Topological order in matrix Ising models

Understand

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.