Generalization of the Fortuin-Kasteleyn transformation and its application to quantum spin simulations,

TL;DR

This paper extends the FK cluster representation to quantum spin models of any spin magnitude and dimension, enabling a versatile cluster algorithm for quantum Monte Carlo simulations.

Contribution

It introduces a generalized FK representation for quantum spins and develops a new cluster algorithm that includes Swendsen-Wang as a special case.

Findings

The new algorithm applies to models with arbitrary exchange range and lattice geometry.

It successfully simulates quantum spin systems like the Ising, S=1 Heisenberg, and general Heisenberg models.

The method improves simulation efficiency for complex quantum spin models.

Abstract

We generalize the Fortuin-Kasteleyn (FK) cluster representation of the partition function of the Ising model to represent the partition function of quantum spin models with an arbitrary spin magnitude in arbitrary dimensions. This generalized representation enables us to develop a new cluster algorithm for the simulation of quantum spin systems by the worldline Monte Carlo method. Because the Swendsen-Wang algorithm is based on the FK representation, the new cluster algorithm naturally includes it as a special case. As well as the general description of the new representation, we present an illustration of our new algorithm for some special interesting cases: the Ising model, the antiferromagnetic Heisenberg model with $S=1$, and a general Heisenberg model. The new algorithm is applicable to models with any range of the exchange interaction, any lattice geometry, and any dimensions.