Localized systems coupled to small baths: From Anderson to Zeno

We investigate what happens if an Anderson localized system is coupled to a small bath, with a discrete spectrum, when the coupling between system and bath is specially chosen so as to never localize the bath. We find that the effect of the bath on localization in the system is a nonmonotonic function of the coupling between system and bath. At weak couplings, the bath facilitates transport by allowing the system to “borrow” energy from the bath. But, above a certain coupling the bath produces localization because of an orthogonality catastrophe, whereby the bath “dresses” the system and hence suppresses the hopping matrix element. We call this last regime the regime of “Zeno localization” since the physics of this regime is akin to the quantum Zeno effect, where frequent measurements of the position of a particle impede its motion. We confirm our results by numerical exact diagonalization.

Figure 1

Figure illustrating the basic setup. There is a band of states on every site $i$, with energies spanning a bandwidth $\omega \sqrt{N}$, and with level spacing $\omega /\sqrt{N}$, which differ only in the state of the bath. In the weak coupling regime $\lambda <\omega /\sqrt{N}$, hopping between sites follows the solid red lines (with no change in the state of the bath), and a typical nearest-neighbor hop changes the energy by an amount of order $W$. Outside the weak coupling regime, the states in the bath start to get hybridized. The size of the energy window over which states in the bath are well hybridized grows with $\lambda$, and becomes of order $W$ at $\lambda \sim \sqrt{\omega W}/N^{1/4}$. The system-bath coupling allows an effective hopping between states that are nearly on shell (blue lines). The system is minimally localized (for small $t$) at $\lambda \sim \sqrt{\omega W}/N^{1/4}$, with larger values of $\lambda$ leading to greater localization because of the quantum Zeno effect.

Figure 2

Schematic phase diagram, with a “phase boundary” that indicates the boundary of stability of the locator expansion. On the small-$t$ side of the phase boundary, the system is strongly localized. Within the strong-localization regime there is a crossover from Anderson localization at small $\lambda$ to quantum Zeno localization at large $\lambda$. Note that the locator expansion is maximally unstable around ${\lambda }_c=\sqrt{\omega W/\left(2\sqrt{2N}\right)}$. For $\lambda >{\lambda }_c$, the phase boundary is approximately linear in $\lambda$, whereas for $\lambda <{\lambda }_c$ it scales as $1/\lambda$. Meanwhile, the large-$t$ side of the phase boundary is the regime of weak localization (in one or two dimensions) or delocalization (in three dimensions). Note that in one or two dimensions, the phase boundary marking breakdown of the locator expansion is really just a crossover to weak localization. Only in three dimensions does it mark a true phase transition to a delocalized phase.

Figure 3

Results of analytic calculations in the forward approximation detailed in the Appendix for fixed $W=3$, $N=300$, $\omega =4$ [substitution into Eq. (17) gives ${\lambda }_c=0.5$]. Top: the $t-\lambda$ phase diagram; the dark region is the weak-localized/delocalized region, the light region is localized. Note the nonmonotonic dependence of the boundary on $\lambda$. The calculation assumes that states in the bath are hybridized according to Eq. (16), and is thus not applicable at $\lambda <\delta /\sqrt{2\pi }$. Bottom: localization length as a function of $\lambda$, along a horizontal slice through the top diagram that stays always on the “strongly localized” side of the phase boundary. Note again the nonmonotonic behavior. There is a pronounced maximum in the localization length close to $\lambda \approx 0.3$, which indicates that the system is least localized at this intermediate value of $\lambda$. Due to the nature of the approximation, $t\left(\lambda \right)$ and $\xi \left(\lambda \right)$ are underestimated, while the ratio $\xi \left(\lambda \right)/\xi \left(0\right)$ is overestimated. The value ${\lambda }_c\approx 0.3$ extracted from the procedure in the Appendix, which includes additional approximations, is reasonably similar to ${\lambda }_c=0.5$ obtained by the simple argument detailed in the main text.

Figure 4

Results of numerics on a one-dimensional system coupled to a small bath. Exact diagonalization is performed for about 50 states in the center of the band for the parameters $t=1$, $\omega =4$, $N=300$, and varying $W$. The values of $W$ are chosen so that we span the phase diagram in Fig. 2. For $W<12$, we slice through the weak-localization region, whereas for $W>12$ we stay always in the strong-localization regime. The entanglement entropy and the participation ratios have a pronounced maximum close to the (same) hybridization threshold ${\lambda }_c$. The peak is sharper when we go through the weak-localization regime. Inset: example of the finite-size scaling of $S^L$ for a given value of $\lambda =0.8$ and $W=6$. To extrapolate the infinite size $S^{\infty }$ we used (27) with $L$ ranging from 10 through 70, taking around 50 disorder realizations for each system size.

Figure 5

Value of ${\lambda }_c$ for $t=1$ and the different disorders $W$. In blue, we plot the numerical data, with their errors coming from the resolution in $\lambda$ at which the plots in Fig. 4 are computed. In dashed green, we plot the analytic estimate of Eq. (17). We note that the exact diagonalization data agree well with the analytic prediction.

Figure 6

Comparisons of the largest eigenvalue of the kernel in (A27) and its approximate form (A38). Top: continuous line is the exact numerical results, dashed lines are analytical approximations. Bottom: relative error, in percent, for $s=-1$.