Testing the Localization Landscape Theory on the Bethe Lattice

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.

Figures (11)

• Figure 1: (a) Bethe lattice centered on site $i$; colored regions mark the subtrees rooted at its $K+1$ nearest neighbors. (b) Same structure with site $i$ removed (dotted link and site), showing only the $k$-th branch. Each disconnected branch $j=1, \dots, K+1$ forms an infinite tree with coordination $K+1$, except at the root.
• Figure 2: Phase diagram of the AM on the Bethe lattice with $t=1$ and $K+1=3$ in the $(E>0,W)$ half-plane (the spectrum is statistically symmetric with respect to $E=0$). Upper bound of the spectrum, $E_{\text{max}} = 2t\sqrt{K} + W/2$ (black dashed line), mobility edge (blue solid line), LLT critical percolation curve (black solid line), and lower bound, $W_{\rm min} \sim 0.3$, below which all $1/u_i$ vanish (black dotted horizontal line). Within the numerical accuracy we have so far, the dashed line for $W_{\rm min} \sim 0.3$ vanishes at a point indistinguishable from the spectral boundary.
• Figure 3: Critical behavior of the Anderson model IPR (blue datapoints and square root fit rizzo2024localized, top axis scale) and LLT percolation inverse cluster size $1/S$ (black datapoints and linear fit, bottom axis scale). $W = 1.5, K = 2$ and $t = 1$.
• Figure 4: The correlation length of the Anderson model, $\xi_{\rm loc}$ (blue, top axis scale), and LLT critical percolation, $\xi_{\rm perc}$ (black, bottom axis scale), respectively. Both diverge with critical exponent $\nu=1$ (inset) at the corresponding transitions. $W = 1.5, K = 2$ and $t = 1$.
• Figure S1: Schematic representation of a Cayley tree and a random regular graph. Left: Cayley tree with $4$ generations and coordination number $K+1 = 3$ (thus having $46$ nodes). Right: realization of a random regular graph also with $K+1=3$ and the same number of nodes.