The critical behaviour of Ising spins on 2D Regge lattices

TL;DR

This study uses high-statistics simulations to analyze Ising spins on 2D Regge lattices coupled with quantum gravity, revealing that critical exponents match static values and are unaffected by certain interactions, challenging previous predictions.

Contribution

It provides the first comprehensive numerical evidence that Ising critical exponents on Regge lattices align with static values and are independent of the $R^2$ coupling, refuting earlier theoretical predictions.

Findings

Critical exponents match static Ising values.

Excludes the Boulatov-Kazakov behavior for this model.

Critical behavior is independent of $R^2$ interaction strength.

Abstract

We performed a high statistics simulation of Ising spins coupled to 2D quantum gravity on toroidal geometries. The tori were triangulated using the Regge calculus approach and contained up to $512^2$ vertices. We used a constant area ensemble with an added $R^2$ interaction term, employing the $dl/l$ measure. We find clear evidence that the critical exponents of the Ising phase transition are consistent with the static critical exponents and do not depend on the coupling strength of the $R^2$ interaction term. We definitively can exclude for this type of model a behaviour as predicted by Boulatov and Kazakov [Phys. Lett. {\bf B186}, 379 (1987)] for Ising spins coupled to dynamically triangulated surfaces.