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.
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.
We propose a method which uses centroid molecular dynamics (CMD) [J. Cao and G. A. Voth, J. Chem. Phys. 100, 5106 (1994)] real-time data in conjunction with the imaginary-time data generated using path integral Monte Carlo simulations in a numerical analytic continuation scheme based on the maximum entropy approach. We show that significant improvement is achieved by including short-time CMD data with the imaginary-time data. In particular, for a particle bilinearly coupled to a harmonic bath, these methods lead to significant improvements over previous calculations and even allow accurate determination of transport coefficients such as the diffusion coefficient and mobility for this system. In addition we show how maximum entropy method can be used to extract accurate dynamic information from short-time CMD data, and that this approach is superior to the direct Fourier transform of long-time data for systems characterized by broad, featureless spectral distributions.
We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.
The quantum harmonic oscillator represents the most common fundamental building block to compute thermal properties of virtually any dielectric crystalline material at low temperatures in terms of phonons, extended further to cases with anharmonic couplings, or even disordered solids. In general, Path Integral Monte Carlo (PIMC) methods provide a powerful numerical tool to stochastically determine thermodynamic quantities without systematic bias, not relying on perturbative schemes. Addressing transport properties, for instance the calculation of thermal conductivity from PIMC, however, has turned out to be substantially more difficult. Although correlation functions of current operators can be determined by PIMC in the form of analytic continuation on the imaginary time axis, Bayesian methods are usually employed for the numerical inversion back to real-time response functions. This task not only relies strongly on the accuracy of the PIMC data, but also introduces noticeable dependence on the underlying model used for the inversion problem. Here, we address both difficulties with great care. In particular, we first devise improved estimators for current correlations which substantially reduce the variance of the PIMC data. Next, we provide a neat statistical approach to the inversion problem, blending into a fresh workflow the classical stochastic maximum entropy method together with more recent notions borrowed from statistical learning theory. We test our ideas on a single harmonic oscillator and a collection of oscillators with a continuous distribution of frequencies, establishing solid grounds for an unbiased, fully quantum mechanical calculation of transport properties in solids.
The quantum harmonic oscillator is the fundamental building block to compute thermal properties of virtually any dielectric crystal at low temperatures in terms of phonons, extended further to cases with anharmonic couplings, or even disordered solids. In general, Path Integral Monte Carlo or Path Integral Molecular Dynamics methods are powerful tools to determine stochastically thermodynamic quantities without systematic bias, not relying on perturbative schemes. Addressing transport properties, for instance calculating thermal conductivity from PIMC, however, is substantially more difficult. Although correlation functions of current operators can be determined by PIMC from analytic continuation on the imaginary time axis, Bayesian methods are usually employed for the numerical inversion back to real time response functions. This task not only strongly relies on the accuracy of the PIMC data but also introduces noticeable dependence on the model used for the inversion. Here, we address both difficulties with care. In particular, we first devise improved estimators for current correlations, which substantially reduce the variance of the PIMC data. Next, we provide a neat statistical approach to the inversion problem, blending into a fresh workflow the classical stochastic maximum entropy method together with recent notions borrowed from statistical learning theory. We test our ideas on a single harmonic oscillator and a collection of oscillators with a continuous distribution of frequencies and provide indications of the performance of our method in the case of a particle in a double well potential. This work establishes solid grounds for an unbiased, fully quantum mechanical calculation of transport properties in solids.
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. This is an extremely ill-posed problem similar to the inversion of a Laplace transform. We suggest an image-reconstruction approach, which has been widely applied to data analysis in experimental research. Specifically, we apply the maximum-entropy method (ME) to the analytic continuation of quantum Monte Carlo data. 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 compare (ME) to alternative methods. We also discuss statistical error propagation for the analytic continuation problem via the likelihood function, which is independent of the choice of image-reconstruction method. This includes the sensitivity of the data to structure in the spectral function, the optimization of Monte Carlo simulations, and how to incorporate covariance in the statistical errors of the Monte Carlo method.
We consider the problem of calculating the electronic absorption spectrum of a chromophore with intramolecular degrees of freedom coupled to a condensed phase environment. We approach this calculation in the framework of the imaginary-time path integral Monte Carlo techniques, and focus on the problem of the analytic continuation of the imaginary-time data to the real-time axis. Two alternative analytic continuation methods are considered: the maximum entropy method and the singular value decomposition method. An exactly solvable model is introduced to test the accuracy of these methods. Exact numerical results for the absorption spectra are compared to the spectra reconstructed by the analytic continuation methods; it is found that the singular value decomposition method gives systematically higher resolution than the maximum entropy method and is capable of reproducing the fine vibronic structure of the absorption spectrum.