Diffusive scaling of R\'enyi entanglement entropy

TL;DR

This paper investigates the diffusive scaling of Rényi entanglement entropy in non-integrable many-body systems, introducing an amplitude measure and proving bounds that explain the observed diffusive growth of entanglement.

Contribution

It introduces an amplitude $A(t)$ to analyze Rényi entropies, proves bounds for its decay in diffusive systems, and connects the decay to a symmetric exclusion process, providing new insights into entanglement dynamics.

Findings

All $n$th Rényi entropies with $n > 1$ grow as $ oot t$ under diffusive transport.

The amplitude $A(t)$ decays as $e^{- oot t}$ in diffusive systems.

Numerical evidence supports the asymptotic decay of $A(t)$ as $e^{- oot t}$ in models with energy diffusion.

Abstract

Recent studies found that the diffusive transport of conserved quantities in non-integrable 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 $n$th R\'enyi 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 $|A(t)| \ge e^{-\sqrt{Dt}}$, which we argue holds true for generic diffusive non-integrable systems, all $n$th R\'enyi 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, $\overline{|A(t)|} \ge e^{ - \sqrt{Dt}} $, in a local spin-$\frac{1}{2}$ random circuit with a $\text{U}(1)$ 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)| \sim e^{ - \sqrt{Dt}} $ in the same random circuit as well as in a prototypical non-integrable model with diffusive energy transport but no disorder.