Diffusive scaling of Rényi entanglement entropy

Abstract

Recent studies found that the diffusive transport of conserved quantities in nonintegrable many-body systems has an imprint on quantum entanglement: while the von Neumann entropy of a state grows linearly in time t under a global quench, all nth Rényi entropies with n>1 grow with a diffusive scaling sqrt[t]. To understand this phenomenon, we introduce an amplitude A(t), which is the overlap of the time evolution operator U(t) of the entire system with the tensor product of the two evolution operators of the subsystems of a spatial bipartition. As long as |A(t)|≥e^{−sqrt[Dt]}, which we argue holds true for generic diffusive nonintegrable systems, all nth Rényi entropies with n>1 (annealed averaged over initial product states) are bounded from above by sqrt[t]. We prove the following inequality for the disorder average of the amplitude, |A(t)|[over ¯]≥e^{−sqrt[Dt]}, in a local spin-1/2 random circuit with a U(1) conservation law by mapping to the survival probability of a symmetric exclusion process. Furthermore, we numerically show that the typical decay behaves asymptotically, for long times, as |A(t)|∼e^{−sqrt[Dt]} in the same random circuit as well as in a prototypical nonintegrable model with diffusive energy transport but no disorder.