Disorder in the Sachdev-Yee-Kitaev Model

TL;DR

This paper explores the mesoscopic and spectral properties of the Sachdev-Yee-Kitaev (SYK) model, revealing its connection to random matrix theory and hydrodynamical behavior in different regimes.

Contribution

It introduces a stochastic deformation of the SYK model, deriving a viscid Burgers equation for its determinant and linking it to universal random matrix phenomena.

Findings

The determinant obeys a viscid Burgers equation with spectral viscosity.

Spectral edge oscillations follow universal Airy patterns.

Hydrodynamical estimates describe relaxation of collective modes.

Abstract

We give qualitative arguments for the mesoscopic nature of the Sachdev-Yee-Kitaev (SYK) model in the holographic regime with $q^2/N\ll 1$ with $N$ Majorana particles coupled by antisymmetric and random interactions of range $q$. Using a stochastic deformation of the SYK model, we show that its characteristic determinant obeys a viscid Burgers equation with a small spectral viscosity in the opposite regime with $q/N=1/2$, in leading order. The stochastic evolution of the SYK model can be mapped onto that of random matrix theory, with universal Airy oscillations at the edges. A spectral hydrodynamical estimate for the relaxation of the collective modes is made.