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. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Correlated pairs in model fermion Monte Carlo calculations

Malvin H. Kalos

Published May 1, 1996• 12 citations
Publisher

Abstract

The issues that prevent the development of efficient and stable algorithms for fermion Monte Carlo calculations in continuum systems are reexamined with special reference to the implications of the ``plus-minus'' symmetry. This is a property of many algorithms that use signed walkers, namely, that the dynamics are unchanged when the signs of the walkers are interchanged. Algorithms that obey this symmetry cannot exhibit the necessary stability. Specifically, estimates of the overlap with any antisymmetric test function cannot be bounded away from zero in the limit of many iterations. Within the framework of a diffusion Monte Carlo treatment of the Schr\"odinger equation, it is shown that this symmetry is easily broken for pairs of walkers while at the same time preserving the correct marginal dynamics for each member of the pair. The key is to create different classes of correlations between members of pairs and to use (at least) two distinct correlations for a pair and for the same pair with signs exchanged. The ideas are applied successfully for a class of simple model problems in two dimensions. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.5420

Related papers

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

Exact Monte Carlo Method for Continuum Fermion Systems

Malvin H. Kalos, Francesco Pederiva

We offer a new proposal for the Monte Carlo treatment of many-fermion systems in continuous space. It is based upon diffusion Monte Carlo with significant modifications: correlated pairs of random walkers that carry opposite signs, different functions "guide" walkers of different signs, the Gaussians used for members of a pair are correlated, and walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 3He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within the reach of contemporary computers.

arXiv:cond-mat/0001193•Physical Review Letters•Oct 23, 2000•39 citations

Cancellation time for correlated random walkers

F. Arias de Saavedra, Malvin H. Kalos, Francesco Pederiva

We have studied the dynamics of the correlated diffusion of pairs of random walkers of opposite signs. The use of populations of such pairs has been proposed for the Monte Carlo treatment of many-fermion systems, where the possibility of their cancellation might prevent the characteristic decay of the signal-to-noise ratio. For four model systems – free fermions, the harmonic oscillator, an N-body system of attractive and repulsive harmonic forces, and an extensive system interacting by Pöschl–Teller potentials – we have explored analytically and by computation the behavior of the time to cancellation as a function of initial conditions and, equally important, as a function of system size. We find that for these systems the computational efficiency does not decay either with large imaginary time or with large N.

Molecular Physics•Aug 4, 2011•1 citation

Model Fermion Monte Carlo with Antithetical Pairs

Zhiping Liu, Shiwei Zhang, Malvin H. Kalos

We consider certain model systems within which we seek solutions of the Schrodinger equation that are antisymmetric on inversion in the origin. We find geometrical constructions that define sequences of correlated random walks for which the overlap with the antisymmetrical test functions are asymptotically constant. The emphasis here is on problems in two dimensions, but the methods generalize to many dimensions and to certain other model problems.

•Feb 1, 1994•0 citations

Model fermion Monte Carlo method with antithetical pairs

Zhiping Liu, Shiwei Zhang, Malvin H. Kalos

We consider certain model systems within which we seek solutions of the Schr\"odinger equation that are antisymmetric on inversion in the origin. We find geometrical constructions that define sequences of correlated random walks for which the overlap with antisymmetrical test functions are asymptotically constant. The emphasis here is on problems in two dimensions, but the methods generalize to many dimensions and to certain other model problems.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1994•19 citations

The fermion Monte Carlo revisited

Roland Assaraf, Michel Caffarel, Anatole Khélif

In this work we present a detailed study of the fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact Fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose–Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett. 85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full ‘sign problem’. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.

arXiv:cond-mat/0611264•Journal of Physics A Mathematical and Theoretical•Jan 23, 2007•19 citations
The issues that prevent the development of efficient and stable algorithms for fermion Monte Carlo calculations in continuum systems are reexamined with special reference to the implications of the ``plus-minus'' symmetry. This is a property of many algorithms that use signed walkers, namely, that the dynamics are unchanged when the signs of the walkers are interchanged. Algorithms that obey this symmetry cannot exhibit the necessary stability. Specifically, estimates of the overlap with any antisymmetric test function cannot be bounded away from zero in the limit of many iterations. Within the framework of a diffusion Monte Carlo treatment of the Schr"odinger equation, it is shown that this symmetry is easily broken for pairs of walkers while at the same time preserving the correct marginal dynamics for each member of the pair. The key is to create different classes of correlations between members of pairs and to use (at least) two distinct correlations for a pair and for the same pair with signs exchanged. The ideas are applied successfully for a class of simple model problems in two dimensions. \textcopyright{} 1996 The American Physical Society.
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 offer a new proposal for the Monte Carlo treatment of many-fermion systems in continuous space. It is based upon diffusion Monte Carlo with significant modifications: correlated pairs of random walkers that carry opposite signs, different functions "guide" walkers of different signs, the Gaussians used for members of a pair are correlated, and walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 3He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within the reach of contemporary computers.
We have studied the dynamics of the correlated diffusion of pairs of random walkers of opposite signs. The use of populations of such pairs has been proposed for the Monte Carlo treatment of many-fermion systems, where the possibility of their cancellation might prevent the characteristic decay of the signal-to-noise ratio. For four model systems – free fermions, the harmonic oscillator, an N-body system of attractive and repulsive harmonic forces, and an extensive system interacting by Pöschl–Teller potentials – we have explored analytically and by computation the behavior of the time to cancellation as a function of initial conditions and, equally important, as a function of system size. We find that for these systems the computational efficiency does not decay either with large imaginary time or with large N.
In this work we present a detailed study of the fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact Fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose–Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett. 85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full ‘sign problem’. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.