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Journal of the Physical Society of Japan

Some Findings on the Minus-Sign Problem for the Hubbard Model in the Projector Quantum Monte Carlo Method

Hans‐Georg Matuttis, Nobuyasu Ito

Published Jun 15, 2001• 2 citations
Publisher

Abstract

The quantum Monte Carlo method for the Hubbard model using the Hubbard-Stratonovich transformation is reexamined. It is observed that the value of the ground state energy for a 3×2 and 4×4 system is estimated correctly by ignoring the minus-sign of the negative sampling weight appearing in the simulation using the projector method. It is also observed that the average of sign is positive and the correlation between the energy and the sign is weak for these systems. The reasons why the ground state energy is estimated correctly is given on a phenomenological base.

DOI
10.1143/jpsj.70.1519

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The quantum Monte Carlo method for the Hubbard model using the Hubbard-Stratonovich transformation is reexamined. It is observed that the value of the ground state energy for a 3×2 and 4×4 system is estimated correctly by ignoring the minus-sign of the negative sampling weight appearing in the simulation using the projector method. It is also observed that the average of sign is positive and the correlation between the energy and the sign is weak for these systems. The reasons why the ground state energy is estimated correctly is given on a phenomenological base.
The minus sign problem, as a problem of the non-positive-definite measure of the path integrals, is examined in the Monte Carlo simulation of the Hubbard model. A new correction method in terms of the ratio of two measured quantities is proposed to calculate physical properties. The momentum distribution and the spin correlation as well as the energy in the ground state are tested against the exactly diagonalized results. The results show that this ratio correction method is a promising prescription to reduce the minus sign problem.
We present an improved version of the projector quantum Monte Carlo method, which has recently been proposed by Sorella et al. This scheme allows a very precise computation of the ground-state energy of fermionic models. The ``minus sign'' has been treated without further approximations and does not influence the accuracy. In this paper we focus on the question of the existence of quasiparticles in the Hubbard model in one dimension and resolve the controversy concerning the discontinuity in the momentum distribution function between recent quantum Monte Carlo and weak-coupling renormalization-group techniques.
In this paper we compare numerical results for the ground state of the Hubbard model obtained by Quantum-Monte-Carlo simulations with results from exact and stochastic diagonalizations. We find good agreement for the ground state energy and superconducting correlations for both, the repulsive and attractive Hubbard model. Special emphasis lies on the superconducting correlations in the repulsive Hubbard model, where the small magnitude of the values obtained by Monte-Carlo simulations gives rise to the question, whether these results might be caused by fluctuations or systematic errors of the method. Although we notice that the Quantum-Monte-Carlo method has convergence problems for large interactions, coinciding with a minus sign problem, we confirm the results of the diagonalization techniques for small and moderate interaction strengths. Additionally we investigate the numerical stability and the convergence of the Quantum-Monte-Carlo method in the attractive case, to study the influence of the minus sign problem on convergence. Also here in the absence of a minus sign problem we encounter convergence problems for strong interactions.
A theorem by Yoshida which states that the discretization error for the Suzuki–Trotter (ST) decomposition is one order smaller than for the corresponding approximated Hamiltonian has long been overlooked with grave consequences for the accuracy of algorithms which use decomposition schemes. In our error analysis for the ground state energy of the Hubbard Hamiltonian by the projector quantum Monte Carlo method, we have used various orders of ST-decompositions, including for the first time in this field, pseudo-symplectic methods. We show that higher order does not necessarily imply better convergence, and identify the condition to obtain good convergence with higher order ST-decompositions. For the first order ST-decomposition, the ground state energy does not converge to the values for numerical diagonalization, in agreement with Yoshida's theorem, which may have caused confusion in connection with the “fermionic sign problem”. We show how the use of the sign of the fermion determinant is in fact a reweighing method which for the ground state energy violates some basic Monte Carlo properties (the Gaussian distribution of observables), in contrast to sampling without sign. Whether sampling with or without the sign deviates less from the exact ground state energy depends on the system parameters (interaction, filling) and on the test wave function, so sampling with sign gives not necessarily better or more physical than sampling without sign, a result which can be supported by arguments on the error compensation. We discuss the implications for related methods.
We present in detail the recently developed multiconfigurational symmetrized projector quantum Monte Carlo method for excited states of the Hubbard model. We describe the implementation of the Monte Carlo method for a multiconfigurational trial wave function. We give a detailed discussion of issues related to the symmetry of the projection procedure that validates our Monte Carlo procedure for excited states. In this context we discuss various averaging procedures for the Green function and present an analysis of the errors incurred in these procedures. We study the ground-state energy and correlation functions of the one-dimensional Hubbard model at half-filling to confirm these analyses. We then study the energies and correlation functions of excited states of Hubbard chains. Hubbard rings away from half-filling are also studied and the pair binding energies for holes of 4n4n4n and 4n+24n+24n+2 systems are compared with the Bethe ansatz results of Fye, Martins, and Scalettar [Phys. Rev. B 42, 6809 (1990)]. Our study of the two-dimensional Hubbard model includes the $4\ifmmode\times\else\texttimes\fi{}2$ ladder and the $3\ifmmode\times\else\texttimes\fi{}4$ lattice with periodic boundary conditions. The $3\ifmmode\times\else\texttimes\fi{}4$ lattice is nonbipartite and amenable to exact diagonalization studies and is, therefore, a good candidate for checks on the method. We are able to reproduce accurately the energies of ground and excited states, both at and away from half-filling. We study the properties of the $4\ifmmode\times\else\texttimes\fi{}2$ Hubbard ladder with bond alternation as the correlation strength and filling are varied. The method reproduces the correlation functions accurately. We also examine the severity of sign problem for one- and two-dimensional systems.
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.