Cluster representations and the Wolff algorithm in arbitrary external fields

TL;DR

This paper extends the Wolff cluster algorithm to systems with arbitrary external fields by introducing a 'ghost transformation' representation, preserving efficiency and enabling simulation of nonlinear fields, with a practical C++ implementation.

Contribution

The authors generalize the Wolff algorithm to arbitrary external fields using a novel 'ghost transformation' approach, maintaining accelerated dynamics and providing a versatile C++ library.

Findings

Preserves scaling of accelerated dynamics in zero-field cases.

Enables simulation of systems with nonlinear and vector external fields.

Provides a practical C++ library for implementation.

Abstract

We introduce a natural way to extend celebrated spin-cluster Monte Carlo algorithms for fast thermal lattice simulations at criticality, like Wolff, to systems in arbitrary fields, be they linear magnetic vector fields or nonlinear anisotropic ones. By generalizing the 'ghost spin' representation to one with a 'ghost transformation,' global invariance to spin symmetry transformations is restored at the cost of an extra degree of freedom which lives in the space of symmetry transformations. The ordinary cluster-building process can then be run on the new representation. We show that this extension preserves the scaling of accelerated dynamics in the absence of a field for Ising, Potts, and $\mathrm O(n)$ models and demonstrate the method's use in modelling the presence of novel nonlinear fields. We also provide a C++ library for the method's convenient implementation for arbitrary models.