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 $1/N^2$ correction of the $2p$-th moments obtained in 1801.02696. Conversely, the $1/N^2$ corrections to the moments give the correction (even $q$) to the ${\beta }^6$ coefficient of the high temperature expansion of the free energy for arbitrary $q$. Our result agrees with the $1/q^3$ 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 $1/N^3$ order and find surprisingly simple enumeration identities for intersection graphs. The $1/N^3$ corrections to the moments, give corrections to the ${\beta }^8$ 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.

References in corpus (41)

• A bound on chaos
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Comments on the Sachdev-Ye-Kitaev model
• Chaos in AdS${}_2$ holography
• The Spectrum in the Sachdev-Ye-Kitaev Model
• Black Holes and Random Matrices
• Delta M_K and epsilon_K in SUSY at the Next-to-Leading order
• Holographic metals and the fractionalized Fermi liquid
• Fermionic Localization of the Schwarzian Theory
• Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles
• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• A Generalization of Sachdev-Ye-Kitaev
• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
• Chaos, Complexity, and Random Matrices
• Quantum Fluctuations of a Nearly Critical Heisenberg Spin Glass
• Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States
• Bi-Local Holography in the SYK Model
• Power-law out of time order correlation functions in the SYK model
• Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N
• All point correlation functions in SYK
• Supersymmetric SYK model and random matrix theory
• Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model
• Quantum Chaos and Holographic Tensor Models
• Towards a 2d QFT Analog of the SYK Model
• Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
• Three Dimensional View of the SYK/AdS Duality
• Phase Transition in the Density of States of Quantum Spin Glasses
• Quantum ergodicity in the SYK model
• Review of the k-Body Embedded Ensembles of Gaussian Random Matrices
• Complete random matrix classification of SYK models with $\mathcal{N}=0$, $1$ and $2$ supersymmetry
• Space-Time in the SYK Model
• Spectral sum rules and Selberg's integral
• Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev Model
• Eigenstate entanglement in the Sachdev-Ye-Kitaev model
• Thouless time for mass-deformed SYK
• Non-Universal Behavior of the k-Body Embedded Gaussian Unitary Ensemble of Random Matrices
• Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$
• Diffusion in higher dimensional SYK model with complex fermions
• Disorder in the Sachdev-Yee-Kitaev Model
• Thermalization in the Two-Body Random Ensemble

Cited by in corpus (15)

• Towards a full solution of the large N double-scaled SYK model
• Krylov complexity in large-$q$ and double-scaled SYK model
• Spectral Fluctuations in the Sachdev-Ye-Kitaev Model
• On large $q$ expansion in the Sachdev-Ye-Kitaev model
• Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
• Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model
• Wilson loops in terms of color invariants
• Chaos on the hypercube
• Violation of the viscosity/entropy bound in translationally invariant non-Fermi liquids
• Near Conformal Perturbation Theory in SYK Type Models
• Parisi's hypercube, Fock-space frustration and near-AdS${}_2$/near-CFT${}_1$ holography
• Multi-trace Correlators in the SYK Model and Non-geometric Wormholes
• Parisi's hypercube, Fock-space fluxes, and the microscopics of near-AdS${}_2$/near-CFT${}_1$ duality
• Superconducting gap ratio from strange metal phase in the absence of quasiparticles
• SYK model with an extra diagonal perturbation: phase transition in the eigenvalue spectrum