Non-equilibrium dynamics of pure states in the Sachdev-Ye-Kitaev model

Understanding the time evolution of pure states leading to the generation of entanglement and ensuing quantum correlations is crucial for understanding thermalization and quantum dynamics. We develop a general Schwinger-Keldysh method to study non-equilibrium dynamics of pure states for fermions. We apply the method to study the time evolution of initial density inhomogeneity or imbalance and multi-point correlations in the Sachdev-Ye-Kitaev (SYK) model in the large N limit. We observe that the disorder averaged imbalance in a non-interacting SYK (SYK2) model exhibits relaxation due to dephasing, albeit with weak long-lived oscillations, independent of initial filling. On the contrary, the interacting SYK (SYK4) model exhibits density-dependent relaxation due to inelastic processes. The relaxation in the interacting model is much slower than the non-interacting SYK2 model due to kinematic constraints. This is in contrast to the non-Fermi-liquid (NFL) to Fermi-liquid (FL) crossover seen for low-temperature mixed states, where the NFL relaxes much faster than the FL. We also observe a dynamical crossover between these two regimes for a model with both SYK4 and SYK2 terms, tuned by their relative coupling strengths. Furthermore, we extend this formalism to study time evolution of an entangled pure state and show that the initial entanglement is encoded in the multi-point correlations that relax rapidly.