# Topological order in matrix Ising models

## Abstract

We study a family of models for an N_1 \times N_2 ${N}_{1}\times {N}_{2}$ matrix worth of Ising spins S_{aB} ${S}_{aB}$ . In the large N_i ${N}_{i}$ 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 to a wide class of Ising Hamiltonians with O(N_1,\mathbb{Z})\times O(N_2,\mathbb{Z}) $O({N}_{1},\mathbb{Z})\times O({N}_{2},\mathbb{Z})$ symmetry. The models can undergo topological large N $N$ phase transitions in which the thermal expectation value of the distribution of singular values of the matrix S_{aB} ${S}_{aB}$ becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.

- Authors:

- Stanford University

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1579979

- Grant/Contract Number:
- SC0018134

- Resource Type:
- Published Article

- Journal Name:
- SciPost Physics Proceedings

- Additional Journal Information:
- Journal Name: SciPost Physics Proceedings Journal Volume: 7 Journal Issue: 6; Journal ID: ISSN 2542-4653

- Publisher:
- Stichting SciPost

- Country of Publication:
- Netherlands

- Language:
- English

### Citation Formats

```
Hartnoll, Sean, Mazenc, Edward, and Shi, Zhengyan. Topological order in matrix Ising models. Netherlands: N. p., 2019.
Web. doi:10.21468/SciPostPhys.7.6.081.
```

```
Hartnoll, Sean, Mazenc, Edward, & Shi, Zhengyan. Topological order in matrix Ising models. Netherlands. doi:10.21468/SciPostPhys.7.6.081.
```

```
Hartnoll, Sean, Mazenc, Edward, and Shi, Zhengyan. Thu .
"Topological order in matrix Ising models". Netherlands. doi:10.21468/SciPostPhys.7.6.081.
```

```
@article{osti_1579979,
```

title = {Topological order in matrix Ising models},

author = {Hartnoll, Sean and Mazenc, Edward and Shi, Zhengyan},

abstractNote = {We study a family of models for an N_1 \times N_2 N 1 × N 2 matrix worth of Ising spins S_{aB} S a B . In the large N_i 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 to a wide class of Ising Hamiltonians with O(N_1,\mathbb{Z})\times O(N_2,\mathbb{Z}) O ( N 1 , ℤ ) × O ( N 2 , ℤ ) symmetry. The models can undergo topological large N N phase transitions in which the thermal expectation value of the distribution of singular values of the matrix S_{aB} S a B becomes disconnected. This topological transition competes with low temperature glassy and magnetically ordered phases.},

doi = {10.21468/SciPostPhys.7.6.081},

journal = {SciPost Physics Proceedings},

number = 6,

volume = 7,

place = {Netherlands},

year = {2019},

month = {12}

}

DOI: 10.21468/SciPostPhys.7.6.081

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