Dynamics of the one-dimensional Anderson insulator coupled to various bosonic baths

We study a particle which propagates in a one-dimensional strong random potential and is coupled to a bosonic bath. We independently test various properties of bosons (hopping term, hard-core effects, and generic boson-boson interaction) and show that bosonic itineracy is the essential ingredient governing the dynamics of the particle. Coupling of the particle to itinerant phonons or hard-core bosons alike leads to delocalization of the particle by virtue of a subdiffusive (or diffusive) spread from the initially localized state. Delocalization remains in effect even when the boson frequency and the bandwidth of itinerant bosons remain an order of magnitude smaller than the magnitude of the random potential. When the particle is coupled to localized bosons, its spread remains logarithmic or even sublogarithmic. The latter result together with the survival probability shows that the particle remains localized despite being coupled to bosons.

Figure 1

The mean square deviation ${\sigma }^2\left(t\right)$ for different values of $t_{\mathrm{b}}$ for the case when the particle is coupled to HCBs using (a) semilog plot and (b) log-log plot; (c) fitting exponents $\alpha$ vs $t_{\mathrm{b}}$ extracted from ${\sigma }^2\left(t\right)=A{t}^{\alpha }$. Fitting was performed in the long-time limit. (d) log-log plot of ${\sigma }^2\left(t\right)$ of the HCB model using different values of nearest and next nearest interaction $V_1$ and $V_2$, respectively. In all cases we have used $\omega =g=1, W=12$, and $N_h=20$.

Figure 2

Semilog plots of ${\sigma }^2\left(t\right)$ of the HCB model for different sizes of the Hilbert spaces generated by $N_h$. Cases with no dispersion, $t_{\mathrm{b}}=0$, are presented in (a) and (c) for two distinct values of disorder $W=12$ and 8, respectively. Results for $t_{\mathrm{b}}=0.5$ are shown as well for two distinct values of $W$ in (b) and (d). Note also substantially different scales used to present results with or without dispersion. Dashed (red) lines represent results after finite-size scaling analysis. The dotted line in (c) represents a fit to the form ${\sigma }^2\left(t\right)=A+Blog\left(t\right).$ Dotted lines in (b) and (d) represent fits to the form ${\sigma }^2\left(t\right)=A{t}^{\alpha }$. The thin (violet in color) line in (a) and (c) represents evolution of ${\sigma }^2\left(t\right)$ for a free particle, i.e., $g=0$, that is subject to Anderson's localization. The inset in (b) displays exponents $\alpha$ (circles) extracted from finite-size scaled results at different values of disorder $W$. A singular square represents the result for the Holstein model with parameters identical to those in the HCB one. Other parameters of the model were $\omega =g=1$ and $V_1=V_2=0$.

Figure 3

(a) and (b) ${\sigma }^2\left(t\right)$ of the Holstein model for different sizes of the Hilbert spaces generated by $N_h$ and $W=12$. The cases with no dispersion, $t_{\mathrm{b}}=0$, are presented in (a) while results for $t_{\mathrm{b}}=0.5$ are shown (b). Dashed (red) lines represent results after finite-size scaling analysis. The dotted line in (a) represents a fit to the form ${\sigma }^2\left(t\right)=A+Blog\left(t\right).$ The dotted line in (b) represents a fit to the form ${\sigma }^2\left(t\right)=A{t}^{\alpha }$. Overlaps $\mathcal{O}$ are shown in (c) ($t_b=0$) and (d) ($t_b=0.5$) for the Holstein model with full lines and for the HCB model with dashed lines. Thin lines (violet in color) in (a) and (c) represent evolution of ${\sigma }^2\left(t\right)$ and $\mathcal{O}$, respectively, for a free particle, i.e., at $g=0$, that undergoes Anderson's localization. The inset in (d) shows the same data as in (d) but using log-log scale. Other parameters of the model were in all cases $\omega =g=1, W=12$, and $V_1=V_2=0$.

Figure 4

${\sigma }^2$ vs $t$ for the Holstein model. $N_{\mathrm{ph}}\left(0\right)$ the number of phonon excitations at $t=0$. The inset represents $\langle N_{\mathrm{ph}}\rangle \left(t\right)$. In both cases we have used $\omega =g=1, t_b=0.5,$ and $W=8$.

Figure 5

${\sigma }^2$ vs $t$ for the Holstein model for different coupling strengths $\lambda$. The inset represents $\langle N_{\mathrm{ph}}\rangle \left(t\right)$. In both cases we have used $\omega =1, t_b=0.5$, and $W=8$.

Figure 6

$\mathcal{O}\left(t\right)$ as given in Eq. (5) of the main text using different system sizes as given by $N_h$, for the HCB model in (a) and (b) and the Holstein model in (b) and (d). In all cases we have used $\omega =g=1, W=12$.

Figure 7

Scaled survival probability $\mathcal{O}\left(t\right)exp\left(a_i{N}_h\right)$ for the case when $t_{\mathrm{b}}=0.5$ for the two models as indicated in the figure. In both cases we have used $\omega =g=1, W=12$.