Many-body self-localization in a translation-invariant Hamiltonian

We study the statistical and dynamical aspects of a translation-invariant Hamiltonian, without quench disorder, as an example of the manifestation of the phenomenon of many-body localization. This is characterized by the breakdown of thermalization and by information preservation of initial preparations at long times. To realize this, we use quasiperiodic long-range interactions, which are now achievable in high-finesse cavity experiments, to find evidence suggestive of a divergent time-scale in which charge inhomogeneities in the initial state survive asymptotically. This is reminiscent of a glassy behavior, which appears in the ground state of this system, being also present at infinite temperatures.

Figure 1

System size dependence of the total energy (a) and order parameter of the incommensurate CDW phase (b) of the classical ground state with $J=0$. (c) Superfluid density ${\rho }_s$ as a function of $V$ for different lattice sizes $L$ and $\beta$ ($L/\beta =1$); the inset is a zoom-in near the phase transition point. (d) The single-particle correlation function $G\left(x\right)$ in the strongly coupling phase for a system with $L=\beta =72$ and $V=10.8J$, highlighting an exponential decay (dashed line) at large separations.

Figure 2

Statistics of adjacent gaps probing ergodicity/ nonergodicity. (a) The distribution of the ratio of adjacent gaps in the central half of the spectrum ($L=26$) and different magnitudes of the interactions. The dashed (dashed-dotted) line depicts the analytical expression of an ergodic (nonergodic) quantum system described by GOE (Poisson) statistics. (b) The average value of this quantity as $V$ is varied: for small and large values, nonergodic behavior sets in, while for intermediate interactions, quantum chaos prediction is obtained. To decrease statistical fluctuations, all values are averaged between the equivalent parity sectors of the system's symmetries (see text). Dashed lines describe the ergodic (nonergodic) regimes with $\langle r\rangle \approx 0.5359$ (0.3863) [72].

Figure 3

(a) Time dependence of the dynamical IPR for a lattice with $L=22$ averaged over all equivalent real parity sectors, and different initial states, for several interaction strengths given by the color code. Panel (b) shows its equilibrium value in the long-time limit ${\langle \overline{\mathcal{I}}\rangle }_{\mathrm{eq}.}\equiv \langle \overline{\mathcal{I}}(\tau \to \infty )\rangle$ as a function of $V/J$, comparing the size dependence. The inset highlights the crossing region marking the beginning of the MBL-like regime; the shaded area represents the confidence interval for the critical interaction.

Figure 4

Time evolution of the integrated charge inhomogeneity $\overline{{\left(\mathrm{\Delta }n\right)}^2}$ averaged over states with similar $N_{\mathrm{dw}}$ in ergodic [nonergodic] regimes in panel (a) [panel (b)] with $V/J={10}^1 [V/J={10}^3]$. Panels (c) and (d) are comparisons of the DE ensemble prediction and the microcanonical result for the corresponding values of interactions.

Figure 5

Normalized density of states [panels (a) and (b)] and eigenstate expectation values of the zero-momentum occupation $n_{\alpha \alpha }^{k=0}$ [panels (c) and (d)] for $V/J=10$ [panels (a) and (c)] and $V/J={10}^3$ [panels (b) and (d)]. The clear ETH-like behavior for $n_{\alpha \alpha }^{k=0}$ in panel (c) is lost once interactions are further strengthened (d).

Figure 6

(a) System size dependence of the ground-state energy per site obtained within the mean-field approximation and (b) the CDW order parameter obtained from the mean-field for an interaction strength $V/J=12$ with incommensurate interactions whose inverse period is $p=1/\sqrt{2}$.

Figure 7

Comparisons of the infinite time-average (diagonal ensemble prediction) of the charge inhomogeneity ${\left(\mathrm{\Delta }\widehat{n}\right)}^2\equiv (1/L){\sum }_i{({\widehat{n}}_{i+1}-{\widehat{n}}_i)}^2$ and the microcanonical (thermal) results ${\left[{\left(\mathrm{\Delta }\widehat{n}\right)}^2\right)}_{\mathrm{mic}.}={\sum }_{\alpha }|\langle \alpha |\psi \left(0\right)\rangle {|}^2{\left(\mathrm{\Delta }\widehat{n}\right)}_{\alpha \alpha }^2]$ in the ergodic (a) and non-ergodic (b) regimes for a lattice with 26 sites. Panel (c) depicts the histogram of the average charge inhomogeneities when $\tau \to \infty$ at infinite temperatures (see text).

Figure 8

Thernodynamic entropy at infinite temperatures vs. the lattice size in the ergodic ($V/J={10}^1$) and non-ergodic regimes ($V/J={10}^3$) for a subsector of the Hamiltonian with zero total momentum and with even parities under particle-hole and reflection symmetries.

Figure 9

Same as in Fig. 3. The crossing region, given by the shaded area in the inset of (b), depicts the confidence interval of the localization transition given the system sizes available: $V_c/J=100-360$. This is consistent with the other periodicity $p=1/\sqrt{2}$ though the interval is reduced due to smaller finite-size effects.

Figure 10

Similar to Figs. 4 and 4 in the main text but for the case of interactions whose period is commensurate with the lattice spacing ($p=1/2$). The time-dependence of the integrated charge inhomogeneity is reported for interactions $V/J={10}^1$[${10}^3$] in (a)[(b)].