Evolution of entanglement after a local quench

TL;DR

This work analyzes entanglement dynamics after a local quench in a critical, half-filled XX chain by removing a single bond defect and tracking the time evolution of the reduced density matrix via the one-particle correlation matrix. The main finding is a logarithmic growth of the entanglement entropy after the defect is removed, followed by slow, non-extensive relaxation toward the homogeneous ground-state value, with the growth and relaxation rates set by the defect strength and defect position. A central defect yields a slow, anomalous level in the single-particle spectrum that governs late-time entropy decay, while a boundary defect produces a pronounced plateau whose shape is captured by log-scaling ansatzes. The results highlight a nonequilibrium entanglement dynamics in a locally perturbed, critical system, showing a form of asymptotic subsystem thermalization with an effective temperature that vanishes for the whole system, and reveal universal features controlled by front propagation at unit velocity.

Abstract

We study free electrons on an infinite half-filled chain, starting in the ground state with a bond defect. We find a logarithmic increase of the entanglement entropy after the defect is removed, followed by a slow relaxation towards the value of the homogeneous chain. The coefficients depend continuously on the defect strength.

Figures (10)

• Figure 1: Time evolution of the low-lying single-particle eigenvalues $\varepsilon_k(t)$ for a subsystem of $L=40$ sites with a central defect $t'=0$. Left: snapshots of the positive eigenvalues for different times, compared with the spectrum in equilibrium. Right: time evolution of the four lowest eigenvalues.
• Figure 2: Large-time behaviour of the eigenvalues $\varepsilon_4(t)$ and $\varepsilon_5(t)$ already seen in Figure \ref{['fig:epsilon_t']}. The avoided level crossing can be fitted by $\ln (t /\tau)$ with $\tau \approx 1$.
• Figure 3: Time evolution of the entanglement entropy for a subsystem of $L=40$ sites with a central defect $t'=0$. A sudden jump is followed by a slow relaxation towards the homogeneous value $S_{h}$. The inset shows the logarithmic correction to the $1/t$ decay.
• Figure 4: Time evolution of the entropy for a subsystem of $L=40$ sites for several defect positions $L_1/L_2$, indicating the number of sites to the left/right of the defect with $t'=0$.
• Figure 5: Snapshots from the time evolution of the eigenvector corresponding to the lowest lying single-particle eigenvalue of a subsystem with $L=100$ sites and with a defect $t'=0$ at the left boundary.