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Physical Review C

Mirror potentials and the fermion problem

Joseph Carlson, Malvin H. Kalos

Published Nov 1, 1985• 25 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.32.1735

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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.
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.
Abstract A method for treating the nodes for small systems was described in this paper. The general procedure involves the use of both positive and negative walkers in Green’s function QMC, along with cancellation of walkers of opposite sign in close proximity. The method is illustrated for the first excited state of a particle in a one-dimensional box and for three model problems with three particles in three dimensions, interacting with each other through square-well potentials with repulsive and attractive parts. The calculations were executed with the Green’s function procedure, made convenient by availability of known Green’s functions satisfying the conditions of the sample problems. A partial cancellation of positive and negative walkers was made on the basis of overlapping Green’s functions, and the distributions of steps for the walkers involved were altered accordingly. Without cancellations, the distributions of positive and negative walkers proceed to the boson ground-state distribution. With cancellation and with control of their numbers, the positive and negative walkers tend to accumulate in separate regions and to commingle in the vicinity of a node. The arithmetic sum of the walkers yields the excited or fermion wavefunction with a node separating positive and negative regions. For the model problems the method successfully produced accuracies within a few percent of the exact energies, where known. It was noted that for larger problems, the total number of walkers required to maintain an adequate density for sufficient cancellation might be prohibitive.
A new Monte Carlo method for computing excited state properties of quantum systems is introduced. It is a generalization of the transient estimate method used for fermion Green’s function Monte Carlo and of subspace projection methods used for computing eigenstates of matrices. The time dependent autocorrelation function of a vector of trial functions is calculated for a random walk generated by the imaginary-time Schrödinger equation and estimates of energy levels are determined by the eigenvalues of the matrix of correlation functions. This method is especially useful for treating states with the same symmetry as it automatically keeps higher states orthogonal to lower states. The estimated energy converges to the exact eigenvalue with a rate which decreases with increasing excitation energy, thus limiting the method to relatively low-lying states. The method is zero variance in the sense that as better trial functions are introduced, the statistical error decreases to zero. The method has a nontrivial bias which is analyzed. As an illustration, the eigenvalues of a double well are computed.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.
A recently developed self-healing diffusion Monte Carlo algorithm [F. A. Reboredo, R. Q. Hood, and P. R. C. Kent, Phys. Rev. B 79, 195117 (2009)] is extended to the calculation of excited states. The formalism is based on an excited-state fixed-node approximation and the mixed estimator of the excited-state probability density. The fixed-node ground-state wave-functions of inequivalent nodal pockets are found simultaneously using a recursive approach. The decay of the wave-function into lower-energy states is prevented using two methods: (i) the projection of the improved trial-wave function into previously calculated eigenstates is removed; and (ii) the reference energy for each nodal pocket is adjusted in order to create a kink in the global fixed-node wave-function, which, when locally smoothed, increases the volume of the higher-energy pockets at the expense of the lower-energy ones until the energies of every pocket become equal. This reference energy method is designed to find nodal structures that are local minima for arbitrary fluctuations of the nodes within a given nodal topology. It is demonstrated in a model system that the algorithm converges to many-body eigenstates in bosonic and fermionic cases.
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.