Cluster representations and the Wolff algorithm in arbitrary external fields

Abstract

We introduce a natural way to extend celebrated spin-cluster Monte Carlo algorithms for fast thermal lattice simulations at criticality, such as the Wolff algorithm, 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 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 modeling the presence of novel nonlinear fields. We also provide a c++ library for the method's convenient implementation for arbitrary models.