SciPost Submission Page
Topological order in matrix Ising models
by Sean A. Hartnoll, Edward A. Mazenc, Zhengyan D. Shi
 Published as SciPost Phys. 7, 081 (2019)
Submission summary
As Contributors:  Sean Hartnoll 
Arxiv Link:  https://arxiv.org/abs/1908.07058v1 (pdf) 
Date accepted:  20191209 
Date submitted:  20190903 02:00 
Submitted by:  Hartnoll, Sean 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
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.
Published as SciPost Phys. 7, 081 (2019)
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 20191122 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1908.07058v1, delivered 20191122, doi: 10.21468/SciPost.Report.1333
Strengths
1. clear and interesting message
2. convincing analytic (matrix integrals) and numerical (Ising spins) analysis
3 discussion with wider context and outlook
Report
Central theme of this study is what the authors call “selferasure of discreteness”. In the concrete model studied, this amounts to a collection of $N_1 N_2$ Ising spins being described by continuous bosonic degrees of freedom with a single constraint. With this spin softening in place, topological large $N$ phase transitions are expected, and the authors indeed establish them in various guises, depending on a choice of interaction potential. Importantly, the topological phase transitions happen at temperatures well above a glassy freeout transition, where the matrix integral loses its relevance for the problem.
The manuscript brings a very clear and interesting message and the underlying analysis (analytics for the matrix integrals and Monte Carlo simulations on the spin systems) is convincing.
The discussion in section 5 reveals some of the true motivations of the authors: a deeper connection with gravitational physics via an extension to the quantum case, building on ref [9] by (in part) the same authors.