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Physical review. B, Condensed matter

Extrapolation to infinite Trotter number in path-integral Monte Carlo simulations of solid-state systems

Alessandro Cuccoli, Alessandro Macchi, Gaia Pedrolli, V. Tognetti, Ruggero Vaia

Published May 1, 1995• 14 citations
Publisher

Abstract

We consider the problem of the extrapolation of path-integral Monte Carlo (PIMC) data to infinite Trotter number P. Finite-P data, being even functions of P, have high-P dependence that is generally well described by a quadratic fit, ${\mathit{a}}_{0}$+${\mathit{a}}_{1}$${\mathit{P}}^{\mathrm{\ensuremath{-}}2}$, where ${\mathit{a}}_{0}$ is the exact quantum value. However, in order to get convergence it is often necessary to run PIMC codes with rather high P values, which implies long computer times and larger statistical errors of the data. It is well known that also for harmonic systems the finite-P data are not exact; nevertheless, they can be easily calculated by Gaussian quadrature. Starting from this observation, we suggest an easy way to correct PIMC data for anharmonic systems in order to take into account the harmonic part exactly, with strong improvement of the extrapolation to P=\ensuremath{\infty}. Lower Trotter numbers are thus required, with the advantages of computer-time saving and much better accuracy of the extrapolated values, without any change in the PIMC code. In order to demonstrate the effectiveness of the approach, we report finite-P data processing for a single anharmonic particle, whose finite-P data are obtained by the matrix-squaring method, and for a chain of atoms with Morse interaction.

DOI
10.1103/physrevb.51.12369

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We consider the problem of the extrapolation of path-integral Monte Carlo (PIMC) data to infinite Trotter number P. Finite-P data, being even functions of P, have high-P dependence that is generally well described by a quadratic fit, a0{\mathit{a}}_{0}a0​+{\mathit{a}}_{1}$${\mathit{P}}^{\mathrm{\ensuremath{-}}2}, where a0{\mathit{a}}_{0}a0​ is the exact quantum value. However, in order to get convergence it is often necessary to run PIMC codes with rather high P values, which implies long computer times and larger statistical errors of the data. It is well known that also for harmonic systems the finite-P data are not exact; nevertheless, they can be easily calculated by Gaussian quadrature. Starting from this observation, we suggest an easy way to correct PIMC data for anharmonic systems in order to take into account the harmonic part exactly, with strong improvement of the extrapolation to P=\ensuremath{\infty}. Lower Trotter numbers are thus required, with the advantages of computer-time saving and much better accuracy of the extrapolated values, without any change in the PIMC code. In order to demonstrate the effectiveness of the approach, we report finite-P data processing for a single anharmonic particle, whose finite-P data are obtained by the matrix-squaring method, and for a chain of atoms with Morse interaction.
Improvements beyond the primitive approximation in the path integral Monte Carlo method are explored both in a model problem and in real systems. Two different strategies are studied: The Richardson extrapolation on top of the path integral Monte Carlo data and the Takahashi-Imada action. The Richardson extrapolation, mainly combined with the primitive action, always reduces the number-of-beads dependence, helps in determining the approach to the dominant power law behavior, and all without additional computational cost. The Takahashi-Imada action has been tested in two hard-core interacting quantum liquids at low temperature. The results obtained show that the fourth-order behavior near the asymptote is conserved, and that the use of this improved action reduces the computing time with respect to the primitive approximation.
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.
We address the possibility of performing numerical Monte Carlo simulations for the thermodynamics of quantum dissipative systems. Dissipation is considered within the Caldeira-Leggett formulation, which describes the system in the path-integral formalism through the inclusion of an influence action that is bilocal and quadratic in the system's coordinates. At a first sight the usual direct approach of discretizing the path integral could seem feasible, but complications arise when one tries to introduce a physically meaningful dissipation kernel: in particular its imaginary-time dependence turns out to be severely singular and difficult to evaluate analytically, in spite of the simple expressions for its Matsubara components. We therefore propose to face the numerical problem using Fourier path-integral Monte Carlo, that can be formulated in two different ways: transforming the continuous paths and then truncating the high Fourier components (with possible improvements upon the truncation procedure), or performing the Fourier transformation upon the discretized paths. The latter choice leads to a simpler formulation and allows for a better control of the extrapolation to the limit of infinite Trotter number. The method is implemented for a single nonlinear particle with Ohmic dissipation and for a phi(4) chain with Drude-like dissipation.
We propose a new method to calculate ground state position dependent observables in quantum many-body systems. The method, which we call the path-integral diffusion Monte Carlo (PI-DMC) method, is essentially a combination of path-integral Monte Carlo (PIMC) and diffusion Monte Carlo (DMC) methods. The distribution resulting from a DMC simulation is further propagated in imaginary time by PIMC sampling. Tests of the new method for simple cases such as the harmonic oscillator, a double well, and a set of ten coupled harmonic oscillators show that the results for several observables converge rapidly to the exact result.
The ab initio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion sign problem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the sign problem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the sign problem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).
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.