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Physics Procedia

Matsubara Green's Function and its Analytic Continuation using Flat Histogram Quantum Monte Carlo Simulation

Nikolaos G. Diamantis, Efstratios Manousakis

Published Jan 1, 2015• 2 citations
Publisher

Abstract

We demonstrate that by incorporating flat histogram techniques in the quantum Monte Carlo simulation, we can calculate the Matsubara Green's function for significantly larger values of the imaginary time. This allows us to obtain more accurate results for the spectral function, particularly for low energy excitations, using known techniques for analytic continuation to real time.

DOI
10.1016/j.phpro.2015.07.113

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We demonstrate that by incorporating flat histogram techniques in the quantum Monte Carlo simulation, we can calculate the Matsubara Green's function for significantly larger values of the imaginary time. This allows us to obtain more accurate results for the spectral function, particularly for low energy excitations, using known techniques for analytic continuation to real time.
The Quantum Monte Carlo (QMC) method can yield the imaginary-time dependence of a correlation function C(τ)C(τ)C(τ) of an operator O^\hat OO^. The analytic continuation to real-time proceeds by means of a "numerical inversion" of these data to find the response function or spectral density A(ω)A(ω)A(ω) corresponding to O^\hat OO^. Such a technique is very sensitive to the statistical errors in C(τ)C(τ)C(τ) especially for large values of τττ, when we are interested in the low-energy excitations. In this paper, we find that if we use the flat histogram technique in the QMC method, in such a way to make the {\it histogram of} C(τ)C(τ)C(τ) flat, the results of the analytic continuation for low-energy excitations improve using the same amount of computational time. To demonstrate the idea we select an exactly soluble version of the single-hole motion in the t−Jt-Jt−J model and the diagrammatic Monte Carlo technique.
We discuss that flat-histogram techniques can be appropriately applied in the sampling of quantum Monte Carlo simulation in order to improve the statistical quality of the results at long imaginary time or low excitation energy. Typical imaginary-time correlation functions calculated in quantum Monte Carlo are subject to exponentially growing errors as the range of imaginary time grows and this smears the information on the low energy excitations. We show that we can extract the low energy physics by modifying the Monte Carlo sampling technique to one in which configurations which contribute to making the histogram of certain quantities flat are promoted. We apply the diagrammatic Monte Carlo (diag-MC) method to the motion of a single hole in the t-J model and we show that the implementation of flat-histogram techniques allows us to calculate the Green's function in a wide range of imaginary-time. In addition, we show that applying the flat-histogram technique alleviates the "sign"-problem associated with the simulation of the single-hole Green's function at long imaginary time.
The diagrammatic Monte Carlo (DiagMC) method is a numerical technique which samples the entire diagrammatic series of the Green's function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte Carlo method, and we term the improved version the "flat histogram diagrammatic Monte Carlo" method. We demonstrate the superiority of this method over the standard DiagMC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron problem.
Analytic continuation is a critical step in quantum many-body computations, connecting imaginary-time or Matsubara Green's functions with real-frequency spectral functions, which can be directly compared to experimental results. However, because of the ill-posed nature of the analytic continuation problems, they have not been completely solved so far. In this paper, we suggest a simple, yet highly efficient method for analytic continuations of Matsubara Green's functions. This method takes advantage of barycentric rational functions to directly interpolate Matsubara Green's functions. At first, the nodes and weights of the barycentric rational functions are determined by the adaptive Antoulas-Anderson algorithm, avoiding reliance on nonconvex optimization. Next, the retarded Green's functions and the related spectral functions are evaluated by the resulting interpolants. We systematically explore the performance of this method through a series of toy models and realistic examples, comparing its accuracy and efficiency with other popular methods, such as the maximum entropy method. The benchmark results demonstrate that the new method can accurately reproduce not only continuous but also discrete spectral functions, irrespective of their positive definiteness. It works well even in the presence of intermediate noise, and outperforms traditional analytic continuation methods in computational speed. We believe that this method should stand out for its robustness against noise, broad applicability, high precision, and ultra-efficiency, offering a promising alternative to the maximum entropy method.
We present two methods for computing two-time correlation functions or Green's functions from real-time bold-line continuous-time quantum Monte Carlo. One method is a formally exact generalized auxiliary lead formalism by which spectral properties may be obtained from single-time observables. The other involves the evaluation of diagrams contributing to two-time observables directly on the Keldysh contour. Additionally, we provide a detailed description of the bold-line Monte Carlo method. Our methods are general and numerically exact, and able to reliably resolve high-energy features such as band edges. We compare the spectral functions obtained from real-time methods to analytically continued spectral functions obtained from imaginary-time Monte Carlo, thus probing the limits of analytic continuation.