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Diagrammatic Monte Carlo method as applied to the polaron problems

Andrei S. Mishchenko

Published Sep 30, 2005• 31 citations
Publisher

Abstract

Numerical methods whereby exact solutions to the problem of a few particles interacting with one another and with several bosonic excitation branches are presented. The diagrammatic Monte Carlo method allows the exact calculation of the Matsubara Green function, and the stochastic optimization technique provides an approximation-free analytic continuation. In this review, results unobtainable by conventional methods are discussed, including the properties of excited states in the self-trapping phenomenon, the optical spectra of polarons in all coupling regimes, the validity range analysis of the Frenkel and Wannier approximations relevant to the exciton, and the peculiarities of photoemission spectra of a lattice-coupled hole in a Mott insulator.

DOI
10.1070/pu2005v048n09abeh002632

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Numerical methods whereby exact solutions to the problem of a few particles interacting with one another and with several bosonic excitation branches are presented. The diagrammatic Monte Carlo method allows the exact calculation of the Matsubara Green function, and the stochastic optimization technique provides an approximation-free analytic continuation. In this review, results unobtainable by conventional methods are discussed, including the properties of excited states in the self-trapping phenomenon, the optical spectra of polarons in all coupling regimes, the validity range analysis of the Frenkel and Wannier approximations relevant to the exciton, and the peculiarities of photoemission spectra of a lattice-coupled hole in a Mott insulator.
We present recent advances in understanding of the ground and excited states of the electron–phonon coupled systems obtained by novel methods of Diagrammatic Monte Carlo and Stochastic Optimization, which enable the approximation-free calculation of Matsubara Green function in imaginary times and perform unbiased analytic continuation to real frequencies. We analyze Rashba–Pekar exciton–polaron as a classic example of a system with self-trapping crossover and demonstrate basic features of the phenomenon. Analysis of unconventional self-trapping cases, such as Jahn–Teller driven crossover in one-dimensional systems and magnon-assisted localization in cuprates, is presented. Results on the optical conductivity of the Fröhlich polaron are also discussed.
We present a numerical solution of the polaron problem by a novel Monte Carlo method. Starting from a conventional diagrammatic expansion for the polaron Green function $G(\mathbf{k},\ensuremath{\tau})$, we construct a process which generates continuous random variables k\mathbf{k}k and $\ensuremath{\tau}$, with a distribution function coinciding exactly with $G(\mathbf{k},\ensuremath{\tau})$. The polaron spectrum is extracted from the asymptotic behavior of the Green function. We compare our results for the polaron energy with the variational treatment of Feynman, and present an accurate dispersion curve which features an end point at finite momentum.
We study the properties of the two-dimensional Fermi polaron model in which an impurity attractively interacts with a Fermi sea of particles in the zero-range limit. We use a diagrammatic Monte Carlo (DiagMC) method which allows us to sample a Feynman diagrammatic series to very high order. The convergence properties of the series and the role of multiple particle-hole excitations are discussed. We study the polaron and molecule energy as a function of the coupling strength, revealing a transition from a polaron to a molecule in the ground state. We find a value for the critical interaction strength which complies with the experimentally measured one and predictions from variational methods. For all considered interaction strengths, the polaron ZZZ factor from the full diagrammatic series almost coincides with the one-particle-hole result. We also formally link the DiagMC and the variational approaches for the polaron problem at hand.
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.
The Fröhlich polaron model describes a ubiquitous class of problems concerned with understanding properties of a single mobile particle interacting with a bosonic reservoir. Originally introduced in the context of electrons interacting with phonons in crystals, this model found applications in such diverse areas as strongly correlated electron systems, quantum information, and high energy physics. In the last few years this model has been applied to describe impurity atoms immersed in Bose-Einstein condensates of ultracold atoms. The tunability of microscopic parameters in ensembles of ultracold atoms and the rich experimental toolbox of atomic physics should allow to test many theoretical predictions and give us new insights into equilibrium and dynamical properties of polarons. In these lecture notes we provide an overview of common theoretical approaches that have been used to study BEC polarons, including Rayleigh-Schrödinger and Green's function perturbation theories, self-consistent Born approximation, mean-field approach, Feynman's variational path integral approach, Monte Carlo simulations, renormalization group calculations, and Gaussian variational ansatz. We focus on the renormalization group approach and provide details of analysis that have not been presented in earlier publications. We show that this method helps to resolve striking discrepancy in polaron energies obtained using mean-field approximation and Monte Carlo simulations. We also discuss applications of this method to the calculation of the effective mass of BEC polarons. As one experimentally relevant example of a non-equililbrium problem we consider Bloch oscillations of Bose polarons and demonstrate that one should find considerable deviations from the commonly accepted phenomenological Esaki-Tsu model. We review which parameter regimes of Bose polarons can be achieved in various atomic mixtures.
We develop a Monte Carlo scheme for sampling series of Feynman diagrams for the proper self-energy which are self-consistently expressed in terms of renormalized particle propagators. This approach is used to solve the problem of a single spin-down fermion resonantly interacting with the Fermi gas of spin-up particles. Though the original series based on bare propagators are sign-alternating and divergent one can still determine the answer behind them by using two strategies (separately or together): (i) using proper series re-summation techniques, and (ii) introducing renormalized propagators which are defined in terms of the simulated proper self-energy, i.e. making the entire scheme self-consistent. Our solution is important for understanding the phase diagram and properties of the BCS-BEC crossover in the strongly imbalanced regime. On the technical side, we develop a generic sign-problem tolerant method for exact numerical solution of polaron-type models, and, possibly, of the interacting many-body Hamiltonians.