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

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 TN. In the localized phase, its logarithm follows the traveling wave form where (i) the disorder-averaged value moves linearly and the localization length diverges as with νloc = 1 and (ii) the variable t* is a fixed random variable with a power-law tail P*(t*) ∼ 1/(t*)1+β(W) for large t* with 0 < β(W) ⩽ 1/2, so that all integer moments of TN are governed by rare events. In the delocalized phase, the transmission TN remains a finite random variable as N → ∞, and we measure near criticality the essential singularity with κT ∼ 0.25. We then consider the statistical properties of normalized eigenstates ∑x|ψ(x)|2 = 1, in particular the entropy S = −∑x|ψ(x)|2ln |ψ(x)|2 and the inverse participation ratios (IPR) Iq = ∑x|ψ(x)|2q. In the localized phase, the typical entropy diverges as with νS ∼ 1.5, whereas it grows linearly as Styp(N) ∼ N in the delocalized phase. Finally for the IPR, 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 toward the existence of several length scales that diverge with different exponents ν at criticality.