Abstract
A fermionic random matrix model, which is a 0-dimensional version of the SYK model with replicas, is considered. The replica-off-diagonal correlation functions vanish at finite N, but we show that they do not vanish in the large N limit due to spontaneous symmetry breaking. We use the Bogoliubov quasi-averages approach to studying phase transitions. The consideration may be relevant to the study of the problem of existence of the spin glass phase in fermionic models.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.D. Landau, On the theory of phase transitions, in Collected Papers. Volume 1, Nauka, Moscow Russia (1969), pp. 234-252 [Zh. Eksp. Teor. Fiz. 7 (1937) 19] [INSPIRE].
N.N. Bogoliubov, Lectures on quantum statistics. Volume 2: Quasi-Averages, Gordon and Breach Science Publishers (1970), republished in Collection of Scientific Works in Twelve Volumes: Statistical Mechanics. Volume 6, Nauka, Moscow Russia (2007).
Y. Nambu, Quasiparticles and Gauge Invariance in the Theory of Superconductivity, Phys. Rev.117 (1960) 648 [INSPIRE].
S. Weinberg, The quantum theory of fields. Volume 2: Modern applications, Cambridge University Press, Cambridge U.K. (1996).
M. Mezard, G. Parisi and M. Virasoro, Spin Glass Theory and beyond, World Scientific (1987).
J.S. Cotler et al., Black Holes and Random Matrices, JHEP05 (2017) 118 [Erratum JHEP09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
I. Aref’eva and I. Volovich, Notes on the SYK model in real time, Theor. Math. Phys.197 (2018)1650 [Teor. Mat. Fiz.197 (2018) 296] [arXiv:1801.08118] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett.70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography, talk given at the Entanglement in Strongly-Correlated Quantum Matter, Santa Barbara, California, U.S.A., 6 April-2 July 2015 and online at http://online.kitp.ucsb.edu/online/entangled15/.
S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev.X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].
W.F. Wreszinski and V.A. Zagrebnov, Bogoliubov quasi-averages: spontaneous symmetry breaking and algebra of fluctuations, Theor. Math. Phys.194 (2018) 157 [arXiv:1704.00190] [INSPIRE].
G. Gur-Ari, R. Mahajan and A. Vaezi, Does the SYK model have a spin glass phase?, JHEP11 (2018)070 [arXiv:1806.10145] [INSPIRE].
A. Georges, O. Parcollet and S. Sachdev, Quantum fluctuations of a nearly critical Heisenberg spin glass, Phys. Rev.B 63 (2001) 134406 [cond-mat/0009388].
J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP04 (2016)001 [arXiv:1601.06768] [INSPIRE].
W. Fu and S. Sachdev, Numerical study of fermion and boson models with infinite-range random interactions, Phys. Rev.B 94 (2016) 035135 [arXiv:1603.05246] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev.D 94 (2016)106002 [arXiv:1604.07818] [INSPIRE].
A.M. García-Garćıa and J.J.M. Verbaarschot, Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, Phys. Rev.D 94 (2016) 126010 [arXiv:1610.03816] [INSPIRE].
I. Aref’eva, M. Khramtsov, M. Tikhanovskaya and I. Volovich, Replica-nondiagonal solutions in the SYK model, JHEP07 (2019) 113 [arXiv:1811.04831] [INSPIRE].
V.V. Belokurov and E.T. Shavgulidze, Simple rules of functional integration in the Schwarzian theory: SYK correlators, arXiv:1811.11863 [INSPIRE].
H. Wang, D. Bagrets, A.L. Chudnovskiy and A. Kamenev, On the replica structure of Sachdev-Ye-Kitaev model, arXiv:1812.02666 [INSPIRE].
J. Kim, I.R. Klebanov, G. Tarnopolsky and W. Zhao, Symmetry Breaking in Coupled SYK or Tensor Models, Phys. Rev.X 9 (2019) 021043 [arXiv:1902.02287] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
A.M. Garc ıa-Garćıa, T. Nosaka, D. Rosa and J.J.M. Verbaarschot, Quantum chaos transition in a two-site Sachdev-Ye-Kitaev model dual to an eternal traversable wormhole, Phys. Rev.D 100 (2019) 026002 [arXiv:1901.06031] [INSPIRE].
V.S. Vladimirov and I. Volovich, Superanalysis. I. Differential Calculus, Theor. Math. Phys.59 (1984)317 [INSPIRE].
V.S. Vladimirov and I. Volovich, Superanalysis. II. Integral calculus, Theor. Math. Phys.60 (1984)743.
P. Deift, T. Kriecherbauer, K.T-R. Mclaughlin, S. Venakides and X. Zhou, Strong Asymptotics of Orthogonal Polynomials with Respect to Exponential Weights, Commun. Pure Appl. Math.52 (1999) 1491.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1902.09970
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Aref’eva, I., Volovich, I. Spontaneous symmetry breaking in fermionic random matrix model. J. High Energ. Phys. 2019, 114 (2019). https://doi.org/10.1007/JHEP10(2019)114
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)114