Out-of-equilibrium correlated systems: Bipartite entanglement as a probe of thermalization

Thermalization plays a central role in out-of-equilibrium physics of ultracold atoms or electronic transport phenomena. On the other hand, entanglement concepts have proven to be extremely useful to investigate quantum phases of matter. Here, it is argued that bipartite entanglement measures provide key information on out-of-equilibrium states and might therefore offer stringent thermalization criteria. This is illustrated by considering a global quench in an (extended) $\mathit{XXZ}$ spin-1/2 chain across its (zero-temperature) quantum critical point. A nonlocal bipartition of the chain preserving translation symmetry is proposed. The time evolution after the quench of the reduced density matrix of the half-system is computed and its associated (time-dependent) entanglement spectrum is analyzed. Generically, the corresponding entanglement entropy quickly reaches a “plateau” after a short transient regime. However, in the case of the integrable $\mathit{XXZ}$ chain, the low-energy entanglement spectrum still reveals strong time fluctuations. In addition, its infinite-time average shows strong deviations from the spectrum of a Boltzmann thermal density matrix. In contrast, when the integrability of the model is broken (by small next-nearest-neighbor couplings), the entanglement spectra of the time average and thermal density matrices become remarkably similar.

Figure 1

(a) An $N$-site (extended) $\mathit{XXZ}$ spin-1/2 chain (drawn here as a “zigzag” on a periodic ribbon) is partitioned into two identical $A$ and $B$ subsystems of $L=N/2$ sites by cutting bonds along the dashed line. (b) Phase diagram of the $\mathit{XXZ}$ spin (or hard-core boson) chain mapped onto a circle assuming $J_{\mathit{xy}}=\cos \theta$ and $J_z=\sin \theta$. (c) The CDW ground state for $\theta =\pi /2$, i.e., $t=0$ ($J_{\mathit{xy}}=0$) prepared before the quench: all bosons (up spins) are located on $A$ and the $B$ sites are empty (down spins).

Figure 2

GS properties of the $\mathit{XXZ}$ chain; (a) VN entanglement entropy vs $\theta$ computed on an $N=24$-site ring. The entropy is normalized by the maximum value $S_{\max }=L\ln 2$ ($L=N/2$). (b)–(e) Typical entanglement excitation spectra [for the four values of $\theta$ shown in (b)] as a function of momentum $K$ along the ribbon. The eigenvalues ${\xi }_{\alpha }$ of $\mathcal{H}=-\ln {\rho }_A$ are labeled according to the $Z$ component of the total spin (i.e., the number of bosons $N_A=L/2-S_A^Z$) of the $A$ subsystem (legend of symbols on graph). Note the CDW (LL), doubly degenerate (unique) GS of $\mathcal{H}$ (of energy ${\xi }_0$) carries $N_A=0$ or $N_A=L$ ($N_A=L/2$) hard-core bosons.

Figure 3

(a)–(c) Squared-fidelity (shaded) and VN entanglement entropy of various nonequilibrium states obtained after different sudden quenches of the $N=24$-site chain Hamiltonian, ${\theta }_{\mathrm{init}}\to {\theta }_f$ as shown in the plots ($t^{\prime }=V^{\prime }=0$). The continuous lines (dashed red line) correspond to a symmetric (nonsymmetric) initial state (see text). The dotted (green) line is the GS entropy for $\theta ={\theta }_f$.

Figure 4

(a) Time-dependent VN entanglement entropy after a sudden ${\theta }_{\mathrm{init}}=\pi /2\to {\theta }_f=\pi /6$ quench ($t^{\prime }=V^{\prime }=0$), for different chain lengths $N$ as shown in the plots. Dashed-dotted (dashed) lines corresponds to $\langle S_{\mathrm{VN}}\rangle$ ($S_{\mathrm{VN}}^{\mathrm{av}}$). (b) Finite-size scaling of the averages shown in (a). (c) Finite-size scaling of the relative time fluctuations of the entanglement entropy (red squares) and quantum fluctuations of $S_A^Z$ (normalized by $L/2$). Time averages are all performed in a $\Delta \tau =40$ time interval using a mesh of 50 points. The data of (b) and (c) are well fitted assuming $~2^{-\alpha L}$ finite-size corrections.

Figure 5

“Snapshots” at different times $\tau$ of the ES of ${\rho }_A$. (a) lies within the transient regime and (b)–(f) lie within the ”entropy plateau”. Symbols are similar to Figs. 2b, 2c, 2d, 2e $(N=24)$.

Figure 6

Entanglement spectra of the time-averaged RDM obtained after a ${\theta }_{\mathrm{init}}=\pi /2\to {\theta }_f=\pi /6$ quench in $\mathit{XXZ}$ chains of different lengths, $N=16$, $20$, and $24$ ($t^{\prime }=V^{\prime }=0$). Symbols are similar to Figs. 2b, 2c, 2d, 2e and time averages are performed as for Fig. 4.

Figure 7

ES of ${\rho }_A^{\mathrm{av}}$ (a,d), ${\rho }_A^{\mathrm{can}}$ (b,e), and ${\rho }_A^{\mathrm{micro}}$ (c,f) obtained by full diagonalization of $N=20$ site integrable ($t^{\prime }=V^{\prime }=0$) and nonintegrable chains (${\theta }_{\mathrm{init}}=\pi /2\to {\theta }_f=\pi /6$ quench). Symbols are similar to Figs. 2b, 2c, 2d, 2e. $1/\beta \simeq 2.07$ (b), $\Delta E=0.3$ (c), $1/\beta \simeq 1.48$ (e), and $\Delta E=0.1$ (f).

Figure 8

VN entanglement entropy of various nonequilibrium states obtained after a ${\theta }_{\mathrm{init}}=\pi /2\to {\theta }_f=\pi /6$ quench realized on a 24-site $\mathit{XXZ}$ chain ($t^{\prime }=V^{\prime }=0$) over different time scales characterized by $T_1$. The arrows indicate the difference between the time-averaged entropy $\langle S_{\mathrm{VN}}\rangle$ (dot-dashed lines) and the VN entropy of ${\rho }_A^{\mathrm{av}}$, $S_{\mathrm{VN}}^{\mathrm{av}}$ (dashed lines). Time averages have been performed using 50 bins in a $\Delta \tau =40$ time interval within the plateaus.

Figure 9

Entanglement spectra of time-averaged reduced density matrices in $\mathit{XXZ}$ chains of different lengths, $N=16$, $20$, and $24$ ($t^{\prime }=V^{\prime }=0$). Comparison between a sudden quench (a)–(c) (data similar to Fig. 6 reproduced here for convenience) and a quasiadiabatic $T_1=4$ quench (d)–(f). In both cases, ${\theta }_{\mathrm{init}}=\pi /2\to {\theta }_f=\pi /6$.