# Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N

## TL;DR

This paper analytically derives the spectral density of the SYK model, showing agreement with Q-Hermite polynomials and random matrix theory, revealing universal features of quantum black holes and holographic duality.

## Contribution

It provides an analytical expression for the spectral density of the SYK model at finite N, connecting it to Q-Hermite polynomials and confirming universality with random matrix theory.

## Key findings

- Spectral density matches Q-Hermite polynomial predictions.
- Numerical results agree with analytical spectral density even at small N.
- Spectral density exhibits a square-root edge and exponential growth, characteristic of black holes.

## Abstract

We show analytically that the spectral density of the $q$-body Sachdeev-Ye-Kitaev (SYK) model agrees with that of Q-Hermite polynomials with Q a non-trivial function of $q \ge 2$ and the number of Majorana fermions $N \gg 1$. Numerical results, obtained by exact diagonalization, are in excellent agreement with the analytical spectral density even for relatively small $N \sim 8$. For $N \gg 1$ and not close to the edge of the spectrum, we find the macroscopic spectral density simplifies to $\rho(E) \sim \exp[2\arcsin^2(E/E_0)/\log \eta]$, where $\eta$ is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions. This spectral density reproduces the known result for the free energy in the large $q$ and $N$ limit. In the infrared region, where the SYK model is believed to have a gravity-dual, the spectral density is given by $\rho(E) \sim \sinh[2\pi \sqrt 2 \sqrt{(1-E/E_0)/(-\log \eta)}]$. It therefore has a square-root edge, as in random matrix ensembles, followed by an exponential growth, a distinctive feature of black holes and also of low energy nuclear excitations. Results for level-statistics in this region confirm the agreement with random matrix theory. Physically this is a signature that, for sufficiently long times, the SYK model and its gravity dual evolve to a fully ergodic state whose dynamics only depends on the global symmetry of the system. Our results strongly suggest that random matrix correlations are a universal feature of quantum black holes and that the SYK model, combined with holography, may be relevant to model certain aspects of the nuclear dynamics.