Testing the Localization Landscape Theory on the Bethe Lattice

TL;DR

This paper critically tests the Localization Landscape Theory (LLT) on the Bethe lattice, revealing that LLT's percolation does not match the true Anderson localization transition and exhibits different critical behavior.

Contribution

The paper provides an exact analytical comparison of LLT and Anderson transition on the Bethe lattice, showing their critical behaviors differ significantly.

Findings

LLT percolation does not coincide with the Anderson transition.

The critical behavior of LLT percolation is mean-field like, unlike the Anderson transition.

LLT does not accurately capture the quantum critical properties of localization.

Abstract

The Localization Landscape Theory (LLT) provides a classical picture of Anderson localization by introducing an effective confining potential whose percolation is proposed to coincide with the mobility edge. Although this proposal shows remarkable numerical agreement in three dimensions, its fundamental validity remains unsettled. Here we test the LLT analytically on the Bethe lattice, where both the Anderson localization transition and the LLT percolation problem are exactly solvable. We find that the two transitions do not coincide, and their critical behaviors differ markedly. In particular, LLT percolation displays standard mean-field percolation criticality that is fundamentally distinct from the peculiar critical behavior of the Anderson transition on the Bethe lattice. Our results provide an exact benchmark showing that, while geometrically intuitive, the LLT does not capture the true quantum critical properties of localization.