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

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 $d\ge 2$ with $T_c\ge 0.4$. 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.

arXiv commentThe proof of the finitude of zero-energy contours in Appendix D.2 contains a technical error, where we failed to notice that opening a red bond in any of the four quadrants can possibly restrict opening of any blue bonds in the contour. This means that the probability that the contour is zero-energy (no blue-bonds) cannot be bounded exponentially, thus the result does not immediately follow