Spectral Fluctuations in the Sachdev-Ye-Kitaev Model

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 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 the scale fluctuations for which we give a simple 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 is 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 using a nonlinear $\sigma$-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlator 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 $\sigma$-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for $q=4$ but also for the $q=2$ SYK model while the correct result in this case should have been Poisson statistics. We analyze the number variance and spectral form factor for $N=32$ and $q=4$ numerically. 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 or exponentially short times, respectively.

References in corpus (51)

• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Localization of interacting fermions at high temperature
• Comments on the Sachdev-Ye-Kitaev model
• Black holes and the butterfly effect
• Black Holes and Random Matrices
• Bekenstein-Hawking Entropy and Strange Metals
• 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
• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
• Eternal traversable wormhole
• A semiclassical ramp in SYK and in gravity
• Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States
• Notes on the complex Sachdev-Ye-Kitaev model
• From chiral Random Matrix Theory to chiral Perturbation Theory
• Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N
• Onset of Random Matrix Behavior in Scrambling Systems
• Towards a full solution of the large N double-scaled SYK model
• Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model
• Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
• Quantum chaos transition in a two-site SYK model dual to an eternal traversable wormhole
• An exact quantization of Jackiw-Teitelboim gravity
• Universal fluctuations in spectra of the lattice Dirac operator
• Eigenstate thermalization in the Sachdev-Ye-Kitaev model
• Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity
• Quantum ergodicity in the SYK model
• Review of the k-Body Embedded Ensembles of Gaussian Random Matrices
• Pure states in the SYK model and nearly-$Ad{S}_2$ gravity
• Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev Model
• SYK non Fermi Liquid Correlations in Nanoscopic Quantum Transport
• Spectral Statistics of the Two-Body Random Ensemble Revisited
• Thouless time for mass-deformed SYK
• Timescales in the quench dynamics of many-body quantum systems: Participation ratio vs out-of-time ordered correlator
• Moments of the Gaussian $\beta$ Ensembles and the large-$N$ expansion of the densities
• Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$
• Thermoelectric power of Sachdev-Ye-Kitaev islands: Probing Bekenstein-Hawking entropy in quantum matter experiments
• Disorder in the Sachdev-Yee-Kitaev Model
• Spectral decoupling in many-body quantum chaos
• Replica-nondiagonal solutions in the SYK model
• Decay of a Thermofield-Double State in Chaotic Quantum Systems
• Majorana Fermion Quantum Mechanics for Higher Rank Tensors
• Probing typical black hole microstates
• On the replica structure of Sachdev-Ye-Kitaev model
• Random Matrix Theory, Chiral Perturbation Theory, and Lattice Data
• Disorder and interaction in chiral chains: Majoranas vs complex fermions
• Replica symmetry breaking in random matrix model: a toy model of wormhole networks
• Large $N$ expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs
• Spectral statistics of the k-body random-interaction model
• Solving the Frustrated Spherical Model with q-Polynomials

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• Towards a String Dual of SYK
• Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials
• Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions
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• Bosonic near-CFT${}_1$ models from Fock-space fluxes