Spectral fluctuations in the Sachdev-Ye-Kitaev model
Abstract
Abstract We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loopmore »
- Authors:
-
- Stony Brook Univ., NY (United States)
- Publication Date:
- Research Org.:
- Stony Brook Univ., NY (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1804000
- Alternate Identifier(s):
- OSTI ID: 2280820
- Grant/Contract Number:
- FG02-88ER40388
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of High Energy Physics (Online)
- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2020; Journal Issue: 7; Journal ID: ISSN 1029-8479
- Publisher:
- Springer Nature
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics; Matrix models; Random systems; Field theories in lower dimensions; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS
Citation Formats
Jia, Yiyang, and Verbaarschot, Jacobus J. M. Spectral fluctuations in the Sachdev-Ye-Kitaev model. United States: N. p., 2020.
Web. doi:10.1007/jhep07(2020)193.
Jia, Yiyang, & Verbaarschot, Jacobus J. M. Spectral fluctuations in the Sachdev-Ye-Kitaev model. United States. https://doi.org/10.1007/jhep07(2020)193
Jia, Yiyang, and Verbaarschot, Jacobus J. M. Mon .
"Spectral fluctuations in the Sachdev-Ye-Kitaev model". United States. https://doi.org/10.1007/jhep07(2020)193. https://www.osti.gov/servlets/purl/1804000.
@article{osti_1804000,
title = {Spectral fluctuations in the Sachdev-Ye-Kitaev model},
author = {Jia, Yiyang and Verbaarschot, Jacobus J. M.},
abstractNote = {Abstract We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings for the number variance or exponentially short times in the case of the spectral form factor.},
doi = {10.1007/jhep07(2020)193},
journal = {Journal of High Energy Physics (Online)},
number = 7,
volume = 2020,
place = {United States},
year = {Mon Jul 27 00:00:00 EDT 2020},
month = {Mon Jul 27 00:00:00 EDT 2020}
}
Works referenced in this record:
Onset of random matrix behavior in scrambling systems
journal, July 2018
- Gharibyan, Hrant; Hanada, Masanori; Shenker, Stephen H.
- Journal of High Energy Physics, Vol. 2018, Issue 7
Universal Fluctuations in Spectra of the Lattice Dirac Operator
journal, May 1995
- Halasz, M. A.; Verbaarschot, J. J. M.
- Physical Review Letters, Vol. 74, Issue 20
Sachdev–Ye–Kitaev model as Liouville quantum mechanics
journal, October 2016
- Bagrets, Dmitry; Altland, Alexander; Kamenev, Alex
- Nuclear Physics B, Vol. 911
Majorana fermion quantum mechanics for higher rank tensors
journal, October 2019
- Klebanov, Igor R.; Pallegar, Preethi N.; Popov, Fedor K.
- Physical Review D, Vol. 100, Issue 8
NO2 jet cooled visible excitation spectrum: Vibronic chaos induced by the X̃ 2A1–Ã 2B2 interaction
journal, October 1991
- Delon, A.; Jost, R.; Lombardi, M.
- The Journal of Chemical Physics, Vol. 95, Issue 8
Notes on the complex Sachdev-Ye-Kitaev model
journal, February 2020
- Gu, Yingfei; Kitaev, Alexei; Sachdev, Subir
- Journal of High Energy Physics, Vol. 2020, Issue 2
Critique of the replica trick
journal, May 1985
- Verbaarschot, J. J. M.; Zirnbauer, M. R.
- Journal of Physics A: Mathematical and General, Vol. 18, Issue 7
Disorder and interaction in chiral chains: Majoranas versus complex fermions
journal, October 2019
- Karcher, J. F.; Sonner, M.; Mirlin, A. D.
- Physical Review B, Vol. 100, Issue 13
An SYK-like model without disorder
journal, October 2019
- Witten, Edward
- Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 47
Black holes and the butterfly effect
journal, March 2014
- Shenker, Stephen H.; Stanford, Douglas
- Journal of High Energy Physics, Vol. 2014, Issue 3
Probing typical black hole microstates
journal, January 2020
- de Boer, Jan; van Breukelen, Rik; Lokhande, Sagar F.
- Journal of High Energy Physics, Vol. 2020, Issue 1
Gapless spin-fluid ground state in a random quantum Heisenberg magnet
journal, May 1993
- Sachdev, Subir; Ye, Jinwu
- Physical Review Letters, Vol. 70, Issue 21
Replica variables, loop expansion, and spectral rigidity of random-matrix ensembles
journal, November 1984
- Verbaarschot, J. J. M.; Zirnbauer, M. R.
- Annals of Physics, Vol. 158, Issue 1
Solving the frustrated spherical model with q -polynomials
journal, April 1998
- Cappelli, Andrea; Colomo, Filippo
- Journal of Physics A: Mathematical and General, Vol. 31, Issue 14
Statistical properties of many-particle spectra
journal, November 1975
- Mon, K. K.; French, J. B.
- Annals of Physics, Vol. 95, Issue 1
Moments of the Gaussian β ensembles and the large- N expansion of the densities
journal, August 2014
- Witte, N. S.; Forrester, P. J.
- Journal of Mathematical Physics, Vol. 55, Issue 8
Statistical properties of many-particle spectra
journal, May 1979
- Pandey, Akhilesh
- Annals of Physics, Vol. 119, Issue 1
Some random-matrix level and spacing distributions for fixed-particle-rank interactions
journal, April 1971
- French, J. B.; Wong, S. S. M.
- Physics Letters B, Vol. 35, Issue 1
Review of the k -body embedded ensembles of Gaussian random matrices
journal, March 2003
- Benet, L.; Weidenm ller, H. A.
- Journal of Physics A: Mathematical and General, Vol. 36, Issue 12
Replica-nondiagonal solutions in the SYK model
journal, July 2019
- Aref’eva, Irina; Khramtsov, Mikhail; Tikhanovskaya, Maria
- Journal of High Energy Physics, Vol. 2019, Issue 7
Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models
journal, February 2017
- Klebanov, Igor R.; Tarnopolsky, Grigory
- Physical Review D, Vol. 95, Issue 4
Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
text, January 2018
- Berkooz, Micha; Narayan, Prithvi; Simon, Joan
- arXiv
Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws
journal, January 1984
- Bohigas, O.; Giannoni, M. J.; Schmit, C.
- Physical Review Letters, Vol. 52, Issue 1
Decay of a thermofield-double state in chaotic quantum systems: From random matrices to spin systems
journal, September 2018
- del Campo, A.; Molina-Vilaplana, J.; Santos, L. F.
- The European Physical Journal Special Topics, Vol. 227, Issue 3-4
On the replica structure of Sachdev-Ye-Kitaev model
journal, September 2019
- Wang, Hanteng; Bagrets, D.; Chudnovskiy, A. L.
- Journal of High Energy Physics, Vol. 2019, Issue 9
Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2
journal, April 2018
- García-García, Antonio M.; Jia, Yiyang; Verbaarschot, Jacobus J. M.
- Journal of High Energy Physics, Vol. 2018, Issue 4
Towards a full solution of the large N double-scaled SYK model
journal, March 2019
- Berkooz, Micha; Isachenkov, Mikhail; Narovlansky, Vladimir
- Journal of High Energy Physics, Vol. 2019, Issue 3
Thermoelectric power of Sachdev-Ye-Kitaev islands: Probing Bekenstein-Hawking entropy in quantum matter experiments
journal, May 2020
- Kruchkov, Alexander; Patel, Aavishkar A.; Kim, Philip
- Physical Review B, Vol. 101, Issue 20
Replica symmetry breaking in random matrix model: A toy model of wormhole networks
journal, April 2020
- Okuyama, Kazumi
- Physics Letters B, Vol. 803
Localization of interacting fermions at high temperature
journal, April 2007
- Oganesyan, Vadim; Huse, David A.
- Physical Review B, Vol. 75, Issue 15
Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs
journal, November 2018
- Jia, Yiyang; Verbaarschot, Jacobus J. M.
- Journal of High Energy Physics, Vol. 2018, Issue 11
Random Matrix Theory, Chiral Perturbation Theory, and Lattice Data
text, January 1999
- Berbenni-Bitsch, M. E.; Göckeler, M.; Hehl, H.
- arXiv
Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
journal, December 2016
- García-García, Antonio M.; Verbaarschot, Jacobus J. M.
- Physical Review D, Vol. 94, Issue 12
Remarks on the Sachdev-Ye-Kitaev model
journal, November 2016
- Maldacena, Juan; Stanford, Douglas
- Physical Review D, Vol. 94, Issue 10
Random-matrix physics: spectrum and strength fluctuations
journal, July 1981
- Brody, T. A.; Flores, J.; French, J. B.
- Reviews of Modern Physics, Vol. 53, Issue 3
Eigenstate thermalization in the Sachdev-Ye-Kitaev model
journal, November 2017
- Sonner, Julian; Vielma, Manuel
- Journal of High Energy Physics, Vol. 2017, Issue 11
Sachdev-Ye-Kitaev Non-Fermi-Liquid Correlations in Nanoscopic Quantum Transport
journal, November 2019
- Altland, Alexander; Bagrets, Dmitry; Kamenev, Alex
- Physical Review Letters, Vol. 123, Issue 22
Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model
journal, May 2018
- García-García, Antonio M.; Jia, Yiyang; Verbaarschot, Jacobus J. M.
- Physical Review D, Vol. 97, Issue 10
The Combinatorics of q-Hermite polynomials and the Askey—Wilson Integral
journal, October 1987
- Ismail, Mourad E. H.; Stanton, Dennis; Viennot, Gérard
- European Journal of Combinatorics, Vol. 8, Issue 4
Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model
journal, June 2018
- García-García, Antonio M.; Loureiro, Bruno; Romero-Bermúdez, Aurelio
- Physical Review Letters, Vol. 120, Issue 24
Sachdev-Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states
journal, March 2017
- You, Yi-Zhuang; Ludwig, Andreas W. W.; Xu, Cenke
- Physical Review B, Vol. 95, Issue 11
From chiral random matrix theory to chiral perturbation theory
journal, February 1999
- Osborn, J. C.; Toublan, D.; Verbaarschot, J. J. M.
- Nuclear Physics B, Vol. 540, Issue 1-2
Holographic Metals and the Fractionalized Fermi Liquid
journal, October 2010
- Sachdev, Subir
- Physical Review Letters, Vol. 105, Issue 15
Disorder in the Sachdev–Ye–Kitaev model
journal, October 2017
- Liu, Yizhuang; Nowak, Maciej A.; Zahed, Ismail
- Physics Letters B, Vol. 773
An Attempt to Calculate the Number of Energy Levels of a Heavy Nucleus
journal, August 1936
- Bethe, H. A.
- Physical Review, Vol. 50, Issue 4
An exact quantization of Jackiw-Teitelboim gravity
journal, November 2019
- Iliesiu, Luca V.; Pufu, Silviu S.; Verlinde, Herman
- Journal of High Energy Physics, Vol. 2019, Issue 11
The Thouless time for mass-deformed SYK
journal, September 2018
- Nosaka, Tomoki; Rosa, Dario; Yoon, Junggi
- Journal of High Energy Physics, Vol. 2018, Issue 9
Quantum ergodicity in the SYK model
journal, May 2018
- Altland, Alexander; Bagrets, Dmitry
- Nuclear Physics B, Vol. 930
Random matrix theory, chiral perturbation theory, and lattice data
journal, November 1999
- Berbenni-Bitsch, M. E.; Göckeler, M.; Hehl, H.
- Physics Letters B, Vol. 466, Issue 2-4
Fermionic localization of the schwarzian theory
journal, October 2017
- Stanford, Douglas; Witten, Edward
- Journal of High Energy Physics, Vol. 2017, Issue 10
Solving the Frustrated Spherical Model with q-Polynomials
text, January 1997
- Cappelli, Andrea; Colomo, Filippo
- arXiv
Spectral statistics of the -body random-interaction model
journal, October 2002
- Srednicki, Mark
- Physical Review E, Vol. 66, Issue 4
Universal fluctuations in spectra of the lattice Dirac operator
text, January 1995
- Halasz, M. A.; Verbaarschot, J. J. M.
- arXiv
Spectral statistics of the two-body random ensemble revisited
journal, January 2001
- Flores, J.; Horoi, M.; Müller, M.
- Physical Review E, Vol. 63, Issue 2
Two-body random hamiltonian and level density
journal, March 1971
- Bohigas, O.; Flores, J.
- Physics Letters B, Vol. 34, Issue 4
Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
journal, August 2018
- Berkooz, Micha; Narayan, Prithvi; Simón, Joan
- Journal of High Energy Physics, Vol. 2018, Issue 8
Works referencing / citing this record: