arXiv:cond-mat/9901055

Study of a generalized Metropolis decision rule in auxiliary field quantum Monte Carlo

Martin, C. L., Richard M. Fye

Published Jan 7, 1999• 0 citations

Abstract

We consider a generalization of the standard Metropolis algorithm acceptance/rejection decision rule and numerically explore its properties using auxiliary field quantum Monte Carlo. The generalization involves a free parameter which, given a criterion for proposing attempted moves, can be used to tune the average acceptance rate in a particular way. Such tuning can also potentially change Monte Carlo autocorrelation times, and the combination of the changing acceptance rate and autocorrelation times raises the possibility of more efficient simulations. We explore these issues using primarily massively parallel quantum Monte Carlo runs of the ``test case'' two-dimensional Hubbard model, and discuss results and applications.

DOI: 10.48550/arxiv.cond-mat/9901055

Related papers

Interfacing Branching Random Walks with Metropolis Sampling: Constraint Release in Auxiliary-Field Quantum Monte Carlo

Zhi-Yu Xiao, Hao Shi, Shiwei Zhang

We present an approach to interface branching random walks with Markov chain Monte Carlo sampling and to switch seamlessly between the two. The approach is discussed in the context of auxiliary-field quantum Monte Carlo (AFQMC) but can be applied to other Monte Carlo calculations or simulations. In AFQMC, the formulation of branching random walks along imaginary-time is needed to realize a constraint to control the sign or phase problem. The constraint is derived from an exact gauge condition and is in practice implemented approximately with a trial wave function or trial density matrix, which can break exactness in the algorithm. We use the generalized Metropolis algorithm to sample a selected portion of the imaginary-time path after it has been produced by the branching random walk. This interfacing allows a constraint release to follow seamlessly from constrained-path sampling, which can reduce the systematic error from the latter. It also provides a way to improve the computation of correlation functions and observables that do not commute with the Hamiltonian. We illustrate the method in atoms and molecules, where improvements in accuracy can be clearly quantified and near-exact results are obtained. We also discuss the computation of the variance of the Hamiltonian and propose a convenient way to evaluate it stochastically without changing the scaling of AFQMC.

arXiv:2305.09575Journal of Chemical Theory and ComputationSep 4, 202310 citations

Metropolis Methods for Quantum Monte Carlo Simulations

David M. Ceperley

Since its first description fifty years ago, the Metropolis Monte Carlo method has been used in a variety of different ways for the simulation of continuum quantum many‐body systems. This paper will consider some of the generalizations of the Metropolis algorithm employed in quantum Monte Carlo: Variational Monte Carlo, dynamical methods for projector monte carlo (i.e. diffusion Monte Carlo with rejection), multilevel sampling in path integral Monte Carlo, the sampling of permutations, cluster methods for lattice models, the penalty method for coupled electron‐ionic systems and the Bayesian analysis of imaginary time correlation functions.

arXiv:physics/0306182AIP conference proceedingsJan 1, 200323 citations

Benchmark study of an auxiliary-field quantum Monte Carlo technique for the Hubbard model with shifted-discrete Hubbard-Stratonovich transformations

Kazuhiro Seki, Sandro Sorella

Within the ground-state auxiliary-field quantum Monte Carlo technique, we introduce discrete Hubbard-Stratonovich transformations (HSTs) that are also suitable for spatially inhomogeneous trial functions. The discrete auxiliary fields introduced here are coupled to local spin or charge operators fluctuating around their Hartree-Fock values. The formalism can be considered a generalization of the discrete HSTs by J. E. Hirsch [Phys. Rev. B 28, 4059 (1983)] or a compactification of the shifted-contour auxiliary-field Monte Carlo formalism by N. Rom et al. [Chem. Phys. Lett. 270, 382 (1997)]. An improvement of the acceptance ratio is found for a real auxiliary field, while an improvement of the average sign is found for a purely imaginary auxiliary field. Efficiencies of the different HSTs are tested in the single-band Hubbard model at and away from half filling by studying the staggered magnetization and energy expectation values, respectively.

arXiv:1902.00321Physical review. B./Physical review. BApr 8, 201910 citations

Markov Chain Monte Carlo Method without Detailed Balance

Hidemaro Suwa, Synge Todo

We present a specific algorithm that generally satisfies the balance condition without imposing the detailed balance in the Markov chain Monte Carlo. In our algorithm, the average rejection rate is minimized, and even reduced to zero in many relevant cases. The absence of the detailed balance also introduces a net stochastic flow in a configuration space, which further boosts up the convergence. We demonstrate that the autocorrelation time of the Potts model becomes more than 6 times shorter than that by the conventional Metropolis algorithm. Based on the same concept, a bounce-free worm algorithm for generic quantum spin models is formulated as well.

arXiv:1007.2262Physical Review LettersSep 17, 2010148 citations

A Simple Monte Carlo Simulation For the Two Dimensional Attractive Hubbard Model

Eberth de Almeida Correa, R. G. M. Rodrigues, L. A. S. Mόl +1 more

Abstract We use the Auxiliary Field Quantum Monte Carlo technique to perform a simple study of the two dimensional attractive Hubbard model and calculate the total ground state energy to compare with another technique in the literature for U = –4t. Our finite size extrapolated result corroborate the approximated lattice density-functional theory method for this value of U at half-filling. To stabilize the Auxiliary Field Quantum Monte Carlo algorithm one must resort to one of the two decomposition methods: the modified Grand-Schmidt decomposition and the singular value decomposition. Two characteristics like accuracy and total execution time were considered to compare the decomposition algorithms, and the results are presented bellow.

Journal of Physics Conference SeriesFeb 1, 20200 citations

AUXILIARY FIELDS AND THE SIGN PROBLEM

John H. Samson

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.

International Journal of Modern Physics CJun 1, 19954 citations