Ising (anti-)ferromagnet on dynamical triangulations and quadrangulations

TL;DR

This paper develops matrix models for Ising spins on dynamical triangulated and quadrangulated surfaces, comparing them with standard dual graph models, and investigates phase transitions including antiferromagnetic order.

Contribution

It introduces matrix models for Ising spins on DTRS and DQRS, and demonstrates their critical temperatures match duality-based predictions, revealing antiferromagnetic phase transitions.

Findings

Critical temperatures agree with duality predictions

Antiferromagnetic Ising model exhibits Neel order on DQRS

Frustration prevents antiferromagnetic order on other surfaces

Abstract

We write down matrix models for Ising spins with zero external field on the vertices of dynamical triangulated random surfaces (DTRS) and dynamically quadrangulated random surfaces (DQRS) and compare these with the standard matrix model approach which places the spins on the dual $\phi^3$ and $\phi^4$ graphs. We show that the critical temperatures calculated in the DTRS and DQRS models agree with those deduced from duality arguments in the standard approach. Using the DQRS model we observe that the Ising antiferromagnet still undergoes a phase transition to a Neel (checkerboard) ordered ground state which is absent because of frustration in the other cases.