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arXiv:cond-mat/9804097 · Physical Review Letters

Polaron Problem by Diagrammatic Quantum Monte Carlo

Nikolay V. Prokof'ev, Boris V. Svistunov

Published Sep 21, 1998• 331 citations
arXivHTMLPDF

Abstract

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 $\mathbf{k}$ 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.

DOI
10.1103/physrevlett.81.2514

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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(k,\ensuremathτ)G(\mathbf{k},\ensuremath{\tau})G(k,\ensuremathτ), we construct a process which generates continuous random variables k\mathbf{k}k and \ensuremathτ\ensuremath{\tau}\ensuremathτ, with a distribution function coinciding exactly with G(k,\ensuremathτ)G(\mathbf{k},\ensuremath{\tau})G(k,\ensuremathτ). 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.
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 results for the solution of the large polaron Fr\"ohlich Hamiltonian in 3 dimensions (3D) and 2 dimensions (2D) obtained via the diagrammatic Monte Carlo (DMC) method. Our implementation is based on the approach by Mishchenko [A. S. Mishchenko et al., Phys. Rev. B 62, 6317 (2000)]. Polaron ground state energies and effective polaron masses are successfully benchmarked with data obtained using Feynman's path integral formalism. By comparing 3D and 2D data, we verify the analytically exact scaling relations for energies and effective masses from $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$, which provides a stringent test for the quality of DMC predictions. The accuracy of our results is further proven by providing values for the exactly known coefficients in weak- and strong-coupling expansions. Moreover, we compute polaron dispersion curves which are validated with analytically known lower and upper limits in the small-coupling regime and verify the first-order expansion results for larger couplings, thus disproving previous critiques on the apparent incompatibility of DMC with analytical results and furnishing useful reference for a wide range of coupling strengths.
These notes are intended as a detailed discussion on how to implement the diagrammatic Monte Carlo method for a physical system which is technically simple and where it works extremely well, namely the Fröhlich polaron problem. Sampling schemes for the Green function as well as the self-energy in the bare and skeleton (bold) expansion are disclosed in full detail. We discuss the Monte Carlo updates, possible implementations in terms of common data structures, as well as techniques on how to perform the Fourier transforms for functions with discontinuities. Control over the variety of parameters, especially in the bold scheme, is demonstrated. Sample codes are made available online along with extensive documentation. Towards the end, we discuss various extensions of the method and their applications. After working through these notes, the reader will be well equipped to explore the richness of the diagrammatic Monte Carlo method for quantum many-body systems.
A detailed study of the Fr\"ohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method [N. V. Prokof'ev and B. V. Svistunov, Phys. Rev. Lett. 81,81,81, 2514 (1998)]. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green's function, within an approach developed in the present paper. We present up to date results for the binding energy, and make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals nontrivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.
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.
A detailed study of the Frohlich polaron model is performed on the basis of diagrammatic quantum Monte Carlo method. The method is further developed both quantitatively (performance) and qualitatively (new estimators), and is enhanced by spectral analysis of the polaron Green function, within a novel approach. We present the best up to date results for the binding energy, and for the first time make available precise data for the effective mass, including the region of intermediate and strong couplings. We look at the structure of the polaron cloud and answer such questions as the average number of phonons in the cloud and their number/momentum distribution. The spectral analysis reveals non-trivial structure of the spectral density at intermediate and large coupling: the spectral continuum features pronounced peaks that we attribute to unstable excited states of the polaron.