Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs

Jia, Yiyang; Verbaarschot, Jacobus M.

doi:10.1007/jhep11(2018)031

Title: Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs

Abstract

In this paper we explain the relation between the free energy of the SYK model for N Majorana fermions with a random q-body interaction and the moments of its spectral density. The high temperature expansion of the free energy gives the cumulants of the spectral density. Using that the cumulants are extensive we find the p dependence of the 1/N² correction of the 2p-th moments obtained in [1]. Conversely, the 1/N² corrections to the moments give the correction (even q) to the β⁶ coefficient of the high temperature expansion of the free energy for arbitrary q. Our result agrees with the 1/q³ correction obtained by Tarnopolsky using a mean field expansion. These considerations also lead to a more powerful method for solving the moment problem and intersection-graph enumeration problems. We take advantage of this and push the moment calculation to 1/N³ order and find surprisingly simple enumeration identities for intersection graphs. The 1/N³ corrections to the moments, give corrections to the β⁸ coefficient (for even q) of the high temperature expansion of the free energy which have not been calculated before. Results for odd q, where the SYK “Hamiltonian” is the supercharge of a supersymmetric theory are discussed as well.

Authors:

Jia, Yiyang ^[1];

^[1]

Stony Brook Univ., NY (United States)

Publication Date:: Tue Nov 06 00:00:00 EST 2018

Research Org.:: State Univ. of New York (SUNY), Albany, NY (United States)

Sponsoring Org.:: USDOE Office of Science (SC)

OSTI Identifier:: 1610128

Grant/Contract Number:: FG02-88ER40388; FAG-88FR40388

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: 2018; Journal Issue: 11; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics; 1/N Expansion; Matrix Models; Random Systems

Citation Formats


                    Jia, Yiyang, and Verbaarschot, Jacobus M. Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs.  United States: N. p., 2018. 
Web.  doi:10.1007/jhep11(2018)031.

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                    Jia, Yiyang, & Verbaarschot, Jacobus M. Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs.  United States.  https://doi.org/10.1007/jhep11(2018)031

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                    Jia, Yiyang, and Verbaarschot, Jacobus M. Tue .  
"Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs".  United States.  https://doi.org/10.1007/jhep11(2018)031.  https://www.osti.gov/servlets/purl/1610128.

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@article{osti_1610128,

  title        = {Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs},

  author       = {Jia, Yiyang and Verbaarschot, Jacobus M.},

  abstractNote = {In this paper we explain the relation between the free energy of the SYK model for N Majorana fermions with a random q-body interaction and the moments of its spectral density. The high temperature expansion of the free energy gives the cumulants of the spectral density. Using that the cumulants are extensive we find the p dependence of the 1/N2 correction of the 2p-th moments obtained in [1]. Conversely, the 1/N2 corrections to the moments give the correction (even q) to the β6 coefficient of the high temperature expansion of the free energy for arbitrary q. Our result agrees with the 1/q3 correction obtained by Tarnopolsky using a mean field expansion. These considerations also lead to a more powerful method for solving the moment problem and intersection-graph enumeration problems. We take advantage of this and push the moment calculation to 1/N3 order and find surprisingly simple enumeration identities for intersection graphs. The 1/N3 corrections to the moments, give corrections to the β8 coefficient (for even q) of the high temperature expansion of the free energy which have not been calculated before. Results for odd q, where the SYK “Hamiltonian” is the supercharge of a supersymmetric theory are discussed as well.},

  doi          = {10.1007/jhep11(2018)031},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 11,

  volume       = 2018,

  place        = {United States},

  year         = {Tue Nov 06 00:00:00 EST 2018},

  month        = {Tue Nov 06 00:00:00 EST 2018}

}

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Title: Large N expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs

Abstract

Citation Formats

All point correlation functions in SYK journal, December 2017

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

Bi-local holography in the SYK model: perturbations journal, November 2016

Supersymmetric SYK model and random matrix theory journal, June 2017

Spacing and individual eigenvalue distributions of two-body random hamiltonians journal, June 1971

The spectrum in the Sachdev-Ye-Kitaev model journal, April 2016

Chaos, complexity, and random matrices journal, November 2017

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

Nonuniversal Behavior of the k -Body Embedded Gaussian Unitary Ensemble of Random Matrices journal, June 2001

Spectral sum rules and Selberg's integral formula journal, June 1994

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

Three dimensional view of the SYK/AdS duality journal, September 2017

Quantum fluctuations of a nearly critical Heisenberg spin glass journal, March 2001

Bi-local holography in the SYK model journal, July 2016

Statistical properties of many-particle spectra journal, November 1975

Validity of random matrix theories for many-particle systems journal, December 1970

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

Quantum chaos and holographic tensor models journal, March 2017

A generalization of Sachdev-Ye-Kitaev journal, February 2017

Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N journal, September 2017

Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2 journal, April 2018

Quantum chaos and thermalization in isolated systems of interacting particles journal, April 2016

Many-body quantum chaos: Recent developments and applications to nuclei journal, March 2011

Power-law out of time order correlation functions in the SYK model journal, August 2017

Phase Transition in the Density of States of Quantum Spin Glasses journal, December 2014

A bound on chaos journal, August 2016

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

Complete random matrix classification of SYK models with N $$ \mathcal{N} $$ = 0, 1 and 2 supersymmetry journal, September 2017

Thermalization in the two-body random ensemble journal, October 2011

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

Chaos in AdS 2 Holography journal, September 2016

Analytic Combinatorics of Chord Diagrams book, January 2000

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model journal, May 2018

Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model journal, June 2018

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

Holographic Metals and the Fractionalized Fermi Liquid journal, October 2010

Systematics of nuclear level density parameters journal, October 2005

Embedded random matrix ensembles for complexity and chaos in finite interacting particle systems journal, June 2001

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

The Thouless time for mass-deformed SYK journal, September 2018

Moments and Cumulants in the Specification of Distributions journal, January 1938

Quantum ergodicity in the SYK model journal, May 2018

Towards a 2d QFT analog of the SYK model journal, October 2017

Fermionic localization of the schwarzian theory journal, October 2017

Space-time in the SYK model journal, July 2018

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

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A bound on chaos text, January 2015

The Spectrum in the Sachdev-Ye-Kitaev Model text, January 2016

Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles text, January 2016

Bi-Local Holography in the SYK Model text, January 2016

A Generalization of Sachdev-Ye-Kitaev text, January 2016

Disorder in the Sachdev-Yee-Kitaev Model text, January 2016

Fermionic Localization of the Schwarzian Theory text, January 2017

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All point correlation functions in SYK text, January 2017

Diffusion in higher dimensional SYK model with complex fermions text, January 2017

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

Systematics of nuclear level density parameters journal, September 2005

Thermalization in the Two-Body Random Ensemble text, January 2011

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Bi-Local Holography in the SYK Model: Perturbations preprint, January 2016

All point correlation functions in SYK
journal, December 2017

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

Bi-local holography in the SYK model: perturbations
journal, November 2016

Supersymmetric SYK model and random matrix theory
journal, June 2017

Spacing and individual eigenvalue distributions of two-body random hamiltonians
journal, June 1971

The spectrum in the Sachdev-Ye-Kitaev model
journal, April 2016

Chaos, complexity, and random matrices
journal, November 2017

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

Nonuniversal Behavior of the $k$ -Body Embedded Gaussian Unitary Ensemble of Random Matrices
journal, June 2001

Spectral sum rules and Selberg's integral formula
journal, June 1994

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

Three dimensional view of the SYK/AdS duality
journal, September 2017

Quantum fluctuations of a nearly critical Heisenberg spin glass
journal, March 2001

Bi-local holography in the SYK model
journal, July 2016

Statistical properties of many-particle spectra
journal, November 1975

Validity of random matrix theories for many-particle systems
journal, December 1970

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

Quantum chaos and holographic tensor models
journal, March 2017

A generalization of Sachdev-Ye-Kitaev
journal, February 2017

Analytical spectral density of the Sachdev-Ye-Kitaev model at finite $N$
journal, September 2017

Exact moments of the Sachdev-Ye-Kitaev model up to order 1/N2
journal, April 2018

Quantum chaos and thermalization in isolated systems of interacting particles
journal, April 2016

Many-body quantum chaos: Recent developments and applications to nuclei
journal, March 2011

Power-law out of time order correlation functions in the SYK model
journal, August 2017

Phase Transition in the Density of States of Quantum Spin Glasses
journal, December 2014

A bound on chaos
journal, August 2016

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

Complete random matrix classification of SYK models with N $$ \mathcal{N} $$ = 0, 1 and 2 supersymmetry
journal, September 2017

Thermalization in the two-body random ensemble
journal, October 2011

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

Chaos in ${AdS}_{2}$ Holography
journal, September 2016

Analytic Combinatorics of Chord Diagrams
book, January 2000

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model
journal, May 2018

Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model
journal, June 2018

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

Holographic Metals and the Fractionalized Fermi Liquid
journal, October 2010

Systematics of nuclear level density parameters
journal, October 2005

Embedded random matrix ensembles for complexity and chaos in finite interacting particle systems
journal, June 2001

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

The Thouless time for mass-deformed SYK
journal, September 2018

Moments and Cumulants in the Specification of Distributions
journal, January 1938

Quantum ergodicity in the SYK model
journal, May 2018

Towards a 2d QFT analog of the SYK model
journal, October 2017

Fermionic localization of the schwarzian theory
journal, October 2017

Space-time in the SYK model
journal, July 2018

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

Quantum ergodicity in the SYK model
text, January 2017

A bound on chaos
text, January 2015

The Spectrum in the Sachdev-Ye-Kitaev Model
text, January 2016

Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles
text, January 2016

Bi-Local Holography in the SYK Model
text, January 2016

A Generalization of Sachdev-Ye-Kitaev
text, January 2016

Disorder in the Sachdev-Yee-Kitaev Model
text, January 2016

Fermionic Localization of the Schwarzian Theory
text, January 2017

Three Dimensional View of the SYK/AdS Duality
text, January 2017

All point correlation functions in SYK
text, January 2017

Diffusion in higher dimensional SYK model with complex fermions
text, January 2017

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

Systematics of nuclear level density parameters
journal, September 2005

Thermalization in the Two-Body Random Ensemble
text, January 2011

Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
text, January 2016

Bi-Local Holography in the SYK Model: Perturbations
preprint, January 2016