Domain size heterogeneity in the Ising model: geometrical and thermal transitions

TL;DR

This paper extends the measure of cluster size heterogeneity to thermal spin models in the Ising model, revealing a double peak in heterogeneity that detects both thermal and percolative transitions.

Contribution

It introduces a new application of heterogeneity measure to thermal clusters in the Ising model, uncovering a double-peak feature indicating two phase transitions.

Findings

Heterogeneity shows a double peak for geometric domains in 2D and 3D.

The measure detects both thermal and percolative transitions.

An alternative scaling interpretation without a new exponent is proposed.

Abstract

A measure of cluster size heterogeneity ($H$), introduced by Lee et al [Phys. Rev. E {\bf 84}, 020101 (2011)] in the context of explosive percolation, was recently applied to random percolation and to domains of parallel spins in the Ising and Potts models. It is defined as the average number of different domain sizes in a given configuration and a new exponent was introduced to explain its scaling with the size of the system. In thermal spin models, however, physical clusters take into account the temperature-dependent correlation between neighboring spins and encode the critical properties of the phase transition. We here extend the measure of $H$ to these clusters and, moreover, present new results for the geometric domains for both $d=2$ and 3. We show that the heterogeneity associated with geometric domains has a previously unnoticed double peak, thus being able to detect both the thermal and percolative transition. An alternative interpretation for the scaling of $H$ that does not introduce a new exponent is also proposed.