Topological order in matrix Ising models
Abstract
We study a family of models for anN_1 \times N_2 ${N}_{1}\times {N}_{2}$matrix worth of Ising spinsS_{aB} ${S}_{aB}$.In the largeN_i ${N}_{i}$limit we show that the spins soften, so that the partition function isdescribed by a bosonic matrix integral with a single ‘spherical’constraint. In this way we generalize the results of to a wide class ofIsing Hamiltonians withO(N_1,\mathbb{Z})\times O(N_2,\mathbb{Z}) $O({N}_{1},Z)\times O({N}_{2},Z)$symmetry. The models can undergo topological largeN $N$phase transitions in which the thermal expectation value of thedistribution of singular values of the matrixS_{aB} ${S}_{aB}$becomes disconnected. This topological transition competes with lowtemperature glassy and magnetically ordered phases.
 Authors:

 Stanford Univ., CA (United States). Dept. of Physics
 Publication Date:
 Research Org.:
 Stanford Univ., CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC)
 OSTI Identifier:
 1579979
 Alternate Identifier(s):
 OSTI ID: 1803258
 Grant/Contract Number:
 SC0018134
 Resource Type:
 Published Article
 Journal Name:
 SciPost Physics
 Additional Journal Information:
 Journal Volume: 7; Journal Issue: 6; Journal ID: ISSN 25424653
 Publisher:
 SciPost Foundation
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics
Citation Formats
Hartnoll, Sean, Mazenc, Edward, and Shi, Zhengyan. Topological order in matrix Ising models. United States: N. p., 2019.
Web. doi:10.21468/scipostphys.7.6.081.
Hartnoll, Sean, Mazenc, Edward, & Shi, Zhengyan. Topological order in matrix Ising models. United States. https://doi.org/10.21468/scipostphys.7.6.081
Hartnoll, Sean, Mazenc, Edward, and Shi, Zhengyan. Thu .
"Topological order in matrix Ising models". United States. https://doi.org/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 anN_1 \times N_2N1×N2matrix worth of Ising spinsS_{aB}SaB.In the largeN_iNilimit we show that the spins soften, so that the partition function isdescribed by a bosonic matrix integral with a single ‘spherical’constraint. In this way we generalize the results of to a wide class ofIsing Hamiltonians withO(N_1,\mathbb{Z})\times O(N_2,\mathbb{Z})O(N1,Z)×O(N2,Z)symmetry. The models can undergo topological largeNNphase transitions in which the thermal expectation value of thedistribution of singular values of the matrixS_{aB}SaBbecomes disconnected. This topological transition competes with lowtemperature glassy and magnetically ordered phases.},
doi = {10.21468/scipostphys.7.6.081},
journal = {SciPost Physics},
number = 6,
volume = 7,
place = {United States},
year = {2019},
month = {12}
}
https://doi.org/10.21468/scipostphys.7.6.081
Web of Science
Works referenced in this record:
TASI Lectures on Complex Structures
conference, March 2012
 Denef, Frederik
 Proceedings of the 2010 Theoretical Advanced Study Institute in Elementary Particle Physics, String Theory and Its Applications
Spherical Model as the Limit of Infinite Spin Dimensionality
journal, December 1968
 Stanley, H. E.
 Physical Review, Vol. 176, Issue 2
Large n Phase Transitions in low Dimensions
journal, May 1986
 Cicuta, G. M.; Molinari, L.; Montaldi, E.
 Modern Physics Letters A, Vol. 01, Issue 02
Possible thirdorder phase transition in the large $N$ lattice gauge theory
journal, January 1980
 Gross, David J.; Witten, Edward
 Physical Review D, Vol. 21, Issue 2
The Spherical Model of a Ferromagnet
journal, June 1952
 Berlin, T. H.; Kac, M.
 Physical Review, Vol. 86, Issue 6
Replica theory of quantum spin glasses
journal, August 1980
 Bray, A. J.; Moore, M. A.
 Journal of Physics C: Solid State Physics, Vol. 13, Issue 24
Planar diagrams
journal, February 1978
 Brézin, E.; Itzykson, C.; Parisi, G.
 Communications in Mathematical Physics, Vol. 59, Issue 1
Replica field theory for deterministic models. II. A nonrandom spin glass with glassy behaviour
journal, December 1994
 Marinari, E.; Parisi, G.; Ritort, F.
 Journal of Physics A: Mathematical and General, Vol. 27, Issue 23
N = ∞ phase transition in a class of exactly soluble model lattice gauge theories
journal, June 1980
 Wadia, Spenta R.
 Physics Letters B, Vol. 93, Issue 4
BekensteinHawking Entropy and Strange Metals
journal, November 2015
 Sachdev, Subir
 Physical Review X, Vol. 5, Issue 4
Matrix models, topological strings, and supersymmetric gauge theories
journal, November 2002
 Dijkgraaf, Robbert; Vafa, Cumrun
 Nuclear Physics B, Vol. 644, Issue 12
Minimal string theory
journal, March 2005
 Seiberg, Nathan; Shih, David
 Comptes Rendus Physique, Vol. 6, Issue 2
Large rectangular random matrices
journal, August 1987
 Cicuta, G. M.; Molinari, L.; Montaldi, E.
 Journal of Mathematical Physics, Vol. 28, Issue 8
Large N matrices from a nonlocal spin system
journal, September 2015
 Anninos, Dionysios; Hartnoll, Sean A.; Huijse, Liza
 Classical and Quantum Gravity, Vol. 32, Issue 19
Breakdown of universality in multicut matrix models
journal, September 2000
 Bonnet, G.; David, F.; Eynard, B.
 Journal of Physics A: Mathematical and General, Vol. 33, Issue 38
Matrix Models as Solvable Glass Models
journal, February 1995
 Cugliandolo, L. F.; Kurchan, J.; Parisi, G.
 Physical Review Letters, Vol. 74, Issue 6