Topological order in matrix Ising models

Abstract

We study a family of models for an N_1 \times N_2N1×N2 matrix worth of Ising spins S_{aB}SaB. In the large N_iNi 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 to a wide class of Ising Hamiltonians with O(N_1,\mathbb{Z})\times O(N_2,\mathbb{Z})O(N1,ℤ)×O(N2,ℤ) symmetry. The models can undergo topological large NN phase transitions in which the thermal expectation value of the distribution of singular values of the matrix S_{aB}SaB becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.