Localization in an imaginary vector potential

TL;DR

This paper studies the localization properties of eigenfunctions in a 1D disordered Hamiltonian with an imaginary vector potential, revealing unusual strong localization with significant fluctuations in wave function amplitude.

Contribution

It introduces and analyzes a novel localization phenomenon in complex eigenvalue regimes of disordered systems with imaginary vector potentials.

Findings

Wave functions remain strongly localized even with complex eigenvalues.

Logarithm of wave functions exhibits large fluctuations similar to Brownian motion.

Localization decay follows a square root law in the logarithmic scale.

Abstract

Eigenfunctions of 1d disordered Hamiltonian with constant imaginary vector potential are investigated. Even within the domain of complex eigenvalues the wave functions are shown to be strongly localized. However, this localization is of a very unusual kind. The logarithm of the wave function at different coordinates $x$ fluctuates strongly (just like the position of Brownian particle fluctuates in time). After approaching its maximal value the logarithm decreases like the square root of the distance $\bar{(\ln|\psi_{max}/\psi|)^2} \sim |x-x_0|$. The extension of the model to the quasi-1d case is also considered.