Quantum entanglement of the Sachdev-Ye-Kitaev models

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 nonchaotic 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 $\mathrm{SYK}q$ 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 $\mathrm{SYK}2$, 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 maximal when one subsystem is small, i.e., $m\ll N$, for 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 $O\left(1\right)$. Finally, we explore the EE for a model with both SYK2 and SYK4 interactions and find a crossover from SYK2 (low-temperature) to SYK4 (high-temperature) behavior as we vary energy.

Figure 1

(a) Scaling of EE of the complex SYK2 model with the subsystem size at different filling factors, numerical versus analytical result, with fixed $N=1000$. (b) EE per site as a function of $\kappa$ and $\lambda$. (c) The Wachter law [Eq. (2)] of the complex SYK2 model, numerical (red) versus analytical (blue) result.

Figure 2

EE scaling of the ground state of both the Majorana fermion ($\chi$) and the complex fermion ($c$) SYK4+SYK2 models, averaged over ten samples. $m=\left|A\right|$ is the subsystem size of complex SYK, while for Majorana SYK the subsystem size is $2m$. The fermion numbers are $N=22$ for $c$ and $N=44$ for $\chi$. The Hamiltonian is $H=H_{4,c\chi }+\frac{g}{N}{H}_{2,c\chi }$. The complex fermion model has fixed particle number $k=4$ (dashed lines) or $k=11$ (dash-dotted lines). Page and Wachter scalings are plotted according to Eqs. (5) and (3). The inset shows finite-size scaling of the EE at subsystem/system = 1/2.

Figure 3

EE as a function of energy for the Majorana SYK4+SYK2 models with different $g$, averaged over ten samples. The energy has been normalized to the interval $[-0.5,0.5]$. We fix subsystem/system=1/2 and $N=32$. The flat dotted line (black) on the top is the Page value.

Figure 4

illustration of the boundary conditions of $G$ and $g$. (a) The argument region of $G(\tau ,{\tau }^{\prime })$ and $g(\tau ,{\tau }^{\prime })$. The horizontal axis and the vertical axis are for $\tau$ and ${\tau }^{\prime }$, respectively. (b) Antiperiodical boundary conditions along $\tau$ and ${\tau }^{\prime }$ [Eq. (8)] for subsystems $A$ and $B$. The $B$ cycles have periodicity $\beta$ and carry the replica index $a,b$ ranging from 1 to $n$, while the $A$ cycle has antiperiodicity $n\beta$ and has only one copy.

Figure 5

Wachter's law for more $(\kappa ,\lambda )$ values.