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 q≥2 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 N∼8. For N1 and not close to the edge of the spectrum, we find that the approximate analytical spectral density simplifies to ρasym(E)=exp[2arcsin2(E/E0)/logη], where η(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 ρ(E)∼sinh[2π2(1-E/E0)/(-logη)]. 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.