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 SYKq models for a bipartition of N real or complex fermions into subsystems containing 2m real/m complex fermions and N - 2m/N - mfermions in the remainder. For the free model SYK2, 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 << 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(1). 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.