Absence of many-body mobility edges

Localization transitions as a function of temperature require a many-body mobility edge in energy, separating localized from ergodic states. We argue that this scenario is inconsistent because local fluctuations into the ergodic phase within the supposedly localized phase can serve as mobile bubbles that induce global delocalization. Such fluctuations inevitably appear with a low but finite density anywhere in any typical state. We conclude that the only possibility for many-body localization to occur is lattice models that are localized at all energies. Building on a close analogy with a model of assisted two-particle hopping, where interactions induce delocalization, we argue why hot bubbles are mobile and do not localize upon diluting their energy. Numerical tests of our scenario show that previously reported mobility edges cannot be distinguished from finite-size effects.

Figure 1

Hopping processes allowed by the Hamiltonian of Eq. (1) in $d=2$. Left panel: single-particle hopping. Right panel: assisted hopping for the left particle due to the presence of the right particle.

Figure 2

Hybridization process. The state with the bubble on the left (top) hybridizes with the state with the bubble on the right (bottom) via an intermediate state. Equation (2) shows that the two transitions depicted here are resonant.

Figure 3

Left: disorder averaged energy per link ${\varepsilon }_i$ at $t=0$ (red) and averaged over time (green) for $L=12$. Initially, a cold region of length $L_c=L/2$ is prepared. The disorder strength is $\delta J=3J$. Right: same protocol, but for $\delta J=J$ and very short cold intervals $\left(L_c=2\right)$, at various $L$. The memory effects diminish with increasing $L$, but the hot region fails to thermalize the system well, even at the largest sizes. Results were averaged over 5000 disorder realizations.

Figure 4

Distribution of $ln\left(\mathrm{IPR}\right)$ associated with matrix elements of ${\sigma }_1^z$ evaluated on eigenstates randomly picked from the middle of the spectrum, for $\delta J/J=0.1,1,3,5$. In the ergodic phase, the typical IPR is exponentially small in the size $L$. In the localized phase, the distribution is size independent. At $\delta J=J$ and the considered $L$, the typical IPR is exponentially small as in a truly ergodic phase, but the distribution is broad, and shows a tail towards localized values. $\delta J=3J$ is nearly critical: the “localized” peak at $\mathrm{IPR}=O\left(1\right)$ slowly decreases with increasing $L$.

Figure 5

Energy imbalance $\mathrm{\Delta }\varepsilon$ as a function of system size, for disorder strengths $J\le \delta J\le 1.5J$. The curves are simple fits to exponentials.

Figure 6

Statistics for the ln(IPRs) of all eigenstates of $H$ for $h_2=0.7$ and different values of $h_1$. From left to right: $h_1=0.01,h_1=0.07,h_1=0.15$. Averages are taken over 500 realizations.