Uncovering the Microscopic Mechanism of Slow Dynamics in Quasiperiodic Many-Body Localized Systems

Abstract

We study the number entropy and quasiparticle width in one-dimensional quasiperiodic many-body localized (MBL) systems and observe slow dynamics that have previously been investigated in detail only in random systems. In contrast, quasiperiodic systems exhibit more structured growth of both observables. We identify the modulation of the Rabi oscillation amplitude of single-particle hoppings as the mechanism underlying the slow growth even deep in the MBL regime. This quantum amplitude modulation and associated beats arise from the interaction between single-particle hopping processes at different positions in the chain. Interestingly, this mechanism is not weakened by increasing the distance between particles and is generic to many-body quantum systems. We develop an analytical model based on the aforementioned mechanism that explains the observed dynamics at all accessible timescales and provides a microscopic picture of the slow dynamics in the MBL regime. Our results are consistent with the stability of the MBL phase in the thermodynamic limit.

Figures (7)

• Figure 1: Sketch of the proposed mechanism behind the growth of $S_n$, and the numerically calculated entropies. (a) Illustration of a chain with QP potential in the Néel state with filled/empty dots representing particles/holes. In the localized regime, dynamics is dominated by hopping between neighbouring sites with modest potential differences, as marked with red arrows. The mechanism involves modulation of a single-particle Rabi oscillation amplitude due to interaction between near resonant hopping processes at distance $d$. Dashed vertical line denotes the boundary between the subsystems relevant for the calculation of entropies. (b) Moving time average of the sample-averaged number entropy $\overline{S_n}(t)$ for $W=6$ scaled with $(N-1)/N$, as explained in Supplemental Material. Short and medium timescales (i) are shaded in pink, long timescales (ii) are shaded in blue and ultra-long timescales (iii) are shaded in green. The grey line is obtained for a non-interacting system ($J_z=0$) and $N=16$. (c) Moving time average of the sample-averaged configurational entropy $\overline{S_c}(t)$ for $W=6$ and $N=16$. The inset shows the same curve on a log-log scale. Dotted lines denote the logarithmic and power law growth at different time intervals in the main figure and the inset, respectively.
• Figure 2: Comparison of exact dynamics with the results obtained from our effective model. (a) A moving time average of $S_n(t)$ for a single configuration of QP potential with $\phi = \phi_{\rm PR}=3.863$ (same as in Fig. \ref{['fig1']} (a) and corresponding to the point-reflection symmetry position between $i=7$ and $i=8$ --left-most site has index $i=0$) and comparison with effective model from Eq. \ref{['Ham4']}. We used the initial Néel state, $N=12$ and $W=10$. The inset shows the same $S_n(t)$ obtained from ED, without time averaging and with a linear time scale. A dashed vertical line marks the onset of beats at $t=\pi/\epsilon$. The dotted horizontal line corresponds to $\ln 2$. (b) A comparison of moving time average of $\overline{S_n}(t)$ for $W=10$ obtained from ED with $N=16$ sites and from our effective model. The grey line shows the results for single-particle dynamics described by Eq. \ref{['S_n_sum']}.
• Figure 3: Dependence of saturation timescales on distance between pairs, $d$, and the connection between the dynamics of $\sigma_x^2$ and $S_n$. (a) Moving time average of $\overline{S_n}(t)$ for $N=12$, $W=10$ and different frequencies $\beta$ corresponding to $d=0$ for $\beta=13/50$, $d=1$ for $\beta=8/25$, $d=2$ for $\beta=2/(1+\sqrt{5})$ and $d=3$ for $\beta=1/\sqrt{2}$. Arrows mark saturation times for different $d$. Note that the choice of a rational $\beta=p/q$ is numerically indistinguishable from the irrational case for $q>N$. $(\cdot)_{sat}$ denotes the saturation value calculated numerically as the average of a given quantity for $t>10^8$. (b) Moving time average of $\overline{\sigma_x^2}(t)$ compared with $\overline{S_n}(t)$ for $W=6$, $N=12$ and $\beta=2/(1+\sqrt{5})$. In this figure, the averages were computed with the initial Néel state.
• Figure A1: $\overline{S_n}(t)$ moving time average for $W=6$ and Néel initial state.
• Figure B1: Comparison of moving time average of $\overline{\sigma_x^2}(t)$ obtained from ED (red line) and from our effective model (blue line), for $W=10$, $N=12$ and the initial Néel state. Grey line denotes the noninteracting case.