Spin models in complex magnetic fields: a hard sign problem

TL;DR

This paper explores the effects of complex external magnetic fields on spin models, particularly the Potts model, revealing phenomena like zeroes of the partition function and proposing methods to mitigate the sign problem.

Contribution

It introduces a coupling approach for the N-state Potts model with complex fields and demonstrates a modified cluster algorithm that reduces the sign problem.

Findings

Analytic calculations in 1D reveal zeroes of the partition function.

Modified cluster algorithm significantly reduces the sign problem.

Coupling methods can mitigate the sign problem in complex magnetic fields.

Abstract

Coupling spin models to complex external fields can give rise to interesting phenomena like zeroes of the partition function (Lee-Yang zeroes, edge singularities) or oscillating propagators. Unfortunately, it usually also leads to a severe sign problem that can be overcome only in special cases; if the partition function has zeroes, the sign problem is even representation-independent at these points. In this study, we couple the N-state Potts model in different ways to a complex external magnetic field and discuss the above mentioned phenomena and their relations based on analytic calculations (1D) and results obtained using a modified cluster algorithm (general D) that in many cases either cures or at least drastically reduces the sign-problem induced by the complex external field.