Quantum quenches, thermalization, and many-body localization

We conjecture that thermalization following a quantum quench in a strongly correlated quantum system is closely connected to many-body delocalization in the space of quasi-particles. This scenario is tested in the anisotropic Heisenberg spin chain with different types of integrability-breaking terms. We first quantify the deviations from integrability by analyzing the level spacing statistics and the inverse participation ratio of the system’s eigenstates. We then focus on thermalization, by studying the dynamics after a sudden quench of the anisotropy parameter. Our numerical simulations clearly support the conjecture, as long as the integrability-breaking term acts homogeneously on the quasiparticle space, in such a way as to induce ergodicity over all the relevant Hilbert space.

Figure 1

A cartoon of the quasi-particle space. For an integrable model all states, represented by the occupations of quasi-particles $\{n(k)\}$, are localized. An integrability-breaking perturbation introduces hopping matrix elements $V$ among different sites, which hybridize, provided $|E(\{n^{\prime }(k)\}-E(\{n^{″}(k)\}|⩽V$. For strong perturbations this may lead to delocalization of wave functions among all points in quasiparticle space in a microcanonical energy shell.

Figure 2

Level spacing indicator for the XXZ model (3) with non-integrable perturbations $(\mathrm{I})$ (left panels) and $(\mathrm{II})$ (right panels). The upper frames show ${\eta }_w(E)$, while the lower ones display ${\eta }_{\mathrm{c}}(E)$. Data are for $L=14$ sites with different values of the integrability-breaking perturbation $\Delta$. The LSI ${\eta }_w(E)$ is evaluated in a microcanonical shell of width $W=2$. Averages are performed over ${10}^3$ disorder instances. In order to exactly recover Poisson and GOE statistics in the two integrable and non-integrable limits, we performed an unfolding of the energy spectrum for each instance, according to standard techniques adopted in quantum chaos.[27]

Figure 3

Inverse participation ratio for the XXZ model (3) with non-integrable perturbations $(\mathrm{I})$ (left panels) and $(\mathrm{II})$ (right panels). Data are for $L=14$ sites and different values of the perturbation strength $\Delta$. The IPR is evaluated in a microcanonical shell of width $W=2\Delta$, in the integrable (upper frames) basis and in the site (lower frames) basis. Data are averaged over all the system eigenstates in the appropriate energy window, and over ${10}^2$ disorder realizations.

Figure 4

IPR in the integrable basis ${\xi }_I$ at $\Delta =0.1$ (upper panels), and at $\Delta =1$ (lower panels), compared to the number of states $N$ in an energy window of width $W=2\Delta$. Results are shown for the integrability-breaking terms $(\mathrm{I})$ (left panels) and $(\mathrm{II})$ (right panels). Notice, however, that all other cases display an identical qualitative behavior. In particular, the disorder is not required to observe the hybridization of the quasiparticle states within the microcanonical energy shell for the case of large $\Delta$.

Figure 5

Effective temperature in the XXZ model with an integrability-breaking perturbation, after a quench in the anisotropy parameter toward a value of $J_z=0.5$. The values of $J_{z0}$ in the $x$-axis denote the initial anisotropies, and stand for different initial conditions [that is, the ground states of the Hamiltonian $\mathcal{H}(J_{z0})$]. Data are for $L=12$ sites; in all panels, except the lower right one, averages are performed over 200 disorder instances.

Figure 6

Comparison between the diagonal (black circle) and canonical (red squares) expectation value of the two-spin correlation function $n^x(k)$ (upper panels) and $n^z(k)$ (lower panels) as a function of the momentum $k$. Data are for a quench from $J_{z0}=10$ to $J_z=0.5$ and two kinds of integrability-breaking perturbation [$(\mathrm{I})$ in the left panels, and $(\mathrm{II})$ in the right panels], with disorder intensity $\Delta =0.4$.

Figure 7

Discrepancies $\delta n_{\pi }^x$ (black circles) and $\delta n_{\pi }^z$ (red squares) between the diagonal and the canonical ensemble predictions, for a quench from $J_{z0}=10$ to $J_z=0.5$. Data in the upper left panel $(\mathrm{I})$ are for different system sizes as depicted in the caption, while in the other panels they are for $L=12$. In the cases of perturbations involving disorder, $(\mathrm{I})$,$(\mathrm{II})$, and $(\mathrm{III})$, averages over 200 instances are performed.