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Physical Review Letters

Method of Gaussian Equivalent Representation: A New Technique for Reducing the Sign Problem of Functional Integral Methods

Stephan A. Baeurle

Published Aug 5, 2002• 31 citations
Publisher

Abstract

We report on the specific features of the sign problem in the classical auxiliary field methodology and the strategies employed for its alleviation. In particular, we focus on a new technique based on the method of Gaussian equivalent representation of Efimov and Nogovitsin [Physica (Amsterdam) 234A, 506 (1996)]] with which we could ameliorate the convergence properties significantly. We believe that this technique can also provide an interesting possibility to reduce the sign problem of other methods of computer simulation based on a functional integral approach.

DOI
10.1103/physrevlett.89.080602

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We report on the specific features of the sign problem in the classical auxiliary field methodology and the strategies employed for its alleviation. In particular, we focus on a new technique based on the method of Gaussian equivalent representation of Efimov and Nogovitsin [Physica (Amsterdam) 234A, 506 (1996)]] with which we could ameliorate the convergence properties significantly. We believe that this technique can also provide an interesting possibility to reduce the sign problem of other methods of computer simulation based on a functional integral approach.
The concept of Lefschetz thimble decomposition is one of the most promising possible modifications of Quantum Monte Carlo (QMC) algorithms aimed at alleviating the sign problem which appears in many interesting physical situations, e.g. in the Hubbard model away from half filling. In this approach one utilizes the fact that the integral over real variables with an integrand containing a complex fluctuating phase is equivalent to the sum of integrals over special manifolds in complex space ("Lefschetz thimbles"), each of them having a fixed complex phase factor. Thus, the sign problem can be reduced if the resulting sum contains terms with only a few different phases. We explore the complexity of the sign problem for the few-site Hubbard model on square lattice combining a semi-analytical study of saddle points and thimbles in small lattices with several steps in Euclidean time with results of test QMC calculations. We check different variants of conventional Hubbard-Stratonovich transformation based on Gaussian integrals. On the basis of our analysis we reveal a regime with minimal number of relevant thimbles in the vicinity of half filling. In this regime we found only two relevant thimbles for the few-site lattices studied in the paper. There is also indirect evidence of the existence of this regime in more realistic systems with large number of Euclidean time slices. In addition, we derive a new non-Gaussian representation of the interaction term, where the number of relevant Lefschetz is also reduced in comparison with conventional Gaussian Hubbard-Stratonovich transformation.
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).