Sub-ballistic Growth of Rényi Entropies due to Diffusion

We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Rényi entropies. We argue that the latter generically grow sub-ballistically, as $\propto \sqrt{t}$, in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic nonintegrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second Rényi entropy and show that it exhibits hydrodynamic tails with three distinct power laws occurring for different classes of initial states.

Figure 1

(a) Growth of (annealed average) second Rényi entropy in a spin-$1/2$, U(1) symmetric random circuit, averaged over all product states. At long times the growth is diffusive ($\propto \sqrt{t}$). Inset: The discrete time derivative $\mathrm{\Delta }{S}_2^{(a)}(t)\equiv S_2^{(a)}(t+1)-S_2^{(a)}(t)$ decays as $t^{-1/2}$. (b) Geometry of the random circuit and block structure of the gates. (c) Rényi entropies of the tilted field Ising model Eq. (1),$S_2$ (solid lines) and $2S_{\infty }$ (dots) show a similar crossover to sub-ballistic growth, while (d) the von Neumann entropy grows mostly linearly.

Figure 2

Effective model at $q\to \infty$. (a) The purity $\mathcal{P}$, written in terms of the state $\rho$ as a matrix product operator. (b) This can be rewritten by introducing the swap operator $\mathcal{F}$. (c) Half-chain “SWAP string” $\mathcal{F}(x)$, along with a few of the terms on the rhs of Eq. (3), with red and blue particles representing local $Z_x$ operators. (d) These particles obey a random walk with hard-core interactions, spreading out diffusively, which (e) leads to a Gaussian density profile.

Figure 3

(a) Saturation of $S_2^{(a)}$ for a four-site subsystem in the spin-$1/2$ random circuit for different initial states. States away or at half filling generically saturate as $t^{-1/2},t^{-1}$, respectively. States with hydrodynamic variables in equilibrium at all times saturate with subleading exponent $t^{-3/2}$. (b) The same $t^{-3/2}$ saturation is present for random product states in the tilted field Ising model Eq. (1) (three-site subsystem, averaged over 50 initial states).