Phase transitions in 2d orders coupled to the Ising model

TL;DR

This paper investigates phase transitions in 2D orders coupled with the Ising model, revealing complex behaviors including mixed and continuous transitions, and analyzing their scaling properties.

Contribution

It provides the first detailed analysis of phase transition orders and scaling behavior in 2D orders coupled to the Ising model, including new geometric phases.

Findings

Transition at positive beta appears mixed order

Transitions at negative beta are continuous and first order for Ising and geometry respectively

Scaling of observables with system size shows fractional scaling

Abstract

The $2$d orders are a sub class of causal sets, which is especially amenable to computer simulations. Past work has shown that the $2$d orders have a first order phase transition between a random and a crystalline phase. When coupling the $2$d orders to the Ising model, this phase transition coincides with the transition of the Ising model. The coupled system also shows a new phase, at negative $\beta$, where the Ising model induces the geometric transition. In this article we examine the phase transitions of the coupled system, to determine their order, as well as how they scale when the system size is changed. We find that the transition at positive $\beta$ seems to be of mixed order, while the two transitions at negative $\beta$ appear continous/ first order for the Ising model/ the geometry respectively. The scaling of the observables with the system size on the other hand is fairly simple, and does, where applicable, agree with that found for the pure $2$d orders. We find that the location of these transitions has fractional scaling in the system size.