Duality relation for frustrated spin models

TL;DR

This paper establishes a duality relation for frustrated spin models on planar graphs, showing how frustration translates into complex external fields in the dual model, with implications for analyzing such systems.

Contribution

It introduces an algebraic method to derive duality relations for frustrated Ising and Potts models, extending known results to arbitrary planar graphs.

Findings

Duality maps frustrated plaquettes to external fields in the dual model

Partition functions can be evaluated in the thermodynamic limit for regular lattices

Results apply to both Ising and Potts spin glass models

Abstract

We consider discrete spin models on arbitrary planar graphs and lattices with frustrated interactions. We first analyze the Ising model with frustrated plaquettes. We use an algebraic approach to derive the result that an Ising model with some of its plaquettes frustrated has a dual which is an Ising model with an external field $i\pi/2$ applied to the dual sites centered at frustrated plaquettes. In the case that all plaquettes are frustrated, this leads to the known result that the dual model has a uniform field $i\pi/2$ whose partition function can be evaluated in the thermodynamic limit for regular lattices. The analysis is extended to a Potts spin glass with analogous results obtained.