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

SWITCHING BOUNDARY CONDITIONS IN THE MANY-BODY DIFFUSION ALGORITHM

F. Luczak, F. Brosens, J. T. Devreese, L. F. Lemmens

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

In this paper we show how the transposition, the basic operation of the permutation group, can be taken into account in a diffusion process of identical particles. Whereas in an earlier approach the method was applied to systems in which the potential is invariant under interchanging the Cartesian components of the particle coordinates, this condition on the potential is avoided here. In general, the potential introduces a switching of the boundary conditions of the walkers. These transitions modelled by a continuous-time Markov chain generate sample paths for the propagator as a Feynman–Kac functional. A few examples, including harmonic fermions with an anharmonic interaction, and the ground-state energy of ortho-helium are studied to elucidate the theoretical discussion and to illustrate the feasibility of a sign-problem-free implementation scheme for the recently developed many-body diffusion approach.

Related papers

SWITCHING BOUNDARY CONDITIONS IN THE MANY-BODY DIFFUSION ALGORITHM

F. LUCZAK, F. BROSENS, J. T. DEVREESE +1 more

In this paper we show how the transposition, the basic operation of the permutation group, can be taken into account in a diffusion process of identical particles. Whereas in an earlier approach the method was applied to systems in which the potential is invariant under interchanging the Cartesian components of the particle coordinates, this condition on the potential is avoided here. In general, the potential introduces a switching of the boundary conditions of the walkers. These transitions modelled by a continuous-time Markov chain generate sample paths for the propagator as a Feynman–Kac functional. A few examples, including harmonic fermions with an anharmonic interaction, and the ground-state energy of ortho-helium are studied to elucidate the theoretical discussion and to illustrate the feasibility of a sign-problem-free implementation scheme for the recently developed many-body diffusion approach.

arXiv:cond-mat/9804131•International Journal of Modern Physics C•Feb 1, 1999•1 citation

Many-body diffusion algorithm: Harmonic fermions

F. Luczak, F. Brosens, J. T. Devreese +1 more

A numerical implementation scheme is presented for the recently developed many-body diffusion approach for identical particles, in the case of harmonic potentials. The procedure is free of the sign problem, by the introduction of the appropriate absorption or reflection conditions for the walkers at the boundary of a state space. These conditions are imposed by the permutation symmetry. The outflow of the walkers at the boundary of the state space contributes substantially to the energy. Furthermore, the implementation of crossing-recrossing effects at absorbing boundaries proves indispensable to sample the antisymmetric states by discrete time steps.

arXiv:cond-mat/9711032•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 1, 1998•6 citations

Many-body diffusion and path integrals for identical particles

L. F. Lemmens, F. Brosens, J. T. Devreese

For distinguishable particles it is well known that Brownian motion and a Feynman-Kac functional can be used to calculate the path integral (for imaginary times) for a general class of scalar potentials. In order to treat identical particles, we exploit the fact that this method separates the problem of the potential, dealt with by the Feynman-Kac functional, from the process which gives sample paths of a noninteracting system. For motion in one dimension, we emphasize that the permutation symmetry of the identical particles completely determines the domain of Brownian motion and the appropriate boundary conditions: absorption for fermions, reflection for bosons. Further analysis of the sample paths for motion in three dimensions allows us to decompose these paths into a superposition of one-dimensional sample paths. This reduction expresses the propagator (and consequently the energy and other thermodynamical quantities) in terms of well-behaved one-dimensional fermion and boson diffusion processes and the Feynman-Kac functional.

arXiv:cond-mat/9601079•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 1, 1996•22 citations

Fermion Monte Carlo

Francesco Pederiva, Malvin H. Kalos

We review the fundamental challenge of fermion Monte Carlo for continuous systems, the “sign problem”. We seek that eigenfunction of the many-body Schrödinger equation that is antisymmetric under interchange of the coordinates of pairs of particles. We describe methods that depend upon the use of correlated dynamics for pairs of correlated walkers that carry opposite signs. There is an algorithmic symmetry between such walkers that must be broken to create a method that is both exact and as effective as for symmetric functions. In our new method, it is broken by using different “guiding” functions for walkers of opposite signs, and a geometric correlation between steps of their walks. With a specific process of cancellation of the walkers, overlaps with antisymmetric test functions are preserved. Finally, we describe the progress in treating free-fermion systems and a fermion fluid with 14 3He atoms.

Computer Physics Communications•Sep 1, 1999•0 citations

A FOURTH ORDER DIFFUSION MONTE CARLO ALGORITHM FOR SOLVING QUANTUM MANY-BODY PROBLEMS

Harald Forbert, Siu A. Chin

We derive a fourth-order diffusion Monte Carlo algorithm for solving quantum many-body problems. The method uses a factorization of the imaginary time propagator in terms of the usual local energy E and Langevin operators L as well as an additional pseudo-potential consisting of the double commutator [EL, [L, EL]]. A new factorization of the propagator of the Fokker-Planck equation enables us to implement the Langevin algorithm to the necessary fourth order. We achieve this by the addition of correction terms to the drift steps and the use of a position-dependent Gaussian random walk. We show that in the case of bulk liquid helium the systematic step size errors are indeed fourth order over a wide range of step sizes.

arXiv:nucl-th/0009068•International Journal of Modern Physics B•May 10, 2001•8 citations

Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. I. Formalism

Michel Caffarel, P. Claverie

This paper presents systematic developments in the previously initiated line of research concerning a quantum Monte Carlo (QMC) method based on the use of a pure diffusion process corresponding to some reference function and a generalized Feynman–Kac path integral formalism. Not only mean values of quantum observables, but also response properties are expressed using suitable path integrals involving the diffusion measure of the reference diffusion process. Moreover, by relying on the ergodic character of this process, path integrals may be evaluated as time-averages along any sample trajectory of the process. This property is of crucial importance for the computer implementation of the method. As concerns the treatment of many-fermion systems, where the Pauli principle must be taken into account, we can use the fixed-node approximation, but we also discuss the potentially exact release-node procedure, whereby some adequate symmetry is imposed on the integrand (of the generalized Feynman–Kac formula), associated with a possibly refined choice of the reference function.

The Journal of Chemical Physics•Jan 15, 1988•105 citations
In this paper we show how the transposition, the basic operation of the permutation group, can be taken into account in a diffusion process of identical particles. Whereas in an earlier approach the method was applied to systems in which the potential is invariant under interchanging the Cartesian components of the particle coordinates, this condition on the potential is avoided here. In general, the potential introduces a switching of the boundary conditions of the walkers. These transitions modelled by a continuous-time Markov chain generate sample paths for the propagator as a Feynman–Kac functional. A few examples, including harmonic fermions with an anharmonic interaction, and the ground-state energy of ortho-helium are studied to elucidate the theoretical discussion and to illustrate the feasibility of a sign-problem-free implementation scheme for the recently developed many-body diffusion approach.
In this paper we show how the transposition, the basic operation of the permutation group, can be taken into account in a diffusion process of identical particles. Whereas in an earlier approach the method was applied to systems in which the potential is invariant under interchanging the Cartesian components of the particle coordinates, this condition on the potential is avoided here. In general, the potential introduces a switching of the boundary conditions of the walkers. These transitions modelled by a continuous-time Markov chain generate sample paths for the propagator as a Feynman–Kac functional. A few examples, including harmonic fermions with an anharmonic interaction, and the ground-state energy of ortho-helium are studied to elucidate the theoretical discussion and to illustrate the feasibility of a sign-problem-free implementation scheme for the recently developed many-body diffusion approach.
A numerical implementation scheme is presented for the recently developed many-body diffusion approach for identical particles, in the case of harmonic potentials. The procedure is free of the sign problem, by the introduction of the appropriate absorption or reflection conditions for the walkers at the boundary of a state space. These conditions are imposed by the permutation symmetry. The outflow of the walkers at the boundary of the state space contributes substantially to the energy. Furthermore, the implementation of crossing-recrossing effects at absorbing boundaries proves indispensable to sample the antisymmetric states by discrete time steps.
For distinguishable particles it is well known that Brownian motion and a Feynman-Kac functional can be used to calculate the path integral (for imaginary times) for a general class of scalar potentials. In order to treat identical particles, we exploit the fact that this method separates the problem of the potential, dealt with by the Feynman-Kac functional, from the process which gives sample paths of a noninteracting system. For motion in one dimension, we emphasize that the permutation symmetry of the identical particles completely determines the domain of Brownian motion and the appropriate boundary conditions: absorption for fermions, reflection for bosons. Further analysis of the sample paths for motion in three dimensions allows us to decompose these paths into a superposition of one-dimensional sample paths. This reduction expresses the propagator (and consequently the energy and other thermodynamical quantities) in terms of well-behaved one-dimensional fermion and boson diffusion processes and the Feynman-Kac functional.
We review the fundamental challenge of fermion Monte Carlo for continuous systems, the “sign problem”. We seek that eigenfunction of the many-body Schrödinger equation that is antisymmetric under interchange of the coordinates of pairs of particles. We describe methods that depend upon the use of correlated dynamics for pairs of correlated walkers that carry opposite signs. There is an algorithmic symmetry between such walkers that must be broken to create a method that is both exact and as effective as for symmetric functions. In our new method, it is broken by using different “guiding” functions for walkers of opposite signs, and a geometric correlation between steps of their walks. With a specific process of cancellation of the walkers, overlaps with antisymmetric test functions are preserved. Finally, we describe the progress in treating free-fermion systems and a fermion fluid with 14 3He atoms.
We derive a fourth-order diffusion Monte Carlo algorithm for solving quantum many-body problems. The method uses a factorization of the imaginary time propagator in terms of the usual local energy E and Langevin operators L as well as an additional pseudo-potential consisting of the double commutator [EL, [L, EL]]. A new factorization of the propagator of the Fokker-Planck equation enables us to implement the Langevin algorithm to the necessary fourth order. We achieve this by the addition of correction terms to the drift steps and the use of a position-dependent Gaussian random walk. We show that in the case of bulk liquid helium the systematic step size errors are indeed fourth order over a wide range of step sizes.
This paper presents systematic developments in the previously initiated line of research concerning a quantum Monte Carlo (QMC) method based on the use of a pure diffusion process corresponding to some reference function and a generalized Feynman–Kac path integral formalism. Not only mean values of quantum observables, but also response properties are expressed using suitable path integrals involving the diffusion measure of the reference diffusion process. Moreover, by relying on the ergodic character of this process, path integrals may be evaluated as time-averages along any sample trajectory of the process. This property is of crucial importance for the computer implementation of the method. As concerns the treatment of many-fermion systems, where the Pauli principle must be taken into account, we can use the fixed-node approximation, but we also discuss the potentially exact release-node procedure, whereby some adequate symmetry is imposed on the integrand (of the generalized Feynman–Kac formula), associated with a possibly refined choice of the reference function.