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Physical Review Letters

Fermion Monte Carlo algorithms and liquid He3

Robert M. Panoff, Joseph Carlson

Published Mar 6, 1989• 121 citations
Publisher

Abstract

Variational Monte Carlo and several many-fermion Green's-function Monte Carlo (GFMC) algorithms are used to study the ground state of liquid $^{3}\mathrm{He}$. We report the first mirror potential GFMC calculations in a many-fermion problem, comparing them with transient estimation and fixed-node studies to illustrate the strengths and weaknesses of each. GFMC results with the Aziz HFDHE2 interaction are in good agreement with experiment, yielding energies within approximately 0.1 K per particle. In addition, each of these calculations predict a kinetic energy per particle of between 12 and 12.5 K.

DOI
10.1103/physrevlett.62.1130

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Variational Monte Carlo and several many-fermion Green's-function Monte Carlo (GFMC) algorithms are used to study the ground state of liquid 3He^{3}\mathrm{He}3He. We report the first mirror potential GFMC calculations in a many-fermion problem, comparing them with transient estimation and fixed-node studies to illustrate the strengths and weaknesses of each. GFMC results with the Aziz HFDHE2 interaction are in good agreement with experiment, yielding energies within approximately 0.1 K per particle. In addition, each of these calculations predict a kinetic energy per particle of between 12 and 12.5 K.
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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.
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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.
We have studied via a Green-function Monte Carlo (GFMC) method the S=(1/2 Heisenberg quantum antiferromagnet in two dimensions. We use a well-known transformation to map the spin problem onto a system of hard-core bosons that allows us to exploit interesting analogies between magnetism and superfluidity. The GFMC method is a zero-temperature stochastic method that projects out the component of the true ground state in a given variational wave function. This method is complementary to previously used finite-temperature Monte Carlo methods and is well suited to studying the ground state and low-lying excited states. Starting with even a simple wave function, e.g., the classical N\'eel state, the GFMC method can obtain the short-range correlations very accurately, and we find the ground-state energy per site E0{\mathit{E}}_{0}E0​/J=-0.6692(2). We show that it is important to include the zero-point motion of the elementary excitations in the ground state and by a spin-wave analysis find that it produces long-range correlations in the wave function. Upon inclusion of such long-range correlations, we obtain a staggered magnetization ${\mathit{m}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=0.31(2) and the structure factor scrS(q)\ensuremath{\sim}q at long wavelengths. Using the Feynman-Bijl relation, from the slope we deduce the renormalization of the spin-wave velocity by quantum fluctuations to be Zc{\mathit{Z}}_{\mathit{c}}Zc​==c/cs{\mathit{c}}_{\mathit{s}}cs​=1.14(5).