Exploration of the stability of many-body localized systems in the presence of a small bath

When pushed out of equilibrium, generic interacting quantum systems equilibrate locally and are expected to evolve towards a locally thermal description despite their unitary time evolution. Systems in which disorder competes with interactions and transport can violate this expectation by exhibiting many-body localization. The strength of the disorder with respect to the other two parameters drives a transition from a thermalizing system towards a nonthermalizing one. The existence of this transition is well established both in experimental and numerical studies for finite systems. However, the stability of many-body localization in the thermodynamic limit is largely unclear. With increasing system size, a generic disordered system will contain with high probability areas of low disorder variation. If large and frequent enough, those areas constitute ergodic grains which can hybridize and thus compete with localization. While the details of this process are not yet settled, it is conceivable that if such regions appear sufficiently often, they might be powerful enough to restore thermalization. We set out to shed light on this problem by constructing potential landscapes with low disorder regions and numerically investigating their localization behavior in the Heisenberg model. Our findings suggest that many-body localization may be more stable than anticipated in other recent theoretical works.

Figure 1

Local expectation values of the eigenstates of the model with three thermal sites for weak ($\mathrm{\Delta }=3$) and strong disorder ($\mathrm{\Delta }=9$). Left: Histograms of $\left.〈E\right|{\sigma }_i^z\left|E\right.〉$ in the Heisenberg model on $L=16$ sites for $i\in \{1,3,...,15\}$ encoded by color. Each histogram is an average over all eigenstates in the zero magnetization sector and 2000 realizations. Right: Standard deviation of the histograms of $\left.〈E\right|{\sigma }_i^z\left|E\right.〉$ in the Heisenberg model for different system sizes. Red hatches show the thermal part of the system. Each data point is an average over at least 2000 realizations.

Figure 2

Local expectation values of the eigenstates of the model with $L/2$ thermal sites for weak ($\mathrm{\Delta }=3$) and strong ($\mathrm{\Delta }=9$) disorder. Left: Histograms of $\left.〈E\right|{\sigma }_i^z\left|E\right.〉$ in the Heisenberg model on $L=16$ sites for $i\in \{1,3,...,15\}$ encoded by color. Each histogram is an average over all eigenstates in the zero magnetization sector and 2000 realizations. Right: Standard deviation of the histograms of $\left.〈E\right|{\sigma }_i^z\left|E\right.〉$ in the Heisenberg model for different system sizes. The red hatched part of each curve indicates the thermal region of the system. Each data point is an average over at least 2000 realizations.

Figure 3

Standard deviation of the histograms of $\left.〈E\right|{\sigma }_i^z\left|E\right.〉$ of the center site in the disordered part positioned at $i=L-\lfloor (L-3)/2\rfloor$ in the Heisenberg model for different system sizes. Each data point is an average over at least 2000 realizations. The number of thermal sites $t$ and the strength of the next-nearest-neighbor hopping $J_{nnn}$ are indicated in the plots.