Remarks on the Sachdev-Ye-Kitaev model

We study a quantum-mechanical model proposed by Sachdev, Ye and Kitaev. The model consists of N Majorana fermions with random interactions of a few fermions at a time. It it tractable in the large-N limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies where it develops an emergent conformal symmetry. We study two-and four-point functions of the fundamental fermions. This provides the spectrum of physical excitations for the bilocal field. The emergent conformal symmetry is a reparametrization symmetry, which is spontaneously broken to SL(2,R), leading to zero modes. These zero modes are lifted by a small residual explicit breaking, which produces an enhanced contribution to the four-point function. This contribution displays a maximal Lyapunov exponent in the chaos region (out-of-time-ordered correlator). We expect these features to be universal properties of large-N quantum mechanics systems with emergent reparametrization symmetry. This article is largely based on talks given by Kitaev, which motivated us to work out the details of the ideas described there.