Disorder in the Sachdev-Yee-Kitaev Model

TL;DR

This work analyzes the Sachdev-Ye-Kitaev (SYK) model across two regimes: the holographic mesoscopic regime with q^2/N << 1 and a high-randomness, random-matrix regime with q^2/N >> 1. In the latter, the authors map the evolution of the characteristic determinant to two chiral copies obeying a viscid Burgers equation with spectral viscosity, revealing semicircular bulk spectra and Airy-edge universality characteristic of finite-size random matrices. They further connect spectral relaxation to hydrodynamic modes and discuss implications for scrambling and thermalization in holographic contexts. The results establish a concrete link between mesoscopic SYK dynamics, random-matrix theory, and edge universality, with potential implications for thermalization times in complex quantum systems.

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.

Figures (1)

• Figure 1: Spectral form factor for the SYK model from POLCHINSKY. We super-imposed on the figure four mesoscopic regimes. An arrow points at the time corresponding to the ergodic (Thouless) time.