Logarithmic entanglement lightcone in many-body localized systems

We theoretically study the response of a many-body localized system to a local quench from a quantum information perspective. We find that the local quench triggers entanglement growth throughout the whole system, giving rise to a logarithmic lightcone. This saturates the modified Lieb-Robinson bound for quantum information propagation in many-body localized systems previously conjectured based on the existence of local integrals of motion. In addition, near the localization-delocalization transition, we find that the final states after the local quench exhibit volume-law entanglement. We also show that the local quench induces a deterministic orthogonality catastrophe for highly excited eigenstates, where the typical wave-function overlap between the pre- and postquench eigenstates decays exponentially with the system size.

Figure 1

Logarithmic lightcone extracted from the local-quench triggered entanglement growth. We begin with highly excited states (obtained by DMRG-X) and calculate $S_x\left(t\right)$ from TEBD (see text). The circled blue curve is determined by investigating when $S_x\left(t\right)$ starts to increase and the red dashed line is a logarithmic fit $x\sim logt$ to the data.

Figure 2

Wave-function overlaps between the pre- and postquench eigenstates in localized systems. (a) Exponential OC of the highly excited eigenstates, at $W=15$ and $v_0=6$. Here, $|{\psi }_I\rangle$ and $|{\psi }_F\rangle$ are chosen to be the excited states in the middle of the spectrum of $H_I$ and $H_F$, respectively. The red dashed lines are linear fits to the data. The deviation of the blue curve ($U=0$) from the linear fit at large $L$ is due to the machine-precision limit. (b) Histogram of the overlaps of the ground and excited states, respectively, at $W=15,v_0=6,U=1$, and $L=10$. Unlike ground-state OC which happens stochastically with a probability $\sim \frac{v_0}{2W}$ [48], OC for a highly excited state occurs deterministically (with probability one in the thermal dynamic limit).

Figure 3

Exact-diagonalization calculations of entanglement dynamics introduced by a local quench and the extracted lightcones for information propagation. (a) Deep localized region with $W=15,U=1,L=12$, and $v_0=6$. We begin with a highly excited state at the middle of the spectrum. $x_c$ indicates the location where we cut the system into two subsystems $A$ and $B$ (cutting the bond between site $x_c$ and $x_c+1$). The local quench is added at the left end (site $-5$ in our notation) of the chain. (b) The extracted logarithmic lightcones for different threshold values. For a given threshold value ${\mathrm{\Delta }}_0$, the corresponding logarithmic lightcone is obtained by studying when $S_x\left(t\right)$ starts to increase to pass ${\mathrm{\Delta }}_0$ (see the text).

Figure 4

The DMRG-X and TEBD calculations of entanglement growth after a local quench starting from a highly excited eigenstate $|{\psi }_I\rangle$ of $H_I$, for large system sizes $L=50$, with $U=1,W=15$, and $v_0=6$. Here, the local quench is added in the middle of the chain (site 0). (a),(b) The entanglement growth $S\left(t\right)-S\left(0\right)$ for the right and left half chains, respectively. The logarithmic lightcone plotted in Fig. 1 is obtained from the same data shown here.

Figure 5

(a) Final entanglement growth vs the distance between the local quench site and the entanglement cut. The results for two different disorder strengths ($W=15,20$) deep in the MBL phases are plotted. In the long-time limit, the net entanglement growths introduced by the local quench decay exponentially with the distance from the quench. Here, the other parameters are $U=1$ and $v_0=6$. (b) Scaling of the saturated entanglement with the system size $L$ for different $W$. Here, we consider a bipartition into two half chains of equal size. Deep in the localized region ($W=14,20$), the final states after the quench exhibit area-law entanglement. In contrast, near the localization-delocalization transition point ($W=6,8$), the local quench gives rise to an unbounded entanglement growth in the thermodynamic limit, $L\to \infty$. For finite $L$, the saturation value of the entanglement scales linearly with the system size $S\left(\infty \right)\sim L$.