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Pauli-Potential and Green Function Monte-Carlo Method for Many-Fermion Systems

B. L. G. Bakker, M. I. Polikarpov, A. I. Veselov

Published Jan 1, 1995• 1 citation
OpenAlex

Abstract

The time evolution of a many-fermion system can be described by a Green's function corresponding to an effective potential, which takes anti-symmetrization of the wave function into account, called the Pauli-potential. We show that this idea can be combined with the Green's Function Monte Carlo method to accurately simulate a system of many non-relativistic fermions. The method is illustrated by the example of systems of several (2-9) fermions in a square well.

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The time evolution of a many-fermion system can be described by a Green's function corresponding to an effective potential, which takes anti-symmetrization of the wave function into account, called the Pauli-potential. We show that this idea can be combined with the Green's Function Monte Carlo method to accurately simulate a system of many non-relativistic fermions. The method is illustrated by the example of systems of several (2-9) fermions in a square well.
The exact treatment of fermion systems by Monte Carlo methods has proved to be difficult. We present a new method based on the concept of a ``mirror potential,'' which is a many-body potential that forces the Monte Carlo iteration to have a stable antisymmetric component. The potential may be determined from the wave function and, within the framework of Green's function Monte Carlo, from the random walk whose density converges to the wave function. In certain limits, the method reduces to the fixed node approximation and to transient estimation, so that it subsumes both of them. As a further consequence it offers an approximation analogous to fixed node for treating systems with noncentral forces. The method may prove to be a general one for treating random walkers with nonpositive or complex weights. In support of that we exhibit a successful calculation of a one-dimensional excited state. In this paper we explore the alternative in which the mirror potential is obtained from trial wave functions. This yields an approximation scheme that proves to be accurate in the experiments described here. We present results for a model problem in which four neutrons interact by a spin-independent potential, and compare the results with those of the fixed node method, and the method of transient estimation.
A novel scheme to solve the quantum eigenvalue problem through the imaginary-time Green function Monte Carlo method is presented. This method is applicable not only to the ground states, but also to the excited states of generic system. We demonstrate the actual utility of the method with the numerical examples on three simple systems. Typeset using REVTEX 1 The stochastic quantization has been successfully applied in recent years to the various problems of physics ranging from quantum chromo dynamics to the plasmon oscillation [1-3]. While it can handle very complex systems which is usually beyond the approach either by the direct diagonalization or by the perturbation, it has been only applicable to the calculation of the ground state properties of the system. In this paper, we propose a new method based on the imaginary-time Green function Monte Carlo approach, which is extendable to the excited states of Hamiltonian system. We have primarily in mind the non-relativistic quantum few body problem such as encountered in nuclear physics [4-6].
On the base of the diffusion Monte-Carlo method we develop the method allowing to simulate the quantum systems with complex wave function. The method is exact and there are no approximations on the simulations of the module and the phase of the system's wave function. In our method averaged value of any quantity have no direct contribution from the phase of distribution function but only from the phase of the Green function of diffusion equation. We test the method by the simulations of the ground state of fermions in two-dimensional parabolic well. Anyons are used for the representation of the two-dimensional (2D) fermions. We vary the number of fermions from two to ten and find a good agreement of the numerical results with analytical ones for the numbers of the particles N>4.
Microscopic models of classical degrees of freedom coupled to noninteracting fermions occur in many different contexts. Prominent examples from solid state physics are descriptions of colossal magnetoresistance manganites and diluted magnetic semiconductors, or auxiliary field methods for correlated electron systems. Monte Carlo simulations are vital for an understanding of such systems, but notorious for requiring the solution of the fermion problem with each change in the classical field configuration. We present an efficient, truncation-free O(N) method on the basis of Chebyshev expanded local Green functions, which allows us to simulate systems of unprecedented size N.
The Green’s function Monte Carlo method used for obtaining exact solutions to the Schrödinger equation of boson systems is generalized to treat systems of several fermions. We show that when it is possible to select eigenfunctions of the Hamiltonian based on physical symmetries, the GFMC method can be used to yield the lowest energy state of that symmetry. In particular, the lowest totally antisymmetric eigenfunction, the fermion ground state, can be obtained. Calculations on several two- and three-body model problems show the method to be computationally feasible for few-body systems.
Enforcing the Pauli principle in many-body systems of fermions to ensure an antisymmetric wave function is typically a numerically expensive task. Numerical methods, such as coupled-cluster, correlated basis function, Monte Carlo, and traditional configuration interaction, scale with powers of NNN that make the determination of the energy spectrum of these systems a challenge. In a recent paper, we successfully obtained energies for large systems of fermions in the unitary regime in which we applied the Pauli principle ``on paper'' without obtaining an explicit wave function. This approach, an extension of the symmetry invariant perturbation method to fermions, applies the Pauli principle by imposing restrictions on the quantum numbers used to obtain the energy. As a followup to this study, we perform an explicit test of the validity of this application of the Pauli principle by comparing our energy results against the energies corresponding to explicitly antisymmetrized wave functions for an exactly solvable system of harmonically confined, harmonically interacting fermions. Our results show that our simple method of applying the Pauli principle selects from a spectrum of possible many-body energies only those energies that correspond to explicitly antisymmetrized wave functions. Our results were tested for values of NNN through $\ensuremath{\approx}10 000$ and for weak and strong interactions both repulsive and attractive.