Topological order in matrix Ising models

TL;DR

This paper investigates matrix Ising models, revealing that in the large size limit they exhibit a topological phase transition characterized by a disconnected distribution of singular values, competing with glassy and ordered phases.

Contribution

It generalizes previous results to a broader class of matrix Ising models with specific symmetries, demonstrating topological large N phase transitions.

Findings

Topological phase transitions occur in matrix Ising models at large N.

The distribution of singular values becomes disconnected during the transition.

The transition competes with glassy and magnetic order phases.

Abstract

We study a family of models for an $N_1 \times N_2$ matrix worth of Ising spins $S_{aB}$. In the large $N_i$ limit we show that the spins soften, so that the partition function is described by a bosonic matrix integral with a single `spherical' constraint. In this way we generalize the results of [1] to a wide class of Ising Hamiltonians with $O(N_1,\mathbb{Z})\times O(N_2,\mathbb{Z})$ symmetry. The models can undergo topological large $N$ phase transitions in which the thermal expectation value of the distribution of singular values of the matrix $S_{aB}$ becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.