The Loop Algorithm

Details The Loop Algorithm The Loop Algorithm

1997-07-21

arxiv.org/abs/cond-mat/9707221v3

Adv.Phys. 52 (2003) 1

Physics

Condensed Matter

Strongly Correlated Electrons

Third Edition, July 2002. (78 pages, 11 figures). To appear in Adv.Phys. Updated. New chapter on Operator Formulation, with co

Scientific paper

10.1080/0001873021000049195

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 which employs nonlocal 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 (SSE). 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 ones. 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 tJ-models. Due to the improved efficiency, precise calculations of asymptotic behavior and of quantum critical exponents have been possible.

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