Dynamics of many-body localization in a translation-invariant quantum glass model

We study the real-time dynamics of a translationally invariant quantum spin chain, based on the East kinetically constrained glass model, in search for evidence of many-body localization in the absence of disorder. Numerical simulations indicate a change, controlled by a coupling parameter, from a regime of fast relaxation-corresponding to thermalization-to a regime of very slow relaxation. This slowly relaxing regime is characterized by dynamical features usually associated with nonergodicity and many-body localization (MBL): memory of initial conditions, logarithmic growth of entanglement entropy, and nonexponential decay of time correlators. We show that slow relaxation is a consequence of sensitivity to spatial fluctuations in the initial state. While numerical results and physical considerations indicate that relaxation time scales grow markedly with size, our finite size results are consistent both with an MBL transition, expected to only occur in disordered systems, and with a pronounced quasi-MBL crossover.

Figure 1

Time evolution of site densities $\langle n_i\left(t\right)\rangle$ starting in the half-filled initial state $|\downarrow \cdots \downarrow \uparrow \cdots \uparrow \rangle$ in the thermal regime, (a) $s=-0.5$, and in the MBL-like regime, (b) $s=0.6$, showing a linear (ballistic) spreading of the state and a logarithmic spread, respectively. (c) Equilibrium value ${\overline{\mathcal{I}}}_{\mathrm{eq}}$ of the initial participation ratio $\mathcal{I}\left(t\right)$, averaged over all half-filling initial states.

Figure 2

(a), (b) Time-averaged magnetization ${\overline{M}}_w\left(t\right)$ resolved by the largest interval $w$ between excitations in the initial state. (c) Initial state dependence quantified by $\mathrm{var}{\left[M\right]}_{\text{eq}}/\mathrm{var}{\left[M\right]}_{\text{max}}$. (d) Comparison between the equilibrium magnetization ${\overline{M}}_{\mathrm{eq}}$, and its microcanonical average ${\langle M\rangle }_{\mathrm{mc}}$. The highlighted points are the equilibrium values ${\overline{M}}_{\mathrm{eq}}$ obtained in (b), corresponding to initial states with energy $E\approx 1.7$.

Figure 3

(a) Entanglement entropy $\overline{S}\left(t\right)$ (with respect to an equal bipartition of the spin chain) showing qualitative difference as $s$ is varied. Inset: Equilibrium value ${\overline{S}}_{\text{eq}}$ of entanglement entropy as a function of $s$ and $N$, suggesting a transition from the thermal regime $s<0$ to the many-body localized regime $s>0$. (c), (d) Average entanglement entropies ${\overline{S}}_w\left(t\right)$ restricted to initial states with domain size $w$, showing that initial state dependence is absent in the thermal phase. (d) Characterization of the logarithmic regime in the growth of the entanglement entropy $\overline{S}\left(t\right)$. The dashed lines show the best logarithmic fit $\overline{S}\left(t\right)\propto logt$ for each value of $N$. Inset: Estimated relaxation time $\tau$ associated with $\overline{S}\left(t\right)$.

Figure 4

Dynamics of the spin polarization $D(k,t)$ for short ($k=\pi$), medium ($k=4\pi /N$), and long ($k=2\pi /N$) wavelengths, respectively, in the thermal ($s=-1$, left panel) and nonergodic phase ($s=1$, right panel) for system size $N=12$.