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arXiv:cond-mat/9711032 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Many-body diffusion algorithm: Harmonic fermions

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

Published Feb 1, 1998• 6 citations
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Abstract

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.

DOI
10.1103/physreve.57.2411

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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.
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.
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.
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.