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 25424653
 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:https://doi.org/10.21468/SciPostPhys.7.6.081
Hartnoll, Sean, Mazenc, Edward, and Shi, Zhengyan. Thu .
"Topological order in matrix Ising models". Netherlands. doi: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 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: https://doi.org/10.21468/SciPostPhys.7.6.081
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
Matrix Models as Solvable Glass Models
journal, February 1995
 Cugliandolo, L. F.; Kurchan, J.; Parisi, G.
 Physical Review Letters, Vol. 74, Issue 6
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
Matrix quantum mechanics from qubits
journal, January 2017
 Hartnoll, Sean A.; Huijse, Liza; Mazenc, Edward A.
 Journal of High Energy Physics, Vol. 2017, Issue 1
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