The Loop Algorithm

TL;DR

The paper reviews the Loop Algorithm, a quantum Monte Carlo method that uses nonlocal updates to efficiently simulate quantum models, reducing autocorrelations and enabling studies of complex systems.

Contribution

It provides a comprehensive overview of the Loop Algorithm's generalizations, its relation to other Monte Carlo techniques, and demonstrates its effectiveness across various quantum models.

Findings

Reduces autocorrelations by orders of magnitude

Enables simulation of infinite systems and overcoming the fermion sign problem in some cases

Allows precise calculations of quantum critical behavior

Abstract

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.