Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions

TL;DR

This paper investigates Anderson localization on the Cayley tree, revealing that the transition behaves like a traveling wave critical point with distinct statistical properties in localized and delocalized phases.

Contribution

It introduces a traveling wave framework to describe the Anderson transition on the Cayley tree, highlighting new critical exponents and the role of rare events in transmission statistics.

Findings

Localization length diverges as (W-W_c)^(-1)

Transmission exhibits power-law tail with exponent beta(W)

Entropy diverges as (W-W_c)^(-1.5) in localized phase

Abstract

For Anderson localization on the Cayley tree, we study the statistics of various observables as a function of the disorder strength $W$ and the number $N$ of generations. We first consider the Landauer transmission $T_N$. In the localized phase, its logarithm follows the traveling wave form $\ln T_N \simeq \bar{\ln T_N} + \ln t^*$ where (i) the disorder-averaged value moves linearly $\bar{\ln (T_N)} \simeq - \frac{N}{\xi_{loc}}$ and the localization length diverges as $\xi_{loc} \sim (W-W_c)^{-\nu_{loc}}$ with $\nu_{loc}=1$ (ii) the variable $t^*$ is a fixed random variable with a power-law tail $P^*(t^*) \sim 1/(t^*)^{1+\beta(W)}$ for large $t^*$ with $0<\beta(W) \leq 1/2$, so that all integer moments of $T_N$ are governed by rare events. In the delocalized phase, the transmission $T_N$ remains a finite random variable as $N \to \infty$, and we measure near criticality the essential singularity $\bar{\ln (T)} \sim - | W_c-W |^{-\kappa_T}$ with $\kappa_T \sim 0.25$. We then consider the statistical properties of normalized eigenstates, in particular the entropy and the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical entropy diverges as $(W-W_c)^{- \nu_S}$ with $\nu_S \sim 1.5$, whereas it grows linearly in $N$ in the delocalized phase. Finally for the I.P.R., we explain how closely related variables propagate as traveling waves in the delocalized phase. In conclusion, both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions, and the Anderson localization/delocalization transition then corresponds to a traveling/non-traveling critical point. Moreover, our results point towards the existence of several exponents $\nu$ at criticality.