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Physical Review Letters

Loop Algorithms for Monte Carlo Simulations of Quantum Spin Systems

Naoki Kawashima, James E. Gubernatis

Published Aug 29, 1994• 59 citations
Publisher

Abstract

We present a new algorithm for world line Monte Carlo simulations of quantum spin systems. The algorithm consists of a stochastic pairing and a stochastic flipping procedure, and the formula for the pairing probability is explicitly given for Heisenberg models. As an illustration of the algorithm, we calculated autocorrelation times for simulations of the $S=1$ antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.

DOI
10.1103/physrevlett.73.1295

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We present a new algorithm for world line Monte Carlo simulations of quantum spin systems. The algorithm consists of a stochastic pairing and a stochastic flipping procedure, and the formula for the pairing probability is explicitly given for Heisenberg models. As an illustration of the algorithm, we calculated autocorrelation times for simulations of the S=1S=1S=1 antiferromagnetic Heisenberg chain. The rapid increase in autocorrelation times which is seen in the conventional world line method as the imaginary-time spacing decreases is eliminated by the present algorithm.
A review of the loop algorithm , its generalizations, and its relation to some other Monte Carlo techniques is given. The loop algorithm is a quantum Monte Carlo procedure that employs non-local changes of worldline configurations, determined by local stochastic decisions. It is based on a formulation of quantum models of any dimension in an extended ensemble of worldlines and graphs, and is related to Swendsen-Wang algorithms. It can be represented directly on an operator level, both with a continuous imaginary time path integral and with the stochastic series expansion. It overcomes many of the difficulties of traditional worldline simulations. Autocorrelations are reduced by orders of magnitude. Grand-canonical ensembles, off-diagonal operators, and variance reduced estimators are accessible. In some cases, infinite systems can be simulated. For a restricted class of models, the fermion sign problem can be overcome. Transverse magnetic fields are handled efficiently, in contrast to strong diagonal fields. The method has been applied successfully to a variety of models for spin and charge degrees of freedom, including Heisenberg and XYZ spin models, hard-core bosons, Hubbard and t - J -models. Owing to the improved efficiency, precise calculations of asymptotic behaviour and of quantum critical exponents have been possible.
Two cluster algorithms, based on constructing and flipping loops, are presented for world-line quantum Monte Carlo simulations of fermions and are tested on the one-dimensional repulsive Hubbard model. We call these algorithms the loop-flip and loop-exchange algorithms. For these two algorithms and the standard world-line algorithm, we calculated the autocorrelation times for various physical quantities and found that the ordinary world-line algorithm, which uses only local moves, suffers from very long correlation times that makes not only the estimate of the error difficult but also the estimate of the average values themselves difficult. These difficulties are especially severe in the low-temperature, large-U regime. In contrast, we find that new algorithms, when used alone or in combinations with themselves and the standard algorithm, can have significantly smaller autocorrelation times, in some cases being smaller by three orders of magnitude. The new algorithms, which use nonlocal moves, are discussed from the point of view of a general prescription for developing cluster algorithms. The loop-flip algorithm is also shown to be ergodic and to belong to the grand canonical ensemble. Extensions to other models and higher dimensions are briefly discussed.
A quantum Monte Carlo method combining update of the loop algorithm with the global flip of the world line is proposed as an efficient method to study the magnetization process in an external field, which has been difficult because of inefficiency of the update of the total magnetization. The method is demonstrated in the one dimensional antiferromagnetic Heisenberg model and the trimer model. We attempted various other Monte Carlo algorithms to study systems in the external field and compared their efficiency.
We propose a generalization of the quantum Monte Carlo loop algorithm to the $t\ensuremath{-}J$ model by a mapping to three coupled six-vertex models. The autocorrelation times are reduced by orders of magnitude compared to the conventional local algorithms. The method is completely ergodic and can be formulated directly in continuous time. We introduce improved estimators for simulations with a local sign problem. Some first results of finite-temperature simulations are presented for a $t\ensuremath{-}J$ chain, a frustrated Heisenberg chain, and $t\ensuremath{-}J$ ladder models.
A world-line Monte Carlo approach is proposed to calculate the elementary excitations of quantum systems. The lowest energies as a function of momentum are successfully extracted from an imaginary-time correlation function evaluated at a sufficiently low temperature. The method is applied to the antiferromagnetic Heisenberg chains of S=12,1,32,S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2},1,\frac{3}{2},S=21​,1,23​, and 222 with length up to 64. The spectra for S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ and 1 coincide well with the result by des Cloizeaux and Pearson and with the result by Takahashi, respectively. The spectra for S=32S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{3}{2}S=23​ and 2 turn out to be massless and massive, respectively. The S=2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2S=2 Haldane gap is estimated to be $0.049\ifmmode\pm\else\textpm\fi{}0.018$.