Global phase diagram of the one-dimensional Sachdev-Ye-Kitaev model at finite $N$

TL;DR

This paper investigates the phase diagram of the one-dimensional SYK model at finite N, revealing localization phenomena, a phase transition, and how these depend on system size and interaction strength.

Contribution

It provides a detailed analysis of the finite-N phase diagram of the 1D SYK model, combining analytical and numerical methods to uncover localization and transition behaviors.

Findings

Localization length scales linearly with N

Existence of a phase transition between many-body localization and thermal diffusion

Critical interaction strength decreases with increasing N, following a specific scaling law

Abstract

Many key features of the higher-dimensional Sachdev-Ye-Kitaev (SYK) model at {\it finite} $N$ remain unknown. Here we study the SYK chain consisting of $N$ ($N$$\ge$$2$) fermions per site with random interactions and hoppings between neighboring sites. In the limit of vanishing SYK interactions, from both supersymmetric field theory analysis and numerical calculations we find that the random hopping model exhibits Anderson localization at finite $N$, irrespective of the parity of $N$. Moreover, the localization length scales linearly with N, implying no Anderson localization \textit{only} at $N\!=\!\infty$. For finite SYK interaction $J$ , from the exact diagonalization we show that there is a dynamic phase transition between many-body localization and thermal diffusion as $J$ exceeds a critical value $J_c$. In addition, we find that the critical value $J_c$ decreases with the increase of $N$, qualitatively consistent with the analytical result of $J_c/t \!\propto\! \frac{1}{N^{5/2}\log N}$ derived from the weakly interacting limit.