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

We derive an approximate analytical formula for the spectral density of the q-body Sachdev-Ye-Kitaev (SYK) model obtained by summing a class of diagrams representing leading intersecting contractions. This expression agrees with that of Q-Hermite polynomials, with Q a nontrivial function of q2 and the number of Majorana fermions N. Numerical results, obtained by exact diagonalization, are in excellent agreement with this approximate analytical spectral density even for relatively small N8. For N1 and not close to the edge of the spectrum, we find that the approximate analytical spectral density simplifies to ₀ₒₘ₌ (E) =exp2arcsin^2 (E/E₀) /log, where (N, q) is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions and E₀ is the ground-state energy per particle. This spectral density reproduces the known result for the free energy in the large-q and large-N limit at arbitrary values of the temperature. In the infrared region, where the SYK model is believed to have a gravity dual, the analytical spectral density is given by (E) 22 (1-E/{E₀) / (-log) }. 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 modeling certain aspects of the nuclear dynamics.