Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: 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 » 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.« less

Authors:
ORCiD logo [1]; ORCiD logo [1]
+ Show Author Affiliations
  1. 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.
Copy to clipboard
Jia, Yiyang, & Verbaarschot, Jacobus J. M. Spectral fluctuations in the Sachdev-Ye-Kitaev model. United States. https://doi.org/10.1007/jhep07(2020)193
Copy to clipboard
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.
Copy to clipboard
@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}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1007/jhep07(2020)193
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

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
  • DOI: 10.1007/JHEP07(2018)124

Universal Fluctuations in Spectra of the Lattice Dirac Operator
journal, May 1995

Sachdev–Ye–Kitaev model as Liouville quantum mechanics
journal, October 2016

Majorana fermion quantum mechanics for higher rank tensors
journal, October 2019

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
  • DOI: 10.1063/1.461620

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
  • DOI: 10.1007/JHEP02(2020)157

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
  • DOI: 10.1088/0305-4470/18/7/018

Disorder and interaction in chiral chains: Majoranas versus complex fermions
journal, October 2019

An SYK-like model without disorder
journal, October 2019

Black holes and the butterfly effect
journal, March 2014

  • Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2014, Issue 3
  • DOI: 10.1007/JHEP03(2014)067

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
  • DOI: 10.1007/JHEP01(2020)062

Gapless spin-fluid ground state in a random quantum Heisenberg magnet
journal, May 1993

Replica variables, loop expansion, and spectral rigidity of random-matrix ensembles
journal, November 1984

Solving the frustrated spherical model with q -polynomials
journal, April 1998

Statistical properties of many-particle spectra
journal, November 1975

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
  • DOI: 10.1063/1.4886477

Statistical properties of many-particle spectra
journal, May 1979

Some random-matrix level and spacing distributions for fixed-particle-rank interactions
journal, April 1971

Review of the k -body embedded ensembles of Gaussian random matrices
journal, March 2003

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
  • DOI: 10.1007/JHEP07(2019)113

Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models
journal, February 2017

Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
text, January 2018

Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws
journal, January 1984

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
  • DOI: 10.1140/epjst/e2018-00083-5

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
  • DOI: 10.1007/JHEP09(2019)057

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
  • DOI: 10.1007/JHEP04(2018)146

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
  • DOI: 10.1007/jhep03(2019)079

Thermoelectric power of Sachdev-Ye-Kitaev islands: Probing Bekenstein-Hawking entropy in quantum matter experiments
journal, May 2020

Localization of interacting fermions at high temperature
journal, April 2007

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
  • DOI: 10.1007/JHEP11(2018)031

Random Matrix Theory, Chiral Perturbation Theory, and Lattice Data
text, January 1999

Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
journal, December 2016

Remarks on the Sachdev-Ye-Kitaev model
journal, November 2016

Random-matrix physics: spectrum and strength fluctuations
journal, July 1981

Eigenstate thermalization in the Sachdev-Ye-Kitaev model
journal, November 2017

Sachdev-Ye-Kitaev Non-Fermi-Liquid Correlations in Nanoscopic Quantum Transport
journal, November 2019

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
  • DOI: 10.1103/PhysRevD.97.106003

The Combinatorics of q-Hermite polynomials and the Askey—Wilson Integral
journal, October 1987

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
  • DOI: 10.1103/PhysRevLett.120.241603

Sachdev-Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states
journal, March 2017

From chiral random matrix theory to chiral perturbation theory
journal, February 1999

Holographic Metals and the Fractionalized Fermi Liquid
journal, October 2010

Disorder in the Sachdev–Ye–Kitaev model
journal, October 2017

An Attempt to Calculate the Number of Energy Levels of a Heavy Nucleus
journal, August 1936

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
  • DOI: 10.1007/JHEP11(2019)091

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
  • DOI: 10.1007/JHEP09(2018)041

Quantum ergodicity in the SYK model
journal, May 2018

Random matrix theory, chiral perturbation theory, and lattice data
journal, November 1999

Fermionic localization of the schwarzian theory
journal, October 2017

  • Stanford, Douglas; Witten, Edward
  • Journal of High Energy Physics, Vol. 2017, Issue 10
  • DOI: 10.1007/JHEP10(2017)008

Solving the Frustrated Spherical Model with q-Polynomials
text, January 1997

Spectral statistics of the k -body random-interaction model
journal, October 2002

Universal fluctuations in spectra of the lattice Dirac operator
text, January 1995

Spectral statistics of the two-body random ensemble revisited
journal, January 2001

Two-body random hamiltonian and level density
journal, March 1971

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
  • DOI: 10.1007/JHEP08(2018)192
    Sort by:
    [ × clear filter / sort ]

    Works referencing / citing this record:

    Towards a String Dual of SYK
    preprint, January 2021

      Sort by:
      [ × clear filter / sort ]

      Similar Records in DOE PAGES and OSTI.GOV collections: