSHod — SSHo with Holstein diagonal coupling  The presence of non-diagonal coupling does not mean that diagonal coupling becomes unimportant. All systems will have at least some diagonal coupling to acoustical phonons. A simpler combination, and the one studied by Liu et al. [58], consist in both a SSH coupling and a Holstein diagonal coupling to the same optical (Einstein) phonon branch. In such a case, a phonon can be emitted by one coupling and absorbed by the other. One might think that the sign problem would be worse for this model because of the possibility of mixed phonon propagators (i.e. phonon propagator with one vertex of each type), and therefore the possibility of a negative sign for diagram of all orders. Fortunately, diagrams with an odd number of SSH vertices cancel with their reflexion image (exact same diagram, but read from right to left with vertex times changed from τi to τ − τi , where τ is the length of the diagram). Once again, no sign problem appears for first order diagrams. For other diagrams, the sign problem is indeed somewhat worse due to the factors of i sin (q/2) of the coupling (4.21). If nSSH is the number of vertices of the SSH type and nSSH is even, we find that we need to add an extra factor of −1 to (4.23) when nSSH /2 is odd. It can also be shown simply by drawing all second order  diagrams that no mixed phonon diagrams contribute up to this order. Up to second order,  we find that only crossed second order diagrams involving exclusively SSH vertices can have a negative value. The sign problem is therefore very similar to the one found in the SSHo model. As for the sharp transition of SSHo, there is little doubt that it will survive the addition of this coupling. If anything, we expect the critical coupling c λo to be lower in the presence of a diagonal coupling. The intermediate Holstein coupling regime was shown to have a reduced bandwidth, therefore the phonon-induced t2 need not be as large to overcome it and shift the minimum away from k = 0. A full investigation of this model is unfortunately outside the scope of this work. The phase space (λo , λd , ω0 ) would now involves two couplings (λo for the non-diagonal SSH coupling, and λd for the Holstein coupling), and the adiabaticity ratio ω0 . We have, however, used BDMC and MAωq to look at the non-adiabatic case of ω0 = 3, with λd between 0.25 and 0.5 and λo between 0.5 and 1.5. We can confirm that the sharp transition remains indeed present. Most properties are only slightly changed, with a critical coupling c λo that varies by less than 5% due to the addition of this weak diagonal coupling. The polaron energy is of course pushed down to lower energy due to the extra coupling, but the shape of the dispersion and the effective mass are only slightly affected. We also find a ground state energy for the polaron in the adiabatic limit that agrees very well with the results from Liu et al. (with ω0 = 0.5, λd = 1 and λo = 2.25 we find a ground state energy only slightly lower and within 0.05% of their result).  138  4.3. SSHa — SSH coupling to one longitudinal acoustic branch  4.3  SSHa — SSH coupling to one longitudinal acoustic branch  Models with acoustic phonons are much less understood, both theoretically and numerically. As pointed out in section 1.5, there is no proof of analyticity available even for g(q) couplings. It is also not obvious whether the average number of phonons remains finite in the cloud, and if a continuum of states is to be found above the polaron ground state. Previous works usually fall into one of three categories: there are variational studies that cannot rule out ansatz-related artifacts, there are semi-classical works with limited domain of applicability, and there are perturbative studies. As mentioned in the introduction of chapter 3, we do not know of any numerical technique that can efficiently study these models other than DMC methods and our extensions to MA. Even then, the absence of a gap in the phonon dispersion leads to a larger number of phonons on average. This means slower convergence for DMC methods, and more significant memory requirements for MAωq . We briefly look at the SSH coupling to acoustic phonons in this section to showcase at least one model with acoustic phonons. The Hamiltonian was presented in section 4.1.3 of this chapter. This is seldom treaded territory and few results are available for this case. Some Rayleigh-Schrödinger perturbative results for this model became available shortly after we started investigating it [57], but they seem to disagree with the earlier Wigner-Brillouin perturbative work of Zoli [101]. Since our own perturbative calculations for SSHa show some quantitative, as well as qualitative, differences compared to the BDMC and mgMAωq results already at small coupling, we will pay special attention to identify the limitations of these perturbative techniques.  4.3.1  Hamiltonian, parameters and normalization  The Hamiltonian is given by (4.20), the phonon dispersion relation by (4.11) and the coupling by (4.21), but we will use the equivalent effective coupling (4.24) since it is real and simplifies the computation. The adiabaticity ratio ω0 /t0 remains the same. For the same reason we could not define an effective coupling in the usual way for SSHo, we also need to define the effective coupling for SSHa by borrowing the definition from the Holstein coupling: λa =  α̃2 . 2t0 ω0  (4.46)  For normalization we need to calculate the limit Σi (τ → 0, k): 8α̃2 Σ (τ → 0, k) = 2π i  Z  0  π  dq sin (q/2) cos2 (k − q/2) =  8 λa ω0 t0 [3 − cos(2k)]. 3π  (4.47) 139  4.3. SSHa — SSH coupling to one longitudinal acoustic branch  4.3.2  Results and discussion  Figure 4.8 shows results obtained with BDMC, MAωq and perturbation theory for ω0 = 3.0. The agreement between BDMC and MAωq is excellent, and a small difference in the energy is only found at larger coupling where the average number of phonons becomes more substantial. In this regime, the maximum number of phonons nmax of MAωq would need to be increased for a better agreement. The dispersion relation E0 (k) (Figure 4.8 (a)) shows no sign of a ground state at finite momentum for the couplings presented. We do, however, find an important peak in the quasiparticle weight Z0 (k) (Figure 4.8 (b)) slightly above k = π. We only show MAωq results for Z0 (k) since achieving a proper convergence for this quantity with BDMC required much more time. The same feature was nevertheless observed with BDMC. Figure 4.8 (c) shows how the ground state energy varies with coupling λa , and that there are no obvious sign of a well defined separation between a weak coupling regime and strong coupling regime in the range studied. We also observe a polaron bandwidth that decreases only very slowly with λa , but it remains to be seen if this holds in the adiabatic regime as well. The effective mass (Figure 4.8 (d)) increases much more slowly with coupling than for the weak coupling SSHo polaron or the Holstein polaron, but it still looks to be doing so exponentially. The BDMC and MAωq data therefore suggest no sharp transition for the SSHa model. The phonon-modulated channel is definitely important for the properties of the polaron, but t2 does not becomes important enough at the couplings investigated for a minimum near π/2 to appear in the polaron dispersion. This is surprising since the coupling is almost the same as for SSHo with only an extra factor : ga (k, q) =  1 go (k, q) with | sin (q/2)|  β̃ = α̃.  (4.48)  This extra factor has little effect near the Brillouin’s zone edges but attenuates the coupling at intermediates couplings. We would expect this to move the critical coupling toward a larger value, but not to remove the transition altogether. In fact, this is exactly what we find if we use ga (k, q) with Einstein phonons; the transition survives. The dispersiveness of the acoustic phonons, or the fact that they are gapless, must therefore be significant. It remains an open question which we plan to address in the near future. The picture presented by perturbation theory is very different. First order RS perturbation st  1 theory predicts a critical coupling c λRS a  ≈ 1 at which the effective mass diverges and a  minimum at finite momentum appears in the dispersion. This value is similar to the critical  coupling for the SSHo model. This is the case studied by Li and al. [57] (the first order RS perturbation theory is therein referred to as second order perturbation theory, since they count 140  4.3. SSHa — SSH coupling to one longitudinal acoustic branch the number of vertices instead of the number of phonons). Second order RS removes the sharp transition, but predicts a very light polaron that becomes lighter with increasing coupling, st  1 very similar to the strong coupling regime of the SSHo polaron. The critical coupling c λRS a  decreases with the adiabaticity ratio, which explains the diverging effective mass observed by Li and al. at fixed λa as ω0 → 0. This result is but a peculiarity of the RS perturbation theory,  and we observe that second order perturbation theory would be better suited to study the SSHa polaron in the perturbative limit. A more thorough investigation for various adiabaticity ratios would be needed, but our results also suggest that at ω0 = 3, the coupling of 0.08 and 0.8 used in their work place them too close to the intermediate coupling, and perturbative results for the effective mass already differ significantly from the BDMC and MAωq results (their definition for the effective coupling and the phonon energy are different from ours, and need to be multiplied by 8 and 2 respectively, to be compared to our values). This only gets worse at smaller ω0 . We also point out that, although the second order RS results for the ground state energy (Figure 4.8 (a) and (c)) are very accurate, the large momentum results are poor at best. This is, of course, closely related to the poor effective mass results away from the very small coupling. WB perturbation theory does not fare much better for the parameters under consideration, and also predicts a critical coupling both at first and second order. The critical couplings, c λWB 1st a  nd  2 , are found at larger coupling, away from the weak coupling regime where and c λWB a perturbative results are expected to apply. Including higher order diagrams moves this critical  coupling to larger values. WB techniques are therefore better suited to look at the effective mass of the SSHa polaron (Figure 4.8 (d)), but produce less accurate results for the ground state st  1 has a peculiar dependence energy than second order RS calculations (Figure 4.8 (c)). c λWB a  on the adiabaticity ratio (not shown here) and seems to have a minimum at intermediate ω0 ≈ 1. This was observed by Zoli [101] who also used WB perturbation theory. The effective st  1 at ω = 3, but the scaling seems very different than the mass increases with λa below c λWB 0 a  coupling-dependent results presented by Zoli for ω0 = 2. This discrepancy will be investigated in further work. Figure 4.9 shows an example of the spectral function for the SSHa polaron. A feature that does not appear to be a lorentzian when calculated at different η, is found at the bare electron energy ǫ(k). We do not know at the moment if this is a continuum or some unphysical artifact of the method. As mentioned in chapter 3, MAωq does not reproduce the shape of the continuum very well for models with optical phonons. The added errors associated with mgMAωq at these larger frequencies can even produce an unphysical negative spectral function in this range. We therefore chose to present only the spectral function below ǫ(k), where we know mgMAωq to be accurate. We can however state that other polaron models with acoustic phonons (the Davydov 141  4.3. SSHa — SSH coupling to one longitudinal acoustic branch model and the deformation coupling to acoustic phonons, for example) have a similar feature at ǫ(k), and this for all adiabaticity ratios tested. If it is indeed a continuum, it would seem that it is very different than the one polaron plus one phonon we are used to, and would be more closely related to the bare electron state. Once again, more work is needed here.  142  4.3. SSHa — SSH coupling to one longitudinal acoustic branch  0.8  1 λ=0.50 λ=1.00 λ=1.50 λ=2.00 λ=2.50  0.6 0.5  -2  0.4 -3  0.3 MAωq BDMC st RS 1 (a) nd RS 2  -4 -5 0  0.2  0.4  0.6  0.8  (b)  0.1 1  0  0  0.2  0.4  k/π 2 (c)  1  (d) 1.5  k=π  BDMC MAωq  -2  st  WB 1 order nd WB 2 order st RS 1 order nd RS 2 order  1 -4  k=0 0.5  -6 -8  0.8  2  0 E0(λa)  0.6 k/π  0  1  2 λa  3  4  0  *  -6  0.2  m0/m  E0(k)  -1  0.7  Z0(k)  0  0  1  2 λa  3  4  Figure 4.8: Results for SSHa with ω0 = 3 from BDMC, MAωq and first and second order RS and WB perturbation theory. Figure (a)-(b) share the same color code and show: (a) E0 (k) and (b) Z0 (k) for the lowest polaron eigenstate. The plain line is for BDMC, the open circles are for MAωq , the dotted line is for first order RS, and the dashed line is second order RS. Perturbation theory results for the dispersion are only shown for λa = 0.50 and λa = 1.50. Figures (c)-(d) share the same color code and show: (c) E0 (k = 0, λa ) and E0 (k = π, λa ), (d) m0 /m∗ (λa ), of the ground state of the lowest polaron eigenstate as a function of the effective coupling.  143  4.3. SSHa — SSH coupling to one longitudinal acoustic branch  A(π, ω)  A(π/2, ω)  A(0,ω)  η=0.001 η=0.0001  -4  -3  -2  -1 ω/t  0  1  2  Figure 4.9: mgMAωq results for the SSHa spectral function for ω0 = 3, λa = 1 and with η = 0.001 (red) and η = 0.0001 (green) . The dotted line drawn at ǫ(k) shows where a feature presumed to be the continuum starts. We only show the spectral function below this value. Many excited states are also shown.  144  4.4. Summary  4.3.3  SSHoa — SSH coupling to longitudinal and acoustic and optical (Einstein) branches  As pointed out earlier, the choice to study only optical phonons is sometimes dictated by simplicity rather than by physical arguments (obvious exceptions include the treatment of electronphonon interactions in ionic crystals, for example, where the coupling to optical phonons clearly dominates). Seeing that the SSHo model has a sharp transition, while the SSHa model does not seem to have one, raises the question of whether the sharp transition would survive the addition of acoustic phonons. It is clear that a small enough coupling will not affect the transition. In the perturbative limit, the first correction from this coupling will simply be proportional to λa , and even the k-dependence of the energy correction should not affect greatly the competition between the energy of the constant hopping term and the energy of the phonon-modulated term for optical phonons. Away from this regime, it is difficult to predict what will happen. This model would warrant its own section to fully study how the sharp transition is affected by the addition of the acoustic branch, and to draw a complete phase diagram. We limit ourselves here to the statement that the transition does indeed survive the addition of a weak coupling to acoustic phonons. We verified this using a BDMC code for an optical phonon frequency of ωo = 3.0, an acoustical phonon frequency of ωa = 1.5, and a coupling to acoustical phonons of λa = 0.1. The dispersion relations found are shifted to smaller energies by this additional coupling, but the critical coupling and the momentum of the ground state were almost unaffected to within the BDMC error bars.  4.4  Summary  1. Phonon-modulated hopping (section 4.1.2) is a process involving the creation or annihilation of a phonon while the electron hops from one site to the other. It couples phonons to the electron’s motion instead of the electron density. It is also referred to as nondiagonal coupling in the site index. Such a coupling arises from terms not considered by the Born-Oppenheimer approximation. With this approximation, and after going to a tight-binding description, the hopping integral is a constant. The SSH coupling restores the dependence of the hopping integral on the actual position of the atoms by expanding it around the equilibrium position of the lattice and keeping the linear contribution only. 2. The SSH hamiltonian (section 4.1.3) has a hopping term to describe the kinetic energy of the electron, the phonon Hamiltonian to describe the lattice, and a non-diagonal electronphonon interaction given by the SSH coupling. The half-filled version of this model was 145  4.4. Summary popularized by Sue, Schrieffer and Heeger [40, 91] to describe trans-polyacetylene. We are interested in the single-electron version in this work. 3. The SSH coupling g(k, q) depends on both the phonon momentum, and the electron momentum (section 4.1.3). Such a coupling has a sign problem (section 4.1.4) because pairs of vertices now depend on the electron momenta k and k′ , which are not necessarily the same for both vertices. With k 6= k′ the contribution of a pair cannot be expressed  as a perfect square, and is therefore not positive definite.  4. The SSH model with coupling to optical (Einstein) phonons, or SSHo, is studied in section 4.2 with BDMC, MA, and perturbation theory. We find two very distinct regimes. At weak coupling, the polaron is very similar to the Holstein polaron (section 4.2.4 and Figures 4.2 to 4.5). We find a similar dispersion relation with a flattening at larger momentum, especially in the adiabatic regime, due to the proximity of the polaron plus one phonon continuum above E0 + ω0 . In the strong coupling regime, the polaron ground state has a finite momentum, and a peak in the quasiparticle weight near π/2. The strong-coupling SSHo polaron is very light compared to the Holstein polaron, and the quasiparticle weight is considerably larger. The strong polaron properties are closely related to the new channel for the electron delocalization opened by the SSH coupling, giving rise to a next nearest neighbour hopping term −2t2 cos(2k) in the polaron disper-  sion, and favouring a minimum at π/2.  5. A sharp transition is found at a critical coupling c λo separating the weak coupling Holstein-like polaron, and the unusual strong coupling polaron. The effective mass diverges at this point. Other properties such as the momentum of the ground state kgs (λo ), and the quasiparticle weight of the ground state Z0 |k=kgs (λ), have non-analyticities (see part (c), (e) and (f) of Figures 4.2 to 4.4 and part (c) of Figure 4.5). Figure 4.6 shows the phase diagram in the (ω0 , c λo ) plane. At ω0 → ∞ we find c λo → 0.5. 6. Perturbation theory (section 4.2.7) also predicts a sharp transition. Both RayleighSchrödinger and Wigner-Brillouin perturbation theory do rather well up to c λo in the non-adiabatic regime, despite the usual large momentum failings. Second order RS usually gives the most accurate results for the ground state. 7. The SSH model with coupling to acoustic phonons, or SSHa, is presented in section 4.3. Although some features due to the phonon-modulated channel are present, such as a peak in the quasiparticle weight near π/2, we do not find a sharp transition in this model up 146  4.4. Summary to the couplings investigated. In fact, there is no obvious separation between the weak and strong coupling regime. The effective mass increases exponentially with the coupling, albeit somewhat more slowly than for the Holstein model. 8. Perturbation theory for SSHa (section 4.3.2) presents a very different picture. Both first order RS perturbation theory and WB perturbation theory to any order find a sharp transition with a diverging effective mass at the critical coupling. Second order RS calculation does not predict a transition, but is only valid at very small couplings. At intermediate coupling, it predicts a rapidly decreasing effective mass with increasing λa . Although second order RS finds an accurate ground state energy even at intermediate couplings, the rest of the dispersion is not well reproduced. Perturbation theory should therefore not be trusted away from the very weak coupling regime and a RS calculation of the effective mass should go at least to second order in the number of phonons. 9. The spectral function for the SSHa polaron (Figure 4.9) clearly shows a quasiparticle peak for the ground state and a number of excited bound states. A feature present at ǫ(k) is likely to be the continuum but more convincing evidence is still needed. 10. The sharp transition found in SSHo survives the addition of other phonon branches, such as an acoustical phonon branch with SSH coupling (section 4.3.3), or other couplings to the same optical (Einstein) phonon branch, such as a diagonal Holstein coupling (section 4.2.9).  147  Chapter 5  Conclusion 5.1  Summary  The contributions of this work can be divided into two categories: technical contributions, and some physical insight into the polaronic properties of models that differ from the usual lattice polaron. The Bold Diagrammatic Monte Carlo (BDMC) was used to study a number of lattice polarons in this work. It was described in chapter 2. Like other Diagrammatic Monte Carlo (DMC) techniques, it relies on a stochastic sampling of a Feynman diagrammatic expansion in imaginary time. This improved technique achieves a more efficient sampling of the self-energy diagrams by using a more complicated propagator to generate new diagrams. By using a selfconsistent procedure based on Dyson’s equation, the drawing element (the bold line) becomes ever more complicated, and the total number of diagrams considered grows much faster than the linear increase with computation time observed for other DMC methods. Proper care needs to be taken to avoid a double-counting of diagrams. This translates into a sampling restricted to strictly-crossing diagrams. The more efficient sampling also allows for models with weak sign problems. The BDMC method is essentially exact, but reaching a proper convergence can take a considerable amount of time. It is versatile enough to consider any type of coupling, any number and type of phonon branches, and this, in any dimension. It does not perform as well in the strongly adiabatic regime, especially with acoustic phonons. The Momentum Average (MA) approximation, is a growing category of analytical approximations, which consists of summing all Feynman diagrams in the self-energy, with each diagram slightly simplified. These techniques were presented in chapter 3. In its simplest version, each propagator is replaced by a momentum-averaged propagator. It can be extended to solve for multiple phonon branches, as well as momentum-dependent coupling g(q). The extension to g(k, q) couplings is done by using a real-space variational interpretation where the phonon operators are Fourier transformed to real space. In this representation, one can select the type of phonon cloud states to keep in the variational basis. Results produced by Berciu with this technique are presented along the BDMC results in chapter 4 on phonon-modulated hopping. 148  5.1. Summary Three novel techniques that admit dispersive phonons are also presented herein. The first is an exact solution for finite systems inspired by the MA formalism. The second, MAωq divides a momentum variable into a coarse component, which is treated exactly, and an offset, which is then averaged out. This technique allows for g(k, q) couplings as well as dispersive phonons, but requires a significant amount of computer memory. Those requirements depend on the number of coarse momentum values kept, and the maximum number of phonons. To treat a larger number of phonons, a further extension is proposed: mgMAωq . It consists of changing the momentum grid during the calculation. It considers fewer momentum values, when the number of phonons is large, and a finer grid, when the number of phonon is less important. With the exception of DMC methods, we believe these are the first techniques that can treat acoustic phonon branches effectively. The polaron models studied in chapter 4 all include a phonon-modulated hopping, or nondiagonal coupling. It is characterized by an interaction where the phonons couple to the electron’s motion instead of the the electron’s density. The SSH coupling is a prominent example in the literature, and we chose to investigate a number of models that include it. In the tight-binding SSH model, the hopping integral is assumed to vary linearly with the distance separating two sites. This lead to a g(k, q) coupling, and an associated sign problem. The SSH polaron was studied with BDMC, MA and perturbation theory, and was contrasted with the Holstein polaron. We recall that the Holstein model is a simple case of lattice polaron with optical phonons, and a momentum-independent coupling. It has distinct properties in the weak coupling regime and the large coupling regime, but those are separated by a smooth crossover. The effective mass as a function of coupling increases as a power-law at small coupling, and as an exponential function in the heavy small polaron regime. Other generic features of lattice polarons with optical phonons include a polaron plus one phonon continuum starting at E0 + ω0 , a very small quasiparticle weight at large coupling, and an average number of phonons increasing rapidly at large couplings. The SSH model with coupling to optical (Einstein) phonons also has two distinct regimes. In the small coupling regime, we find a behaviour not unlike the usual lattice polaron. The strong coupling regime, however, sees the appearance of a finite-momentum ground state due to the phonon-modulated channel for the delocalization of the electron. This channel, opened by the SSH coupling, is associated with a next-nearest neighbour hopping term of the form −2t2 cos(2k) with a negative t2 . A peak in the quasiparticle weight is also observed near π/2.  The quasiparticle weight is significantly larger in this regime than the exponentially suppressed one found for Holstein. The effective mass actually decreases with an increasing coupling. The 149  5.2. Future work transition between the two regimes is shown to be a sharp transition with non-analyticities in the quasiparticle weight, the momentum of the ground state and the effective mass. The latter diverges at the critical coupling, suggesting a possible experimental confirmation. The sharp transition is seen to survive the addition of a weak diagonal coupling to optical phonons, and the addition of a non-diagonal coupling to acoustic phonons, The SSH model with coupling to acoustic phonons does not show two regimes, and does not seem to have a sharp transition, up to the largest coupling investigated. It still has some features associated with the phonon-modulated channel, such as a peak in the quasiparticle weight near π/2. We also point out the dangers of relying on perturbation theory for this model. First order Rayleigh-Schrödinger perturbation theory, and first and second order Wigner-Brillouin perturbation theory, all predict a sharp transition. A second order Rayleigh-Schrödinger calculation is needed to remove this artifact in the effective mass. Second order RS results for the dispersion are relatively accurate at small momentum, but differ notably at large momentum.  5.2  Future work  An important part of the work we propose to undertake, is to revisit a number of electronphonon couplings to acoustic phonons and study the polaron properties with these couplings. We already have preliminary results for the deformation coupling, and the Davydov model. A number of questions remain, especially regarding the properties and nature (and existence) of the continuum. A number of previous semi-classical and variational studies also claim to find non-analyticities in those models (see references mentioned in the last section of [34]), but our preliminary results do not support this. Regarding the SSH model and phonon-modulated hopping, we have stated that the sharp transition found in the SSHo polaron survives the addition of other couplings (diagonal coupling to optical phonon), and the addition of other phonon branches (SSH coupling to acoustic phonons). This is an important result, but a more thorough study is needed to draw the phase diagram of those models. The SSHa model also require some more work, as few values of the parameter space have been investigated so far. The BDMC method can already be considered a mature technique as we have applied it successfully to the many variants of the SSH Hamiltonian, to the polaron with the deformation potential, to the Davydov model, to the Holstein model and breathing mode Hamiltonian. The MAωq approximation, on the other hand, was a late addition to this work, and a more systematic comparison to other MA techniques is needed. 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When applying perturbation theory to polaron problems in the weak coupling regime, the unperturbed Hamiltonian is given by the electron hopping term and the phonon hamiltonian. The perturbation is assumed to be the electron-phonon interaction Ĥel-ph , which can  only create or absorb phonons. Only even corrections to the energy need to be considered, such that we usually label the order of the perturbation calculation by the number of phonons involved. In this section we will, however, use the number of vertices instead.  A.1  Rayleigh-Schrödinger  We assume that the set of unperturbed states is given by {|n0 i}, that the eigenstates are all  non-degenerate, and that they are numbered by the integer n. In practice, the index will simply be a momentum index. The full eigenstate is given by |ni = |n0 i + |n1 i + |n2 i + . . . ,  (A.1)  where the exponent is the order of the correction, and the eigenenergy is En = En0 + En1 + En2 + · · · = En0 + En2 + . . .  (A.2)  The corrections to the eigenstates of j th order are found while requiring that the perturbed state be normalized up to that order. We require hn|ni = 1, but we keep only contributions up  to order j such as: hnj |n0 i, hnj−1 |n1 i, etc. We also use hn0 |Ĥel-ph |n0 i = 0, and the fact that P odd powers of Ĥel-ph also vanish to simplify the results. Finally, we define ′k as the sum over all states except the one we are calculating corrections for (i.e. a sum over all k except where k = n).  158  A.1. Rayleigh-Schrödinger We find the first two non-vanishing corrections to the energy to be En2 =  X′ hk0 |Ĥel-ph |n0 i 2 , En0 − Ek0  (A.3)  k  En4 =  X′ X′ X′ hn0 |Ĥel-ph |m0 ihm0 |Ĥel-ph |k0 ihk0 |Ĥel-ph |l0 ihl0 |Ĥel-ph |n0 i 0 ) (En0 − Ek0 )(En0 − El0 )(En0 − Em  m  l  k  X′ X′ hk0 |Ĥel-ph |n0 i 2 hl0 |Ĥel-ph |n0 i 2 − , (En0 − Ek0 )2 (En0 − El0 ) k  (A.4)  l  while the first two non-vanishing corrections to the states are |n1 i = 2  |n i =  X′ k  |k0 i  X′ X′ k  l  hk0 |Ĥel-ph |n0 i , En0 − Ek0  (A.5) 2  hk0 |Ĥel-ph |l0 ihl0 |Ĥel-ph |n0 i 1 X′ 0 hk0 |Ĥel-ph |n0 i |n i . − |k i 2 (En0 − Ek0 )(En0 − El0 ) (En0 − Ek0 )2 0  (A.6)  k  The quasiparticle weight, defined as Zn = |hn0 |ni|2 , is given to zeroth order by Zn0 = |hn0 |n0 i|2 =  1. The first order correction vanishes since hn0 |n1 i = 0. The first non-vanishing contribution  is given by  2  Zn2  1 X′ hk0 |Ĥel-ph |n0 i . = hn |n i = 2 (En0 − Ek0 )2 0  2  (A.7)  k  0 P The average number of phonons is given by nph = h q bq † bq i. To zeroth order we find nph n =  hn0 |nph |n0 i = 0, and since nph cannot create or absorb phonons, we also have hn0 |nph |n1 i =  hn1 |nph |n0 i = 0. We take the index k to be a phonon momentum index and find the first non-vanishing correction to be nph  2  n  X′ hk0 |Ĥel-ph |n0 i 2 . = hn |nph |n i + 2hn |nph |n i = {z } | (En0 − Ek0 )2 1  1  0  2  ∝hn0 |nph |n0 i=0  (A.8)  k  To understand the large momentum failings we need to look at the denominator of the above 0 for models with optical (Einstein) phonons (part (a)), expressions. Figure A.1 shows En0 − Em  and acoustic phonons (part (b)). This is given by ǫ(k)−ǫ(k −q)−ω(q) = −2t cos(k)+2t cos(k −  q) − ω0 . For optical phonons this means that below ω0 = 4 there will be a momentum value  at which the denominator becomes zero, and perturbative corrections will diverge. For the  perturbed energy up to second order, this is seen as an unphysical maximum in the dispersion, 159  A.2. Wigner-Brillouin and a negative diverging energy at large momentum. For the perturbed energy up to fourth order, we find instead a positive diverging energy at large momentum. For models with acoustic phonons, on the other hand, things are more complicated since the denominator −2t cos(k) +  2t cos(k − q) − 2ω0 sin(q/2) can go to zero for any phonon energy ω0 , with q = 0 or q = π.  At these values, however, the coupling to acoustic phonons is expected to vanish, making the singularity removable in some expressions. We can still calculate the second and fourth order corrections to the energy, but the expressions for the eigenstates, the quasiparticle weight, and the average number of phonons, all diverges due to the denominator being elevated to a greater power than the couplings in the numerator. For ω0 > 2 the second and fourth order correction  to the energy can be calculated for all electron’s momentum. Below this value, they can be calculated at momenta smaller than some limit at which the denominator vanishes.  (a)  (b)  0 of RS perturbation theory corrections for models with: (a) Figure A.1: Denominator En0 − Em optical (Einstein) phonons, and (b) acoustic phonons. The denominator is presented for the limiting case of ω0 = 4 in (a) and ω0 = 2 in (b).  A.2  Wigner-Brillouin  Wigner-Brillouin (WB) perturbation theory, also known as Green’s function perturbation theory, usually converges more slowly with increasing order than RS perturbation theory. It comes, however, with a number of advantages that make it well suited to study polaron models with acoustic phonons. The denominator found in the WB corrections depends self-consistently on the perturbed energy En∗ instead of the unperturbed En0 . Since En∗ < En0 , the denominator does 160  A.2. Wigner-Brillouin not vanish and there is no divergence. The correction to the energy up to fourth order is given by En∗ =En0 +  X′ hk0 |Ĥel-ph |n0 i En∗ − Ek0  2  (A.9)  k  +  X′ X′ X′ hn0 |Ĥel-ph |m0 ihm0 |Ĥel-ph |k0 ihk0 |Ĥel-ph |l0 ihl0 |Ĥel-ph |n0 i k  l  0 ) (En∗ − Ek0 )(En∗ − El0 )(En∗ − Em  m  + ...,  while the perturbed eigenstate up to second order is |ni = |n0 i +  X′ k  |k0 i  hk0 |Ĥel-ph |n0 i X′ X′ 0 hk0 |Ĥel-ph |l0 ihl0 |Ĥel-ph |n0 i + + ... |k i En∗ − Ek0 (En∗ − Ek0 )(En∗ − El0 ) k  (A.10)  l  The perturbed state |ni is not normalized with itself, but is chosen such that hn0 |ni = 1.  Knowing this, we find the quasiparticle weight with its first correction: Z=  |hn0 |ni|2 = hn|ni  1 1+  P  0 0 ′ hk |Ĥel-ph |n i ∗ −E 0 )2 k (En k  .  2  (A.11)  + ...  The average number of phonons is found in a similar fashion  nph    n  =  |hn0 |ni|2 = hn|ni  P  1+  0 0 ′ hk |Ĥel-ph |n i 0 ∗ 2 k (En −Ek )  P  2  0 0 ′ hk |Ĥel-ph |n i 0 2 ∗ k (En −Ek )  + ... .  2  (A.12)  + ...  161  


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