The directed-loop algorithm

TL;DR

The paper introduces the directed-loop algorithm, a generalized quantum Monte Carlo update scheme that extends traditional loop algorithms by allowing self-intersecting loops, improving simulation flexibility.

Contribution

It presents the directed-loop equations and demonstrates their application to the anisotropic S=1/2 Heisenberg model in a magnetic field, generalizing the conventional loop algorithm.

Findings

Directed-loop equations provide probabilities for loop-building steps.

The scheme allows self-intersecting loops to satisfy detailed balance.

Application to the Heisenberg model shows the method's effectiveness.

Abstract

The directed-loop scheme is a framework for generalized loop-type updates in quantum Monte Carlo, applicable both to world-line and stochastic series expansion methods. Here, the directed-loop equations, the solution of which gives the probabilities of the various loop-building steps, are discussed in the context of the anisotropic $S=1/2$ Heisenberg model in a uniform magnetic field. This example shows how the directed-loop concept emerges as a natural generalization of the conventional loop algorithm, where the loops are selfavoiding, to cases where selfintersection must be allowed in order to satisfy detailed balance.