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Topological order in matrix Ising models

by Sean A. Hartnoll, Edward A. Mazenc, Zhengyan D. Shi

Submission summary

As Contributors: Sean Hartnoll
Arxiv Link: (pdf)
Date accepted: 2019-12-09
Date submitted: 2019-09-03 02:00
Submitted by: Hartnoll, Sean
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


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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Submission 1908.07058v1 on 3 September 2019

Reports on this Submission

Anonymous Report 1 on 2019-11-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1908.07058v1, delivered 2019-11-22, doi: 10.21468/SciPost.Report.1333


1. clear and interesting message
2. convincing analytic (matrix integrals) and numerical (Ising spins) analysis
3 discussion with wider context and outlook


Central theme of this study is what the authors call “self-erasure 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 free-out 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.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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