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Physical review. B, Condensed matter

Monte Carlo diagonalization of many-body problems: Application to fermion systems

Hans A. De Raedt, Wolfgang von der Linden

Published Apr 15, 1992• 31 citations
Publisher

Abstract

Existing quantum Monte Carlo algorithms suffer the so-called minus-sign problem. We propose a stochastic algorithm to compute ground-state properties of quantum lattice models that is free of minus-sign problems. Illustrative results for the two-dimensional Hubbard model are presented.
DOI
10.1103/physrevb.45.8787

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All known Quantum-Monte-Carlo algorithms for fermions suffer from the so-called “minus-sign-problem” which is detrimental to the application of these simulation methods to fermion systems at very low temperatures and/or of very many lattice sites. We identify the origin of this fundamental problem, demonstrate that it is a very general feature, not necessarily related to the presence of fermionic degrees of freedom. We describe an novel algorithm which does not suffer from the minus-sign problem. Illustrative results for the two-dimensional Hubbard model are presented
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.
For important classes of many-fermion problems, quantum Monte Carlo (QMC) methods allow exact calculations of ground-state and finite-temperature properties without the sign problem. The list spans condensed matter, nuclear physics, and high-energy physics, including the half-filled repulsive Hubbard model, the spin-balanced atomic Fermi gas, and lattice quantum chromodynamics calculations at zero density with Wilson Fermions, and is growing rapidly as a number of problems have been discovered recently to be free of the sign problem. In these situations, QMC calculations are relied on to provide definitive answers. Their results are instrumental to our ability to understand and compute properties in fundamental models important to multiple subareas in quantum physics. It is shown, however, that the most commonly employed algorithms in such situations have an infinite variance problem. A diverging variance causes the estimated Monte Carlo statistical error bar to be incorrect, which can render the results of the calculation unreliable or meaningless. We discuss how to identify the infinite variance problem. An approach is then proposed to solve the problem. The solution does not require major modifications to standard algorithms, adding a "bridge link" to the imaginary-time path integral. The general idea is applicable to a variety of situations where the infinite variance problem may be present. Illustrative results are presented for the ground state of the Hubbard model at half-filling.
We provide an extension to lattice systems of the reptation quantum Monte Carlo algorithm, originally devised for continuous Hamiltonians. For systems affected by the sign problem, a method to systematically improve upon the so-called fixed-node approximation is also proposed. The generality of the method, which also takes advantage of a canonical worm algorithm scheme to measure off-diagonal observables, makes it applicable to a vast variety of quantum systems and eases the study of their ground-state and excited-state properties. As a case study, we investigate the quantum dynamics of the one-dimensional Heisenberg model and we provide accurate estimates of the ground-state energy of the two-dimensional fermionic Hubbard model.
In this study we present an optimization method based on the quantum Monte Carlo diagonalization for many-fermion systems. Using the Hubbard-Stratonovich transformation, employed to decompose the interactions in terms of auxiliary fields, we expand the true ground-state wave function. The ground-state wave function is written as a linear combination of the basis wave functions. The Hamiltonian is diagonalized to obtain the lowest energy state using the variational principle within the selected subspace of the basis functions. This method is free from the difficulty known as the negative sign problem. We can optimize a wave function using two procedures. The first procedure is to increase the number of basis functions. The second improves each basis function through the operators ${e}^{\ensuremath{-}\ensuremath{\Delta}\ensuremath{\tau}H}$ using the Hubbard-Stratonovich decomposition. We present an algorithm for the quantum Monte Carlo diagonalization method using a genetic algorithm and the renormalization method. We compute the ground-state energy and correlation functions of small clusters to compare with available data.
We describe and discuss a recently proposed quantum Monte Carlo algorithm to compute the ground-state properties of various systems of interacting fermions. In this method, the ground state is projected from an initial wave function by a branching random walk in an overcomplete basis of Slater determinants. By constraining the determinants according to a trial wave function |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉 is exact. We illustrate the method by describing in detail its implementation for the two-dimensional one-band Hubbard model. We show results for lattice sizes up to 16\ifmmode\times\else\texttimes\fi{}16 and for various electron fillings and interaction strengths. With simple single-determinant wave functions as |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, the method yields accurate (often to within a few percent) estimates of the ground-state energy as well as correlation functions, such as those for electron pairing. We conclude by discussing possible extensions of the algorithm.