Population imbalance for a family of one-dimensional incommensurate models with mobility edges

In this paper, we look at four generalizations of the one-dimensional Aubry-André-Harper (AAH) model which possess mobility edges. We map out a phase diagram in terms of population imbalance and look at the system size dependence of the steady-state imbalance. We find nonmonotonic behavior of imbalance with the system parameters, which contradicts the idea that the relaxation of an initial imbalance is fixed only by the ratio of the number of extended states to the number of localized states. We propose that there exist dimensionless parameters, which depend on the fraction of single-particle localized states, single-particle extended states, and the mean participation ratio of these states. These ingredients fully control the imbalance in the long time limit and we present numerical evidence of this claim. Among the four models considered, three of them have interesting duality relations and their locations of mobility edges are known. One of the models (next-nearest-neighbor hopping) has no known duality but a mobility edge exists and the model has been experimentally realized. Our findings are an important step forward to understanding nonequilibrium phenomena in a family of interesting models with incommensurate potentials.

Figure 1

(a) Phase diagram of model 1 for $\beta =\frac{\sqrt{5}+1}{2}$ as a function of $\lambda$ and $\alpha$ in terms of the fraction of localized states $\eta$ (single disorder realization). (b) Steady-state imbalance for model 1 ($L=256$, averaged over 1000 disorder configurations).

Figure 2

(a) Phase diagram of model 2 for $\beta =\frac{\sqrt{5}+1}{2}$ as a function of $\lambda$ and $\alpha$ in terms of the fraction of localized states $\eta$ (single disorder realization). (b) Steady-state imbalance for model 2 ($L=256$ sites, averaged over 1000 disorder configurations).

Figure 3

(a) Phase diagram of model 3 for $\beta =\frac{\sqrt{5}+1}{2}$ as a function of $\lambda$ and $\mu$ in terms of $\text{MPR}/L$. $\text{MPR}/L\sim 0$ means a localized phase, and $\text{MPR}/L\sim O\left(1\right)$ means a delocalized phase. (b) Steady-state imbalance for model 3 ($L=256$ sites, averaged over 1000 disorder configurations).

Figure 4

(a) Phase diagram of model 4 for $\beta =\frac{\sqrt{5}+1}{2}$ as a function of $\lambda$ and $\alpha$ in terms of the fraction of localized states $\eta$ (single disorder realization). (b) Steady-state imbalance for model 4 ($L=256$ sites, averaged over 1000 disorder configurations).

Figure 5

(a) Steady-state imbalance vs $1/L$ ($L$ is the system size) for model 1 (averaged over 100 realizations). The residual value at the limit $L\to \infty$, $I_0$, has been subtracted to make the plots converge at $L\to \infty$, which makes the $1/L$ linear scaling clear. (b) Thermodynamic limit of steady-state imbalance $I_0$ for model 1, averaged over 100 realizations, as a function of $\alpha$ for fixed $\lambda$'s, obtained from the linear fits for the top panel. Notice that there is a change in trend of $I_0$ whenever there is a phase transition from a localized to mobility edge phase, for $\lambda =1.2$ and $\lambda =1.4$.

Figure 6

(a) $\epsilon$ [Eq. (23)] for model 1, for $L=256$, averaged over 100 realizations. $\epsilon$ is calculated only in the mobility edge phase, along a constant $\lambda$ line in the phase diagram. (b) ${\epsilon }^{\prime }$ [Eq. (24)] for different values of $\lambda$. In the localized phase, it is calculated over the entire spectrum, whereas in the mobility edge phase it is calculated only over states below the mobility edge that are localized. Note that both $\epsilon$ and ${\epsilon }^{\prime }$ are 0 in the delocalized phase.

Figure 7

(a) $\epsilon$ for model 4, for $L=256$, averaged over 100 realizations. Note the relative peaks in $\epsilon$ around $\alpha =0.5$, in the mobility edge phase, although the imbalance phase diagram is structureless in this region and does not show any region of relatively high imbalance. (b) ${\epsilon }^{\prime }$ for model 4 averaged over 100 disorder realizations, calculated for different values of $\lambda$ in the localized phase. Note the dip in the value for ${\epsilon }^{\prime }$ for all $\lambda$'s around $\alpha =0.9$. This corresponds to a fingerlike projection of the low imbalance region into the localized phase [Fig. 4].

Figure 8

(a) System size dependence of steady-state imbalance, averaged over 100 disorder realizations for (a) model 2, (b) model 3, and (c) model 4. The residual value at the thermodynamic limit has been subtracted to make the plots converge at $L\to \infty$ and to make the $1/L$ linear scaling clear.

Figure 9

Thermodynamic limit of steady-state imbalance $I_0$ for different $\lambda$'s with increasing $\alpha$, from similar linear fits (averaged over 100 disorder realizations) up to leading order in $1/L$ as in Fig. 8. All the linear fits from which we extract $I_0$ have not been presented in Fig. 8 for the sake of brevity. Plots are for (a) model 2, (b) model 3, and (c) model 4. The various regions explored in Figs. 8 and 9 can be looked up in Table 2 and the phase diagrams in Figs. 1–4.

Figure 10

PR of single-particle states, defined in Eq. (15) for model 3, scaled by system size. We recollect that PR is $O\left(1\right)$ for localized states and $O\left(L\right)$ for delocalized states [see Eq. (21)]. All the plots are for $\lambda =0.9$, showing a transition from a delocalized to mobility edge to a delocalized phase again through appearances and the merger of multiple mobility edges. (a) $\alpha =0$: The AAH limit, where all states are delocalized. (b) $\alpha =0.15$: A single mobility edge appears separating the $O\left(L\right)$ extended states from $O\left(1\right)$ localized states. For (c) $\alpha =0.35$ and (d) $\alpha =0.4$, two mobility edges appear separating a region of localized states in the spectrum around $E=2$ from extended states on either sides. In other words, around $E=2$ there is a region of localized states. (e) $\alpha =0.45$: The mobility edges merge to give a special band of states near $E=1.6$. (f) $\alpha =0.6$: The mobility edges have merged, giving a delocalized phase again.