Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System

We study the time evolution of the entanglement entropy of a one-dimensional nonintegrable spin chain, starting from random nonentangled initial pure states. We use exact diagonalization of a nonintegrable quantum Ising chain with transverse and longitudinal fields to obtain the exact quantum dynamics. We show that the entanglement entropy increases linearly with time before finite-size saturation begins, demonstrating a ballistic spreading of the entanglement, while the energy transport in the same system is diffusive. Thus, we explicitly demonstrate that the spreading of entanglement is much faster than the energy diffusion in this nonintegrable system.

Figure 1

(a) Spreading of entanglement entropy $S(t)$ for chains of length $L$. Initially, the entanglement grows linearly with time for all cases, with the same speed $v\cong 0.70$. Then, the entanglement saturates at long time. This saturation begins earlier for smaller $L$, as expected. The linear fit function is $f(t)=0.70t$. Standard error is less than 0.04 for all points, and thus the error bars are only visible at early times. (b) Same data scaled by the infinite-time entropy for each $L$. Note that we use logarithmic scales both here and in Fig. 2.

Figure 2

(a) The average energy spreading $R(t)$ (defined in the main text) vs time. Before saturation, its behavior does not depend on the system size. As we increase the system size, diffusive $\sqrt{t}$ behavior becomes more apparent. (b) Direct comparison of $S(t)$ and $R(t)$ for $L=16$. It is clear that the entanglement spreads faster than energy diffuses in the scaling regime before saturation.