Spectral Properties of the SYK Model

Abstract

The Sachedev-Ye-Kitaev model provides us with a new window on strongly interacting many-body systems. In particular, it is a variant of the two-body random ensemble which captures the main features of nuclear interactions. One of main successes of recent work on the SYK model is that it shows that the Bethe formula for the nuclear level density is a direct consequence of the conformal symmetry of the low-energy sector of this model. We study the spectrum of the q-body SYK model by means of the moment method, and have obtained analytical results for all moments up to order 1/ N^3 in the number of Majorana fermions N. We elucidate the structure of the moments and how they relate to the free energy including $1/q$ corrections. For fixed $q^2/N$, the spectral density is given by the weight function of the Q-Hermite polynomials, which for large $N$ and $q$ simplifies to $exp( -N arcsin^2(E/E_0) / q^2)$. This spectral form reproduces the free energy obtained by path integral methods in the same limit. From applications to nuclear physics, it is clear that the SYK model has to be chaotic, which is also one of the main reasons why it is of interest to black hole physics. We study the chaotic dynamics by means of the number variance and the closely related spectral form factor. We show that the asymptotic quadratic dependence of the number variance results in a $1/t^2$ peak of the spectral form factor for short times, while random matrix spectral statistics is found for longer time scales.