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The Journal of Chemical Physics

Dynamic path integral methods: A maximum entropy approach based on the combined use of real and imaginary time quantum Monte Carlo data

Dongsup Kim, J. D. Doll, David L. Freeman

Published Mar 8, 1998• 30 citations
Publisher

Abstract

A new numerical procedure for the study of finite temperature quantum dynamics is developed. The method is based on the observation that the real and imaginary time dynamical data contain complementary types of information. Maximum entropy methods, based on a combination of real and imaginary time input data, are used to calculate the spectral densities associated with real time correlation functions. Model studies demonstrate that the inclusion of even modest amounts of short-time real time data significantly improves the quality of the resulting spectral densities over that achievable using either real time data or imaginary time data separately.

DOI
10.1063/1.475790

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A new numerical procedure for the study of finite temperature quantum dynamics is developed. The method is based on the observation that the real and imaginary time dynamical data contain complementary types of information. Maximum entropy methods, based on a combination of real and imaginary time input data, are used to calculate the spectral densities associated with real time correlation functions. Model studies demonstrate that the inclusion of even modest amounts of short-time real time data significantly improves the quality of the resulting spectral densities over that achievable using either real time data or imaginary time data separately.
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We report the details of an application of the method of maximum entropy to the extraction of spectral and transport properties from the imaginary-time correlation functions generated from quantum Monte Carlo simulations of the nondegenerate, symmetric, single-impurity Anderson model. We find that these physical properties are approximately universal functions of temperature and frequency when these parameters are scaled by the Kondo temperature. We also found that important details for successful extractions included the generation of statistically independent, Gaussian-distributed data, and a good choice of a default model to represent the state of our prior knowledge about the result in the absence of data. We suggest that our techniques are not restricted to the Hamiltonian and quantum Monte Carlo algorithm used here, but that maximum entropy and these techniques lay the general groundwork for the extraction of dynamical information from imaginary-time data generated by other quantum Monte Carlo simulations.
An outstanding problem in the simulation of condensed matter phenomena is how to obtain dynamical information. We consider the numerical analytic continuation of imaginary time Quantum Monte Carlo data to obtain real frequency spectral functions. We suggest an image reconstruction approach which has been widely applied to data analysis in experimental research, the Maximum Entropy Method (MaxEnt). We report encouraging preliminary results for the Fano-Anderson model of an impurity state in a continuum. The incorporation of additional prior information, such as sum rules and asymptotic behavior, can be expected to significantly improve results. We also compare MaxEnt to alternative methods. 17 refs., 4 figs.
A major problem still confronting molecular simulations is how to determine time-correlation functions of many-body quantum systems. In this paper the results of the maximum entropy (ME) and singular value decomposition (SVD) analytic continuation methods for calculating real time quantum dynamics from path integral Monte Carlo calculations of imaginary time time-correlation functions are compared with analytical results for quantum mechanical vibrational relaxation processes. This system studied is an exactly solvable system: a harmonic oscillator bilinearly coupled to a harmonic bath. The ME and SVD methods are applied to exact imaginary-time correlation functions with various level of added random noise, and also to imaginary-time data from path integral Monte Carlo (PIMC) simulations. The information gathered in the present benchmark study is valuable for the application of the analytic continuation of PIMC data to complex systems.
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