A Finite-temperature Phase Transition for the Ising Spin-glass in $d\geq 2$

TL;DR

This paper rigorously proves the existence of a finite-temperature phase transition in the Ising spin-glass model for dimensions two and higher, challenging previous beliefs about its ordering behavior.

Contribution

It introduces a graphical and contour-based method to establish a finite-temperature transition in the $ ext{±}J$ Ising spin-glass for $d extgreater= 2$, with a critical temperature $T_c extgreater= 0.4$.

Findings

Existence of a finite-temperature phase transition in $d extgreater= 2$

Coexistence of multiple infinite clusters with aligned spins

Clusters are entropically stable and negatively correlated

Abstract

It is believed that the $\pm J$ Ising spin-glass does not order at finite temperatures in dimension $d=2$. However, using a graphical representation and a contour argument, we prove rigorously the existence of a finite-temperature phase transition in $d\geq 2$ with $T_c \geq 0.4$. In the graphical representation, the low-temperature phase allows for the coexistence of multiple infinite clusters each with a rigidly aligned spin-overlap state. These clusters correlate negatively with each other, and are entropically stable without breaking any global symmetry. They can emerge in most graph structures and disorder measures.