Relaxation and thermalization after a quantum quench: Why localization is important

We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/$XY$ spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.

Figure 1

Time evolution of $G_{jj}\left(t\right)$, with $j=L/2$ (solid line), and its running-time average $t^{-1}{\int }_0^{t}d{t}^{\prime }{G}_{jj}\left(t^{\prime }\right)$ (dashed line), for two different values of $\alpha$ and two different sizes. The horizontal dash-dotted line is the GGE average of ${\widehat{n}}_j$. The data are obtained using a single realization, with $\alpha =0.5$ for the extended case, and $\alpha =2$ for the localized one (which shows no size effect). Disorder realizations for the smaller $L$ chain are obtained, here and in the following, by removing sites from the right and left edges of the larger chain.

Figure 2

Average value of ${\delta }_{G_{jj}}^2$ (top) and ${\delta }_{G_{kk}}^2$ (bottom) for the disordered long-range hopping model ${\widehat{H}}_{\text{hop}}$ as a function of the chain size $L$ for different values of $\alpha$. Here $G_{jj}\left(t\right)=\langle \Psi \left(t\right)|{\widehat{c}}_j^{†}{\widehat{c}}_j|\Psi \left(t\right)\rangle$ with $j=L/2$, and ${\mathcal{G}}_{kk}\left(t\right)=\langle \Psi \left(t\right)|{\widehat{c}}_k^{†}{\widehat{c}}_k|\Psi \left(t\right)\rangle$ with $k=0$. The data are obtained starting from the ground state of a clean fermionic chain with nearest-neighbor hopping and quenching to ${\widehat{H}}_{\text{hop}}$ with different values of $\alpha$. The “Localized” points are for $\alpha =2$, while the “Extended” points are for $\alpha =0.5$. We used 50 realizations of disorder. In all cases we report the usual (arithmetic) mean (the error bar, when visible, is the standard deviation), except for ${\delta }_{G_{jj}}^2$ in the localized phase, where we plot the median (the geometric mean; see Sec. 3 for details). In the inset, the IPR [see Eqs. (28) and (31)] as a function of size. Notice how for all cases (“Localized” and “Extended” quenches) the eigenstates of the final Hamiltonian are “extended” in reciprocal space and consequently ${\delta }_{G_{k_1{k}_2}}^2$ goes to zero for $L\to \infty$. For a smoother size scaling, each disorder realization of the largest $L$ generated is employed, by removing the same amount of sites from the two edges, to generate realizations for smaller $L$.

Figure 3

(a) Plot of $|u_{j\mu }{|}^2$ versus the site index $j$, where $u_{j\mu }$ is a typical extended eigenstate of ${\widehat{H}}_{\text{hop}}$ for $\alpha =0.5$ and $L=2048$, with energy ${\epsilon }_{\mu }$ in the middle of the band. (b) The corresponding momentum space $|u_{k\mu }{|}^2$, with $u_{k\mu }=\frac{1}{\sqrt{L}}{\sum }_j{e}^{-ikj}{u}_{j\mu }$, quite clearly extended. (c) Plot of $|u_k{|}^2$ for a toy extended wave function $u_j=w_j/\sqrt{L}$, where $w_j=\pm 1$ is a random sign.

Figure 4

Time evolution of ${\rho }_{j_1{j}_2}\left(t\right)$, with $j_1=L/2$ and $j_2=j_1+1$ (solid line), and its running-time average $t^{-1}{\int }_0^{t}d{t}^{\prime }{\rho }_{j_1{j}_2}\left(t^{\prime }\right)$ (dashed line), for two different values of $\alpha$ and two different sizes. The horizontal dash-dotted line is the GGE average for ${\widehat{\rho }}_{j_1{j}_2}$. The data are obtained using a single realization (see caption of Fig. 1 for details).

Figure 5

Average value of ${\Delta }_{j_1{j}_2}={\langle {\widehat{\rho }}_{j_1{j}_2}\rangle }_{\mathrm{GGE}}-{\langle {\widehat{\rho }}_{j_1{j}_2}\rangle }_{\mathrm{time}}$, the discrepancy between the GGE average and the time average, for the density-density correlation ${\widehat{\rho }}_{j_1{j}_2}={\widehat{n}}_{j_1}{\widehat{n}}_{j_2}$, as a function of the chain size, for with $j_1=L/2$ and two values of $j_2-j_1$. The data are obtained using the same quenches of Fig. 2 (see its caption for details). In the localized phase ($\alpha =2$) we plot the median ${\left[{\Delta }_{j_1{j}_2}\right]}_{\mathrm{av}}^{\mathrm{G}}=\exp \left({\left[\log {\Delta }_{j_1{j}_2}\right]}_{\mathrm{av}}\right)$ because ${\Delta }_{j_1{j}_2}$ is there roughly log-normal distributed.

Figure 6

Value of ${\delta }_O^2$, with $\widehat{O}$ the local transverse magnetization ${\widehat{\sigma }}_j^z$ or the total magnetization ${\widehat{m}}_z$, as a function of the chain size. The initial state is the ground state of a clean Ising chain at the critical point ($\epsilon =0$, $\gamma =1$, $J_j=1$, and $h_j=1$) and the final Hamiltonian is a disordered Ising chain at the infinite-randomness critical point ($\epsilon =1$, $\gamma =1$, $J_j\in [0,2]$, and $h_j\in [0,2]$). Data obtained with 50 disorder realizations. For a smoother size scaling, each disorder realization of the largest $L$ generated is employed, by removing the same amount of sites from the two edges, to generate realizations for smaller $L$. For $\widehat{O}={\widehat{\sigma }}_j^z$ we plot the median ${\left[{\delta }_{{\sigma }_j^z}^2\right]}_{\mathrm{av}}^{\mathrm{G}}=\exp \left({\left[\log {\delta }_{{\sigma }_j^z}^2\right]}_{\mathrm{av}}\right)$ because ${\delta }_{{\sigma }_j^z}^2$ is roughly lognormal distributed.