Perturbed Sachdev-Ye-Kitaev model: a polaron in the hyperbolic plane

Abstract

We study the SYK${}_4$ model with a weak SYK${}_2$ term of magnitude $\Gamma$ beyond the simplest perturbative limit considered previously. For intermediate values of the perturbation strength, $J/N\ll \Gamma \ll J/\sqrt{N}$, fluctuations of the Schwarzian mode are suppressed, and the SYK${}_4$ mean-field solution remains valid beyond the timescale $t_0\sim N/J$ up to $t_{\ast }\sim J/{\Gamma }^2$. Out-of-time-order correlation function displays at short time intervals exponential growth with maximal Lyapunov exponent $2\pi T$, but its prefactor scales as $T$ at low temperatures $T\le \Gamma$.

12 pages