Many-body localization and delocalization in the two-dimensional continuum

We discuss whether localization in the two-dimensional continuum can be stable in the presence of short-range interactions. We conclude that, for an impurity model of disorder, if the system is prepared below a critical temperature $T<T_c$, then perturbation theory about the localized phase converges almost everywhere. As a result, the system is at least asymptotically localized and perhaps even truly many-body localized, depending on how certain rare regions behave. Meanwhile, for $T>T_c$, perturbation theory fails to converge, which we interpret as interaction-mediated delocalization. We calculate the boundary of the region of perturbative stability of localization in the interaction-strength-temperature plane. We also discuss the behavior in a speckle disorder (relevant for cold-atom experiments) and conclude that perturbation theory about the noninteracting phase diverges for arbitrarily weak interactions with speckle disorder, suggesting that many-body localization in the two-dimensional continuum cannot survive away from the impurity limit.

Figure 1

Schematic phase diagram for the two-dimensional continuum with impurity-model disorder. Perturbation theory about the localized phase converges almost everywhere for temperatures $T<T_c$, indicating that (at least asymptotic) many-body localization can occur at low enough temperature. For $T>T_c$, perturbation theory about the localized phase diverges for arbitrarily weak interactions. What happens in this regime cannot be answered within our current calculation scheme, but the breakdown of localization may represent interaction-mediated delocalization. These results should be contrasted with one-dimensional continuum systems (dashed line), where localization can be perturbatively stable at arbitrary temperature, at least if we ignore the potentially delocalizing effect of rare regions (Appendix pp2).