Polaron physics beyond the Holstein model by Dominic Marchand B.Sc. in Computer Engineering, Universit´eLaval, 2002 B.Sc. in Physics, Universit´eLaval, 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 intro- duced 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 devel- oped 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 quasipar- ticle 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. Svis- tunov 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 Abstract ........................................... ii Preface ............................................ iii Table of Contents ...................................... vi List of Figures ........................................ ix Acknowledgements ..................................... xi Dedication .......................................... xiii Foreword ........................................... xiv 1 Polaron Physics ..................................... 1 1.1 Early days: Pekar’s large polaron . ...... 1 1.1.1 Pekar’spolaron ............................... 2 1.1.2 The effective mass of Pekar’s polaron . ... 5 1.2 Fr¨ohlich’s large polaron, a first microscopic model . ............. 7 1.2.1 The Fr¨ohlich Hamiltonian . 7 1.2.2 Weakcouplingregime .. .. .. .. .. .. .. .. 10 1.2.3 Other treatments of the Fr¨ohlich polaron . ....... 11 1.3 Thelatticepolaron ............................... 14 1.3.1 The lattice polaron Hamiltonian . 15 1.4 TheHolsteinmodel ................................ 17 1.4.1 Weak coupling perturbation . 18 1.4.2 Strongcouplingperturbation . 19 1.4.3 Ground state properties as a function of the coupling . ......... 21 1.4.4 Momentum-dependent properties of the Holstein polaron ........ 22 1.4.5 Spectral properties of the Holstein polaron . ........ 22 1.5 From large polaron to small polaron: transition versus crossover . 26 1.6 Measuring polaron properties experimentally . ........... 28 1.6.1 ARPES — Angle-Resolved Photoemission Spectroscopy . ........ 28 1.6.2 Effectivemassandmobility . 29 1.6.3 Optical conductivity and Photo-Induced Infrared Absorption . 31 vi Table of Contents 1.7 Summary ....................................... 32 2 Bold Diagrammatic Monte Carlo .......................... 34 2.1 GenericMonteCarlotechnique . ..... 34 2.1.1 Weighted averages . 34 2.1.2 Size of the configuration space and Monte Carlo average ........ 35 2.1.3 Thestochasticsumasarandomwalk . 36 2.1.4 Explorationofagraph ........................... 37 2.1.5 Balanceequation .............................. 38 2.1.6 Metropolis algorithm . 41 2.1.7 Continuousvariables . 42 2.1.8 Signproblem ................................. 42 2.1.9 Erroranalysis ................................ 43 2.2 DiagrammaticMonteCarlo . 47 2.2.1 Hamiltonian ................................