Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Fourier path-integral Monte Carlo methods: Partial averaging

Jimmie D. Doll, Rob D. Coalson, David L. Freeman

Published Jul 1, 1985• 144 citations
Publisher

Abstract

Monte Carlo Fourier path-integral techniques are explored. It is shown that fluctuation renormalization techniques provide an effective means for treating the effects of high-order Fourier contributions. The resulting formalism is rapidly convergent, is computationally convenient, and has potentially useful variational aspects.
DOI
10.1103/physrevlett.55.1

Related papers

Hamiltonians, path integrals, and a new renormalization group

Marvin Weinstein

New nonperturbative methods for dealing with Hamiltonian systems are introduced. The derivation of these methods requires identities for rewriting exponentials of sums of operators which are different from the usual Campbell-Hausdorff formula. These identities allow one to derive approximations to ${e}^{\ensuremath{-}\ensuremath{\delta}H}$ which are correct to higher order in $\ensuremath{\delta}$ and which contain fewer terms than the Campbell-Hausdorff formula. This allows one to generate path-integral actions which are more accurate for finite-size steps in time and which can be exploited to improve the rate of convergence of Monte Carlo calculations. To show that these methods allow one to include effects which show up in the stationary-phase approximation to the path integral, e.g., instantons, solitons, etc., I not only derive the connection between the Hamiltonian and path-integral formalism but the relationship between a specific stationary-phase approximation and the corresponding Hamiltonian calculation. My focus, however, is the direct application of these new identities to the study of nonperturbative Hamiltonian dynamics. I show that these methods are easier to apply and give better results than those based upon the naive $t$-expansion, block mean-field, or real-space renormalization ideas. Comparison of these older methods with the computational tools introduced in this paper are discussed in the context of simple examples. It is shown that the new methods allow one to extrapolate answers to finite $t$ without the use of Pad\'e approximants.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jun 15, 1993•11 citations

Fourier path integrals, partial averaging, and the polaron problem

Constantia Alexandrou, Wolfgang Fleischer, R. Rosenfelder

We evaluate the Fourier path integral for an optical polaron by the method of partial averaging. Higher Fourier coefficients are taken into account by averaging with Feynman's trial action, whereas the first N coefficients are treated explicitly by numerical integration. This method removes the Coulomb singularity in the effective action, yields lower bounds to the partition function, and systematically improves Feynman's variational results. We present numerical results for the exact ground-state energy for small and intermediate values of the coupling constant.

Physical Review Letters•Nov 19, 1990•28 citations

Variational Monte Carlo

Tao Pang

Monte Carlo methods have been very prominent in computer simulation of various systems in physics, chemistry, biology, and materials science. This book focuses on the discussion and path-integral quantum Monte Carlo methods in many-body physics and provides a concise but complete introduction to the Metropolis algorithm and its applications in these two techniques. To explore the schemes in clarity, several quantum many-body systems are analysed and studied in detail. The book includes exercises to help digest the materials covered. It can be used as a tutorial to learn the discussion and path-integral Monte Carlo or a recipe for developing new research in the reader's own area. Two complete Java programs, one for the discussion Monte Carlo of 4He clusters on a graphite surface and the other for the path-integral Monte Carlo of cold atoms in a potential trap, are ready for download and adoption.

Morgan & Claypool Publishers eBooks•Dec 1, 2016•4 citations

Approximate Functional Integral Methods in Statistical Mechanics. I. Moment Expansions

Armand Siegel, Terence Burke

In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.

Journal of Mathematical Physics•Nov 1, 1972•13 citations

Quantum Monte Carlo Method

Tarō Kashiwa, Yoshio Ohnuki, Masuo Suzuki

Abstract The path integral discussed in the preceding chapters can be exactly calculated only in a few restricted systems such as harmonic oscillators. The WKB method and perturbational expansions based on the above path integral method are in fact applicable to systems close to classical ones and weakly interacting systems, respectively. However, in order to study interesting phenomena such as high Tc superconductivity, we have to treat strongly correlated systems in which the quantum effect plays an important role. It is effective in these systems to extract some physically crucial contributions numerically. This will be possible if we use the quantum Monte Carlo method. Although there are many different types of quantum Monte Carlo methods, we explain here mainly the quantum statistical Monte Carlo method based on a Trotter-like formula and on the important sampling method applied to transformed classical systems.

•Jul 3, 1997•1 citation

Fast converging path integrals for time-dependent potentials: I. Recursive calculation of short-time expansion of the propagator

Antun Balaž, Ivana Vidanović, Aleksandar Bogojević +2 more

In this and the subsequent paper (Balaž et al 2011 J. Stat. Mech. P03005) we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte Carlo simulations for path integrals, or to set up various analytic approximation techniques to study properties of quantum systems in time-dependent potentials. The analytically derived results are numerically verified by treating several simple models.

arXiv:0912.2743•Journal of Statistical Mechanics Theory and Experiment•Mar 4, 2011•9 citations
New nonperturbative methods for dealing with Hamiltonian systems are introduced. The derivation of these methods requires identities for rewriting exponentials of sums of operators which are different from the usual Campbell-Hausdorff formula. These identities allow one to derive approximations to ${e}^{\ensuremath{-}\ensuremath{\delta}H}$ which are correct to higher order in $\ensuremath{\delta}$ and which contain fewer terms than the Campbell-Hausdorff formula. This allows one to generate path-integral actions which are more accurate for finite-size steps in time and which can be exploited to improve the rate of convergence of Monte Carlo calculations. To show that these methods allow one to include effects which show up in the stationary-phase approximation to the path integral, e.g., instantons, solitons, etc., I not only derive the connection between the Hamiltonian and path-integral formalism but the relationship between a specific stationary-phase approximation and the corresponding Hamiltonian calculation. My focus, however, is the direct application of these new identities to the study of nonperturbative Hamiltonian dynamics. I show that these methods are easier to apply and give better results than those based upon the naive ttt-expansion, block mean-field, or real-space renormalization ideas. Comparison of these older methods with the computational tools introduced in this paper are discussed in the context of simple examples. It is shown that the new methods allow one to extrapolate answers to finite ttt without the use of Pad\'e approximants.
We evaluate the Fourier path integral for an optical polaron by the method of partial averaging. Higher Fourier coefficients are taken into account by averaging with Feynman's trial action, whereas the first N coefficients are treated explicitly by numerical integration. This method removes the Coulomb singularity in the effective action, yields lower bounds to the partition function, and systematically improves Feynman's variational results. We present numerical results for the exact ground-state energy for small and intermediate values of the coupling constant.
Monte Carlo methods have been very prominent in computer simulation of various systems in physics, chemistry, biology, and materials science. This book focuses on the discussion and path-integral quantum Monte Carlo methods in many-body physics and provides a concise but complete introduction to the Metropolis algorithm and its applications in these two techniques. To explore the schemes in clarity, several quantum many-body systems are analysed and studied in detail. The book includes exercises to help digest the materials covered. It can be used as a tutorial to learn the discussion and path-integral Monte Carlo or a recipe for developing new research in the reader's own area. Two complete Java programs, one for the discussion Monte Carlo of 4He clusters on a graphite surface and the other for the path-integral Monte Carlo of cold atoms in a potential trap, are ready for download and adoption.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.
Abstract The path integral discussed in the preceding chapters can be exactly calculated only in a few restricted systems such as harmonic oscillators. The WKB method and perturbational expansions based on the above path integral method are in fact applicable to systems close to classical ones and weakly interacting systems, respectively. However, in order to study interesting phenomena such as high Tc superconductivity, we have to treat strongly correlated systems in which the quantum effect plays an important role. It is effective in these systems to extract some physically crucial contributions numerically. This will be possible if we use the quantum Monte Carlo method. Although there are many different types of quantum Monte Carlo methods, we explain here mainly the quantum statistical Monte Carlo method based on a Trotter-like formula and on the important sampling method applied to transformed classical systems.
In this and the subsequent paper (Balaž et al 2011 J. Stat. Mech. P03005) we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte Carlo simulations for path integrals, or to set up various analytic approximation techniques to study properties of quantum systems in time-dependent potentials. The analytically derived results are numerically verified by treating several simple models.