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Physical Review E

Remedy for the fermion sign problem in the diffusion Monte Carlo method for few fermions with antisymmetric diffusion process

Yuriy Mishchenko

Published Feb 22, 2006• 4 citations
Publisher

Abstract

We suggest an exact approach to help remedy the fermion sign problem in diffusion quantum Monte Carlo simulations. The approach is based on an explicit suppression of symmetric modes in the Schrödinger equation by means of a modified stochastic diffusion process (antisymmetric diffusion process). We introduce this algorithm and illustrate it on potential models in one dimension (1D) and show that there it solves the fermion sign problem exactly and converges to the lowest antisymmetric state of the system. Then, we discuss extensions of this approach to many-dimensional systems on examples of quantum oscillator in 2D-20D and a toy model of three and four fermions on harmonic strings in 2D and 3D. We show that in all these cases our method shows a performance comparable to that of a fixed-node approximation with an exact node.

DOI
10.1103/physreve.73.026706

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We suggest an exact approach to help remedy the fermion sign problem in diffusion quantum Monte Carlo simulations. The approach is based on an explicit suppression of symmetric modes in the Schrödinger equation by means of a modified stochastic diffusion process (antisymmetric diffusion process). We introduce this algorithm and illustrate it on potential models in one dimension (1D) and show that there it solves the fermion sign problem exactly and converges to the lowest antisymmetric state of the system. Then, we discuss extensions of this approach to many-dimensional systems on examples of quantum oscillator in 2D-20D and a toy model of three and four fermions on harmonic strings in 2D and 3D. We show that in all these cases our method shows a performance comparable to that of a fixed-node approximation with an exact node.
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015E_{h} of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10^{-5}E_{h} without an importance sampling or Jastrow factor.
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.
An approach is presented to compute properties of excited states in path integral Monte Carlo simulations of quantum systems. The approach is based on the introduction of several images of the studied system which have the total wavefunction antisymmetric over permutations of these images, and a simulation of the whole system at low enough temperature. The success of the approach relies, however, on the solution of the sign problem for the corresponding system. In the cases when the sign problem may be resolved (as for fermions in one dimension), properties of excited states can be computed with high precision. This was demonstrated by a simulation of excited states of a single particle in one- and two-dimensional harmonic oscillators. An example of the simulation of excited states of several interacting electrons in one dimension is also given.
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.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.