Analytical spectral density of the Sachdev-Ye-Kitaev model at finite N

By Antonio M. Garc'ia-Garc'ia, J. Verbaarschot. 2017

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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.

Subfields: Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c146b4cbcdd3bf6432064b587c050e0bdffc6203

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Notes on the complex Sachdev-Ye-Kitaev model (2019, background): “The density of states (DOS) and the partition function for the pure Schwarzian model are as follows [33–36], where the energy E and the temperature β−1 are measured in units of (αSN) −1:”