Cluster Percolation in the Two-Dimensional Ising Spin Glass

TL;DR

This study uses Monte Carlo simulations to analyze percolation properties of clusters in the 2D Edwards-Anderson Ising spin glass, linking cluster percolation to the spin-glass transition and providing new insights into geometric ordering phenomena.

Contribution

It demonstrates the percolation behavior of various clusters in the 2D spin glass and connects these findings to the zero-temperature spin-glass transition.

Findings

FKCK clusters percolate at a non-zero temperature.

Overlap-based clusters' percolation thresholds decrease with system size.

Spin-glass transition linked to density difference of largest clusters.

Abstract

Suitable cluster definitions have allowed researchers to describe many ordering transitions in spin systems as geometric phenomena related to percolation. For spin glasses and some other systems with quenched disorder, however, such a connection has not been fully established, and the numerical evidence remains incomplete. Here we use Monte Carlo simulations to study the percolation properties of several classes of clusters occurring in the Edwards-Anderson Ising spin-glass model in two dimensions. The Fortuin-Kasteleyn-Coniglio-Klein clusters originally defined for the ferromagnetic problem do percolate at a temperature that remains non-zero in the thermodynamic limit. On the Nishimori line, this location is accurately predicted by an argument due to Yamaguchi. More relevant for the spin-glass transition are clusters defined on the basis of the overlap of several replicas. We show that various such cluster types have percolation thresholds that shift to lower temperature by increasing the system size, in agreement with the zero-temperature spin-glass transition in two dimensions. The overlap is linked to the difference in density of the two largest clusters, thus supporting a picture where the spin-glass transition corresponds to an emergent density difference of the two largest clusters inside the percolating phase.