Rényi entanglement entropy of Fermi and non-Fermi liquids: Sachdev-Ye-Kitaev model and dynamical mean field theories

Abstract

We present a method for calculating Rényi entanglement entropies for fermionic field theories originating from microscopic Hamiltonians. The method builds on an operator identity, which leads to the representation of traces of operator products, and thus Rényi entropies of a subsystem, in terms of fermionic-displacement operators. This allows for a very transparent path-integral formulation, both in and out of equilibrium, having a simple boundary condition on the fermionic fields. The method is validated by reproducing well-known expressions for entanglement entropy in terms of the correlation matrix for noninteracting fermions. We demonstrate the effectiveness of the method by explicitly formulating the field theory for Rényi entropy in a few zero-and higher dimensional large-N interacting models akin to the Sachdev-Ye-Kitaev (SYK) model and for the Hubbard model within the dynamical mean field theory (DMFT) approximation. We use the formulation to compute Rényi entanglement entropy of interacting Fermi liquid (FL) and non-Fermi liquid (NFL) states in the large-N models and compare the results successfully with those obtained via exact diagonalization for finite N. We elucidate the connection between Rényi entanglement entropy and residual entropy of the NFL ground state in the SYK model and extract sharp signatures of quantum phase transition in the entanglement entropy across an NFL to FL transition. Furthermore, we employ the method to obtain nontrivial system-size scaling of entanglement in an interacting diffusive metal described by a chain of SYK dots.

Entanglement entropy and its quench dynamics for pure states of the Sachdev-Ye-Kitaev model

“…Numerically, the subsystem entropy has been studied in [36] by exact diagonalization for the ground state, and in [37,38] by path-integral approach for thermal ensembles. There are also studies for two copies of (coupled) SYK models prepared in the thermofield double state, which purifies the thermal density matrix [20,39,40].…”

Non-unitary dynamics of Sachdev-Ye-Kitaev chain

“…Many variants of the SYK model have been constructed to study quantum chaos, quantum gravity and non-Fermi liquid analytically [22-25, 25, 26, 26-31]. In particular, there are studies on the entanglement entropy of the SYK model [32][33][34][35][36][37][38], where transitions to the replica wormhole [39,40] solution are found in the long time limit.…”

Measurement-Induced Phase Transition in the Monitored Sachdev-Ye-Kitaev Model

“…With this motivation in mind, we consider Brownian Sachdev-Kitaev-Ye (SYK) chains [34][35][36][37][38][39][40] subjected to continuous monitoring, and explore possible phases and transitions. Previously, the SYK model has been extensively used to analytically understand quantum chaos and quantum information dynamics [41][42][43][44][45][46][47][48]. We find that varying the monitoring rate in the nonunitary SYK dynamics causes a measurementinduced phase transition corresponding to the spontaneous breaking of C 4 or Oð2Þ symmetry depending on whether the model is interacting or not (see Fig.…”