Quantum Entanglement of the Sachdev-Ye-Kitaev Models

TL;DR

This paper investigates the entanglement entropy in SYK models, deriving analytic results for free cases, demonstrating maximal entanglement for small subsystems, and exploring crossover behaviors in mixed models.

Contribution

It provides the first analytic expression for EE in SYK2, shows EE is maximal for small subsystems across all q, and analyzes EE behavior in mixed SYK2 and SYK4 models.

Findings

Analytic EE expression for SYK2 derived from random matrix theory.

EE is maximal for small subsystems regardless of q.

EE in SYK4 is smaller than Page value when subsystems are comparable.

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical model of fermions interacting with $q$-body random couplings. For $q=2$, it describes free particles, and is non-chaotic in the many-body sense, while for $q>2$ it is strongly interacting and exhibits many-body chaos. In this work we study the entanglement entropy (EE) of the SYK$q$ models, for a bipartition of $N$ real or complex fermions into subsystems containing $2m$ real/$m$ complex fermions and $N-2m$/$N-m$ fermions in the remainder. For the free model SYK$2$, we obtain an analytic expression for the EE, derived from the $\beta$-Jacobi random matrix ensemble. Furthermore, we use the replica trick and path integral formalism to show that the EE is {\em maximal} for when one subsystem is small, i.e. $m\ll N$, for {\em arbitrary} $q$. We also demonstrate that the EE for the SYK4 model is noticeably smaller than the Page value when the two subsystems are comparable in size, i.e. $m/N$ is $O(1)$. Finally, we explore the EE for a model with both SYK2 and SYK4 interaction and find a crossover from SYK2 (low temperature) to SYK4 (high temperature) behavior as we vary energy.