Information propagation and equilibration in long-range Kitaev chains

We study the propagation of information through a Kitaev chain with long-range pairing interactions. Although the Lieb-Robinson bound is violated in the strict sense for long-range interacting systems, we illustrate that a major amount of information in this model still propagates ballistically on a light cone. We find a pronounced effect of the interaction range on the decay of the mutual information between spatially disconnected subsystems. A significant amount of information is shared at timelike separations. This regime is accompanied by very slow equilibration of local observables. As the Kitaev model is quasifree, we illustrate how the distribution of quasiparticle group velocities explains the physics of this system qualitatively.

Figure 1

Spectrum of the LRK model for (a) $\mu =-1$ and (b) $\mu =1$. (c) and (d) show the quasiparticle group velocity distribution for $\mu =-1$ and 1, respectively.

Figure 2

Mutual information in the LRK chain after a global quantum quench as a function of subsystem separation ($\mathrm{\Delta }r$) and time ($t$), as schematically depicted in (a). Subsystems are composed of 16 sites. All quenches start from the noninteracting ground state and go to (b) $\alpha =1,\mu =-1$, (c) $\alpha =0.7,\mu =1$, (d) $\alpha =1.5,\mu =-1$, and (e) $\alpha =10,\mu =1$. The side panels show the mutual information at a fixed separation of 30 sites. Note the very slow dynamics observed in (d), which is caused by the corresponding peak at very small $c$ in Fig. 1.

Figure 3

(a) Entanglement entropy as a function of time after a global quench in a LRK chain. Results are shown for subsystem sizes from 128 to 2048 sites. All quenches start from the noninteracting ground state to a value of $\mu =1$ and $\alpha =0.7$. For comparison, the dashed lines show the result for $\alpha =10$. (b) The time ${\tau }_{\text{sat}}$ it takes to reach 95% of the GGE entropy as a function of $\alpha$, for $\mu =-1$ (blue solid line) and $\mu =1$ (red dashed line). The subsystem size is 200.

Figure 4

The scaling of entanglement entropy as a function of subsystem size (left) and time (right) for $\mu =-1$ and different values of $\alpha$. The black dotted line marks the GGE result.

Figure 5

The correlation function $G_r^{\left(n\right)}=\langle c_j^{†}{c}_{j+r}\rangle$ (blue) and the approximation (red) from pp3 at $t=1$ for (a) $\alpha =0.3$ and (b) $\alpha =0.7$, both with $\mu =-1$.