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

TL;DR

This paper investigates the effects of a weak SYK$_2$ perturbation on the SYK$_4$ model, revealing a regime where mean-field solutions hold longer and the Lyapunov exponent behavior is temperature-dependent.

Contribution

It extends the understanding of the SYK model by analyzing the impact of a weak SYK$_2$ term beyond perturbation theory, identifying a regime with suppressed fluctuations and modified chaos.

Findings

Mean-field solution remains valid up to longer timescales.

Exponential growth of out-of-time-order correlator with maximal Lyapunov exponent.

Prefactor of chaos scales with temperature at low T.

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 $t_* \sim J/\Gamma^2$. 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 $T \leq \Gamma$.