Many-body localization in a quasiperiodic system

Recent theoretical and numerical evidence suggests that localization can survive in disordered many-body systems with very high energy density, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in extended quantum systems far from the zero-temperature limit. It separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present numerical evidence that many-body localization also occurs in models without disorder but rather a quasiperiodic potential. In one dimension, these systems already have a single-particle localization transition, and we show that this transition becomes a many-body localization transition upon the introduction of interactions. We also comment on possible relevance of our results to experimental studies of many-body dynamics of cold atoms and nonlinear light in quasiperiodic potentials.

Figure 1

The proposed phase diagram of our interacting Aubry-André model at high energy density. Interactions convert the localized and extended phases of the AA model into many-body localized and ergodic phases and induce an expansion of the many-body ergodic phase. The phases of the interacting model differ qualitatively from their noninteracting counterparts. The differences are explained in Sec. 4.

Figure 2

Three characteristic time series for the temporal autocorrelator with $u=0.32$ and $\Delta t=1$. In each panel, we show time series for a particular value of the hopping $g$. Only a few representative error bars are displayed in each time series. The legend refers to different lattice sizes $L$. The reference lines in panels (b) and (c) show diffusive $t^{-\frac{1}{2}}$ decay.

Figure 3

The value of $\chi$ in the latest time bin ($t=9980...9999$) plotted against $g$. In panels (a)–(c), $u=0$, $0.04$, and $0.64$, respectively. The legend refers to different lattice sizes $L$.

Figure 4

The value of $\eta$ in the latest time bin ($t=9980...9999$) plotted against $g$. In panels (a)–(c), $u=0$, $0.04$, and $0.64$, respectively. The legend refers to different lattice sizes $L$. See Eq. (11) for the definition of $\eta$. In the ergodic phase $\eta \approx 0.5$.

Figure 5

Estimates of $\kappa$ from a fit of $\eta \propto e^{-\kappa L}$ in the latest time bin ($t_{\text{bin}}=9980–9999$). The legend refers to different values of the interaction strength $u$.

Figure 6

Example time series of the Rényi entropy for two values of the tuning parameter $g$. The legend refers to different values of the interaction strength $u$. Panel (a) shows data for $L=10$ lattices at $g=0.2$. Panel (b) shows data for $L=20$ lattices at $g=1.1$. In the localized regime, we need to use smaller lattices to see convergence Rényi entropy.

Figure 7

The value of $\frac{S_2}{L}$ at $t=9999$ plotted against $g$. In panels (a)–(c), $u=0$, $0.04$, and $0.64$, respectively. The legend refers to different lattice sizes $L$. In panel (a), the inset plot shows $S_2$ vs $g$ in the low-$g$ regime. In panels (b) and (c), the insets show $\frac{S_2}{L}$ vs $g$ for low $L$ in the low-$g$ regime.

Figure 8

The estimated slope of $S_2$ vs $L$ at late times as a function of $g$. The legend refers to different values of the interaction strength $u$.

Figure 9

Collapse of single-particle IPR vs $g$, using the scaling hypothesis (A2). The legend refers to different lattice sizes $L$. In panel (a), we show data for the usual AA Hamiltonian (1). In panel (b), we show data obtained from diagonalizing the unitary evolution operator for one time step in the modified dynamics (5). We use potential wave number $k=\frac{1}{\varphi }$ and 50 samples for all lattice sizes. The insets show magnified views of the curves for the three largest lattice sizes in the vicinity of the transition.

Figure 10

The density of states vs quasienergy for $L=12$ systems at half-filling with interaction strength $u=0.16$. The legend refers to different values of $\Delta t$; the time-independent, parent model is referred to as “PM.” In panels (a)–(c), $g=0.25$, $0.4$, and $0.9$, respectively.

Figure 11

The mean of the ratio between adjacent gaps in the spectrum, defined in (A4). These data were obtained by exact diagonalization of the parent model (1) for $L=12$ systems. All data points have been averaged over 50 samples, and the legend refers to different values of the interaction strength $u$. The mean value of $\langle r_n\rangle$ shows a crossover from Poisson statistics (indicated by the bottom reference line) to Wigner-Dyson statistics (indicated by the top reference line), for the largest values of $u$. Representative error bars have been included in the plot; the absent error bars have roughly the same size.