Topological order in matrix Ising models

We study a family of models for an N-1 x 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, Z) x O(N-2, 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 SaB becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.