Hybridization of wave functions in one-dimensional localization

Abstract

A quantum particle can be localized in a disordered potential, the effect known as Anderson localization. In such a system, correlations of wave functions at very close energies may be described, due to Mott, in terms of a hybridization of localized states. We revisit this hybridization description and show that it may be used to obtain quantitatively exact expressions for some asymptotic features of correlation functions, if the tails of the wave functions and the hybridization matrix elements are assumed to have log-normal distributions typical for localization effects. Specifically, we consider three types of one-dimensional systems: a strictly one-dimensional wire and two quasi-one-dimensional wires with unitary and orthogonal symmetries. In each of these models, we consider two types of correlation functions: the correlations of the density of states at close energies and the dynamic response function at low frequencies. For each of those correlation functions, within our method, we calculate three asymptotic features: the behavior at the logarithmically large "Mott length scale", the low-frequency limit at length scale between the localization length and the Mott length scale, and the leading correction in frequency to this limit. In the several cases, where exact results are available, our method reproduces them within the precision of the orders in frequency considered.Comment: 10 pages, 5 figures. Several references added, minor corrections corresponding to the journal versio

Anderson localization of a Majorana fermion

“…Following the argument of Ref. 16, we conclude that this hybridization mechanism produces a crossover between ρ E (x) = 0 and ρ E (x) = 1 at x = L M , in agreement with the result (10) of our cal- …”

“…We further explain our results in terms of the Mott hybridization of localized states [15,16] and compare our findings to those in Ref. 14.…”

“…The average intensity of the Majorana mode decays away from the interface at the length scale 4ξ, similar to the statistics of a single localized wave function [26,28,32]. On general grounds, we believe that the statistics of the tails of this localized state is lognormal [16] with the typical Majorana state decaying at the length scale ξ. Note however that the exact form of the average intensity of the Majorana state (11) We can also remark a similarity of the LDOS profile (10) to the two-point LDOS correlation function in the quasi-one-dimensional wire [16,28].…”

“…The difference from the two-point-correlation case considered in Ref. 16 is that now three states hybridize: the Majorana state and the pair of particle-hole (Bogoliubov-de-Gennes) symmetric [18,20] states at energies ε and −ε (see Fig. 3).…”

“…On general grounds, we believe that the statistics of the tails of this localized state is lognormal [16] with the typical Majorana state decaying at the length scale ξ. Note however that the exact form of the average intensity of the Majorana state (11) We can also remark a similarity of the LDOS profile (10) to the two-point LDOS correlation function in the quasi-one-dimensional wire [16,28]. Namely, at low energies, it consists of two parts: the "short-range" part localized at the length scale ξ and proportional to the singleparticle statistics [Φ M (x) in our case] and the "longrange" part at the length scale L M .…”