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Physical review. B, Condensed matter

Diffusive behavior of states in the Hubbard-Stratonovitch transformation

Stephen Fahy, Donald R. Hamann

Published Jan 1, 1991• 42 citations
Publisher

Abstract

The Hubbard-Stratonovitch auxiliary-field approach to projection of the ground state of an interacting fermion system from a trial state is examined. It is shown that the method is equivalent to solving a differential equation with diffusion, drift, and branching terms on the manifold of normalized Slater determinants. Explicit general expressions are given for the coefficients of the diffusion equation in terms of the Hubbard-Stratonovitch fields. The form of the equation is somewhat similar to that obtained in the continuum Green's-function Monte Carlo method, although its interpretation and relation to the physical many-body problem is quite different. The character of the representation of the many-body ground state arising in the auxiliary-field projection approach is discussed within this framework. The diffusion process is normally found to concentrate the states representing the ground state near the classical mean-field solutions. The consequences of this picture of the auxiliary-field approach for the ``fermion minus-sign'' problem in this context are discussed, and some conclusions are reached concerning the range of validity of a recently suggested approximation where minus signs are ignored.

DOI
10.1103/physrevb.43.765

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Physical review. B, Condensed matter•Jun 1, 1990•40 citations
The Hubbard-Stratonovitch auxiliary-field approach to projection of the ground state of an interacting fermion system from a trial state is examined. It is shown that the method is equivalent to solving a differential equation with diffusion, drift, and branching terms on the manifold of normalized Slater determinants. Explicit general expressions are given for the coefficients of the diffusion equation in terms of the Hubbard-Stratonovitch fields. The form of the equation is somewhat similar to that obtained in the continuum Green's-function Monte Carlo method, although its interpretation and relation to the physical many-body problem is quite different. The character of the representation of the many-body ground state arising in the auxiliary-field projection approach is discussed within this framework. The diffusion process is normally found to concentrate the states representing the ground state near the classical mean-field solutions. The consequences of this picture of the auxiliary-field approach for the ``fermion minus-sign'' problem in this context are discussed, and some conclusions are reached concerning the range of validity of a recently suggested approximation where minus signs are ignored.
An algorithm is developed for determining the exact ground state properties of quantum many-body systems which is equally applicable to bosons and fermions. The Schroedinger eigenvalue equation for the ground state energy is recast into the form of a many-dimensional integral through the use of the Hubbard-Stratonovitch representation of the imaginary time many- body evolution operator. The resulting functional integral is then evaluated stochastically. The algorithm is tested for an exactly soluble boson system and is then extended to include fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Improved computational efficiency is attained by performing the calculations in momentum space.
The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.
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.
We introduce an efficient and numerically stable technique to make use of a BCS trial wave function in the computation of correlation functions of strongly correlated quantum fermion systems. The technique is applicable to any projection approach involving paths of independent-fermion propagators, for example, in mean-field or auxiliary-field quantum Monte Carlo (AFQMC) calculations. Within AFQMC, in the absence of the sign problem, the methodology allows the use of a BCS reference state which can greatly reduce the required imaginary time of projection and improves Monte Carlo sampling efficiency and statistical accuracy for systems where pairing correlations are important. When the sign problem is present, the approach provides a powerful generalization of the constrained-path AFQMC technique which usually uses Slater determinant trial wave functions. As a demonstration of the capability of the methodology, we present benchmark results for the attractive Hubbard model, both spin balanced (no sign problem) and with a finite spin polarization (with sign problem).
The ground state of a many-electron system can be projected from a trial state using an integral of one-electron projectors over a set of Hubbard-Stratonovich auxiliary fields. We have explored several alternative formulations of this method and find that the convergence of the ground-state energy with the number of field variables can be dramatically improved by choosing the appropriate one. This is illustrated using the two-dimensional Hubbard model as an example. These results are also used to examine the error associated with neglecting the so-called ``sign problem'' common to quantum Monte Carlo many-fermion calculations.