Marginal Anderson localization and many-body delocalization

We consider $d$-dimensional systems which are localized in the absence of interactions, but whose single-particle localization length diverges near a discrete set of (single-particle) energies, with critical exponent $\nu$. This class includes disordered systems with intrinsic or symmetry protected topological bands, such as disordered integer quantum Hall insulators. We show that such marginally localized systems exhibit anomalous properties intermediate between localized and extended, including vanishing dc conductivity but subdiffusive dynamics, and fractal entanglement (an entanglement entropy with a scaling intermediate between area and volume law). We investigate the stability of marginal localization in the presence of interactions, and argue that arbitrarily weak short-range interactions trigger delocalization for partially filled bands at nonzero energy density if $\nu \ge 1/d$. We use the Harris-Chayes bound $\nu \ge 2/d$ to conclude that marginal localization is generically unstable in the presence of interactions. Our results suggest the impossibility of stabilizing quantized Hall conductance at nonzero energy density.

Figure 1

Schematic dependence of SP localization length $\xi$ on single-particle energy $E$ for a fully localized system (a), marginally localized system (b), and extended system with a mobility edge (c).

Figure 2

Diagrammatic representation of mediated hopping, to lowest order.

Figure 3

Diagram illustrating flip-flop process.

Figure 4

The flip-flop assisted hopping process.

Figure 5

Some high-order diagrams contributing to $\widehat{T}$.

Figure 6

A diagram of the class shown in Fig. 5, which appears at third order in perturbation theory, and is used to illustrate the calculation of high-order diagrams.