Entanglement spreading after a geometric quench in quantum spin chains

We investigate the entanglement spreading in the anisotropic spin-$1/2$ Heisenberg $\left(XXZ\right)$ chain after a geometric quench. This corresponds to a sudden change of the geometry of the chain or, in the equivalent language of interacting fermions confined in a box trap, to a sudden increase of the trap size. The entanglement dynamics after the quench is associated with the ballistic propagation of a magnetization wave front. At the free fermion point ($XX$ chain), the von Neumann entropy $S_A$ exhibits several intriguing dynamical regimes. Specifically, at short times a logarithmic increase is observed, similar to local quenches. This is accurately described by an analytic formula that we derive from heuristic arguments. At intermediate times partial revivals of the short-time dynamics are superposed with a power-law increase $S_A\sim t^{\alpha }$, with $\alpha <1$. Finally, at very long times a steady state develops with constant entanglement entropy, apart from oscillations. As expected, since the model is integrable, we find that the steady state is nonthermal, although it exhibits extensive entanglement entropy. We also investigate the entanglement dynamics after the quench from a finite to the infinite chain (sudden expansion). While at long times the entanglement vanishes, we demonstrate that its relaxation dynamics exhibits a number of scaling properties. Finally, we discuss the short-time entanglement dynamics in the $XXZ$ chain in the gapless phase. The same formula that describes the time dependence for the $XX$ chain remains valid in the whole gapless phase.

Figure 1

(a), (b) Geometric quench in $XXZ$ spin chain: quench setup for open boundary conditions. (a) $t<0$: Two independent chains ($A$ and $B$) of length $L_A$ and $L_B\equiv L-L_A$ are prepared in the ground state of the $XXZ$ chain in the sectors with zero and maximal magnetization, respectively. (b) $t=0$: $A$ and $B$ are glued together. Here we consider geometric quenches with several aspect ratios $\omega \equiv L_A/L$, i.e., $0\le \omega \le 1$, focusing on the entanglement between $A$ and $B$. (c) Example of typical local quench protocol: The initial state at $t=0$ is obtained connecting two identical chains prepared in the ground state at zero magnetization.

Figure 2

Magnetization wave front after the geometric quench with aspect ratio $\omega \equiv L_A/L=1/3$ (cf. Fig. 1) in the open $XX$ chain: local magnetization $\langle S_i^z\rangle$ at site $0\le i<3L_A$ in the chain. Data are exact results for $L_A=60$ and several times. At $t=0$ a steplike profile is present. At $t>0$ a wave front forms propagating symmetrically (with $v_s=1$) in parts $A$ and $B$ (see horizontal arrows). The dashed line is the flat profile expected on average at $t\to \infty$. At $t=L_A/v_s$ the wave front is reflected at the (left) boundary of the chain. Inset: rescaled dynamics, $\langle S_i^z\rangle$ versus $(i-L_A)/t$. All data for different times collapse on the same scaling function (dashed line).

Figure 3

Entanglement spreading after the geometric quench with aspect ratio $\omega \equiv L_A/L=1/2$ (cf. Fig. 1) in the open $XX$ chain. (a) von Neumann entropy (dashed-dotted line) $S_A$ for part $A$ of the chain as a function of time: exact results for $L_A=200,0\le t\le t_s^{*}$ (short times), with $t_s^{*}\equiv 2L_A/v_s$ where $v_s=1$ is the spinon velocity. Dotted vertical lines mark the times $t=t_s^{*}/2,t_s^{*},2t_s^{*}$. (b) Semiclassical interpretation: Entanglement growth is understood in terms of the ballistic propagation (with velocity $v_s=1$) of free effective excitations (the lines denote their trajectories). These are created at $t=0$ at the interface between $A$ and $B$. At $t=t_s^{*}/2$, perfect reflection at the boundary of the chain occurs. Entanglement “jumps” at $t=m{t}_s^{*},m=1,2\cdots$ correspond to excitations crossing the center of the chain. (c) Long-time behavior: At $t\sim t_{\ell }^{*}\equiv L_A^2/v_s$ [i.e., after $O\left(L_A\right)$ crossings] the system reaches a steady state with constant entropy (apart from superimposed oscillations). Inset: approach to the steady-state entanglement (data for the same parameters as in the main figure and $0\le t\le t_{\ell }^{*}$). A logarithmic scale is used on both axes. The dashed line is $S_A\sim t^{\alpha }$, with $\alpha \approx 0.6$, whereas the dashed-dotted one is $S_A\sim t^{1/2}log\left(t\right)$.

Figure 4

von Neumann entropy $S_A$ for part $A$ after the geometric quench (with $\omega \equiv L_A/L=1/2,1/3$) in the $XX$ chain: short-time behavior (at $t\le t_s^{*}\sim L_A/v_s$, and $v_s=1$ the spinon velocity). (a) $XX$ chain with open boundary conditions, $S_A$ versus $v_{s}t/\left(2L_A\right)$ (exact results for several sizes $L_A$). The crosses are data for $L=150$ and $\omega =1/3$. Dashed-dotted lines are one-parameter fits to $S_{\mathrm{ansatz}}$ [see Eq. (12)]. Inset: shifted entropy, $S_A-1/4log{L}_A$ versus $v_{s}t/\left(2L_A\right)$. Note the perfect data collapse for all chain sizes and times. (b) The same as in (a) for periodic boundary conditions: Now $S_A$ is plotted versus $v_{s}t/L_A$. Dashed-dotted lines are fits to Eq. (12) (notice the dependence on boundary conditions). Inset: shifted entropy, $S_A-1/2log{L}_A$ versus $v_{s}t/L_A$.

Figure 5

Geometric quench ($\omega \equiv L_A/L=1/2$; cf. Fig. 1) in the $XX$ spin chain: short-time dynamics of the von Neumann entropy $S_A$ for part $A$ of the chain. Deviations from $S_{\mathrm{ansatz}}$ [see Eq. (12)] for (a) open and (b) periodic boundary conditions. (a) $S_A-S_{\mathrm{ansatz}}$ plotted versus $v_{s}t/\left(2L_A\right)$. Data are exact results for $L_A=50,100$ (full and dashed line, respectively). Note that $k_{\nu }^{\prime }$ [see Eq. (12)] has been fitted and subtracted from the curves. Inset: same as in the main panel (circles are data for $L_A=100$), the continuous line is a fit to $[a_{1}cos\left(t\right)+a_{2}cos\left(2t\right)]/t$. (b) Same as in (a) yet for periodic boundary conditions: $S_A-S_{\mathrm{ansatz}}$ versus $v_{s}t/L_A$. Inset: same as in (a). The full line is a fit to $[a_{1}cos\left(t\right)+a_{2}cos\left(2t\right)]/t$.

Figure 6

Momentum distribution function $n_k\equiv \langle c_k^{†}{c}_k\rangle$ restricted to part $A$ of the chain (cf. Fig. 1) after a geometric quench in the periodic $XX$ chain. Here $c_k,c_k^{†}$ are the fermionic operators of the corresponding free-fermion chain (cf. Appendix pp1). (a) $n_k$ in the steady state ($t\gg t_{\ell }^{*}$; cf. Fig. 3) plotted versus the single-particle momentum $0\le k\le 2\pi$. Data are for $L_A=50$. (b) Long-time average of $n_k$ (over the interval $20000\le t\le 40000$). Full and dashed-dotted lines are for the geometric quench with aspect ratios $\omega =1/3$ and $\omega =1/2$, respectively (Fig. 1). The plateaux in the central region ($\pi /2\le k\le 3/2\pi$) correspond to $n_k=1-\omega$ (arrows in the figure).

Figure 7

Entanglement in the steady state after a geometric quench with $\omega =1/3$ in the periodic $XX$ chain. Single-particle entanglement spectrum levels ${\zeta }_l \left(l=1,2,\cdots ,L_A\right)$ at long times $t>t_{\ell }^{*}$ (see Fig. 3). Data are exact results for a chain with $L_A=100$ and several times $8000\le t\le 20000$. Note the first $L_A/2$ levels with ${\zeta }_l=1$. The full line is the long-time average. Inset: Long-time average of the single-particle entanglement spectrum for quenches with several values of $\omega \equiv L_A/L=2/3,1/2,1/3,1/4$. The arrow indicates decreasing $\omega$. Notice that the time-averaged ${\zeta }_l$ are in general different from the time-averaged $n_k$ in Fig. 6.

Figure 8

Extensive entanglement in the steady state after a geometric quench. Data are for the periodic $XX$ chain. von Neumann entropy $S_A$ for part $A$ of the chain (cf. Fig. 1) at large times ($t\gg t_{\ell }^{*}$; see Fig. 3): $S_A/L_A$ plotted versus time $t$ for quenches with several aspect ratios $\omega =L_A/L=1/5,1/8,1/20,1/40$ and $L_A=50$ (the same scale is used on both axes in all the panels). The dashed line is $S_A/L_A=-[\omega log\omega +(1-\omega )log(1-\omega )]/2$.

Figure 9

Fluctuations of the entanglement entropy in the steady state after a geometric quench with aspect ratio $\omega \equiv L_A/L=1/2$. Data are for the open $XX$ chain with $L=100,200,400$: rescaled von Neumann entropy $S_A/L_A$ plotted versus the rescaled time $t/L_A^2$. Notice that partial revivals (oscillations) persist in the long-time regime, and do not decay with the chain size.

Figure 10

Entanglement relaxation after the geometric quench with aspect ratio $\omega \equiv L_A/L\ll 1$ in the periodic $XX$ chain. Symbols are exact numerical data for $L_A=10,20,25,50,100$. Rescaled entropy dynamics: $S_A$ versus $t/L_A^2$. Perfect data collapse is observed for all $L_A$ and times. The dashed line is the analytic result at $t\to \infty$. Inset: same as in the main figure at $t/L_A^2<1$. Note the logarithmic scale on the $x$ axis. The dashed dotted line is $-0.4log(t/L_A^2)$.

Figure 11

Magnetization wave front after a geometric quench with aspect ratio $\omega \equiv L_A/L=1/3$ in the $XXZ$ spin chain. Symbols are tDMRG data for $L_A=60$ and anisotropies $\Delta =-0.5,-0.3,-0.2,0.5$ [panels (a)–(d) in the figure]. The same scale is used on the $x$ axis in all panels. The local magnetization $\langle S_i^z\rangle$ is plotted versus $(i-L_A)/t$, with $i$ being the position in the chain. All the data collapse on the same $\Delta$-dependent scaling function. The dashed vertical line corresponds to $(i-L_A)/t=1$, while the dashed-dotted one is $(i-L_A)/t=-v_s\left(\Delta \right)$, with $v_s\left(\Delta \right)$ the spinon velocity. The continuous lines are fits to $s_0+(i-L_A)/(2\pi v_s\left(\Delta \right)t)$, with $s_0\approx 1/4$ the fitting parameter.

Figure 12

Entanglement spreading after the geometric quench with aspect ratio $\omega \equiv L_A/L=1/3$ (see Fig. 1) in the $XXZ$ spin chain at $\Delta =-1/2$ (a) and $\Delta =1/2$ (b). (a) Shifted von Neumann entropy $S_A-1/4log{L}_A$ versus $t/\left(2L_A\right)$. Symbols are tDMRG data for $L_A=20,30,40$. Notice the perfect data collapse for all sizes and times. The dashed lines are fits to $S_{\mathrm{ansatz}}$ [Eq. (25)], with $v_e$ (entanglement spreading rate) and $k_{\nu }^{\prime }$ the fitting parameters. The vertical dotted line marks the point $2L_A/t=v_e$. The fit gives $v_e\approx v_s\left(\Delta \right)$. (b) Same as in (a) for $\Delta =1/2$ and $L_A=20,30,60$. The fit to Eq. (25) now yields $v_e\approx 1.13<v_s\approx 1.3$. In both panels, the vertical dashed line is $1/v_s\left(\Delta \right)$.

Figure 13

Ground-state single-particle ES levels for the $XX$ chain. Symbols are exact results for a chain of length $L=300$. The ES levels are for a subsystem with $L_A=75$ (rhombi) and $L_A=150$ (circles). Only few ES levels (with ${\zeta }_l\ne 0,1$) contribute to the entanglement entropy.