Dual Geometric Worm Algorithm for Two-Dimensional Discrete Classical Lattice Models

TL;DR

This paper introduces a dual geometric worm algorithm for 2D Ising models that is efficient, easy to implement, and generalizable to other bond-variable models, offering advantages over existing algorithms.

Contribution

The paper presents a novel dual geometric worm algorithm for 2D Ising models, with explicit proofs of detailed balance and improved computational efficiency.

Findings

Comparable efficiency to Swendsen-Wang and Wolff algorithms

Simpler implementation of the dual algorithm

Facilitates calculation of domain wall free energy

Abstract

We present a dual geometrical worm algorithm for two-dimensional Ising models. The existence of such dual algorithms was first pointed out by Prokof'ev and Svistunov \cite{ProkofevClassical}. The algorithm is defined on the dual lattice and is formulated in terms of bond-variables and can therefore be generalized to other two-dimensional models that can be formulated in terms of bond-variables. We also discuss two related algorithms formulated on the direct lattice, applicable in any dimension. These latter algorithms turn out to be less efficient but of considerable intrinsic interest. We show how such algorithms quite generally can be "directed" by minimizing the probability for the worms to erase themselves. Explicit proofs of detailed balance are given for all the algorithms. In terms of computational efficiency the dual geometrical worm algorithm is comparable to well known cluster algorithms such as the Swendsen-Wang and Wolff algorithms, however, it is quite different in structure and allows for a very simple and efficient implementation. The dual algorithm also allows for a very elegant way of calculating the domain wall free energy.