Theory of directed localization in one dimension

TL;DR

This paper provides an analytical solution to the delocalization transition caused by an imaginary vector potential in a disordered one-dimensional chain, enhancing understanding of non-Hermitian localization phenomena.

Contribution

It introduces an analytical approach to the delocalization transition in 1D disordered systems with imaginary vector potentials, complementing previous numerical studies.

Findings

Analytical relation between real and imaginary energy parts in the thermodynamic limit.

Finite-size effects on the delocalization transition.

Good agreement with numerical simulations for weak disorder.

Abstract

We present an analytical solution of the delocalization transition that is induced by an imaginary vector potential in a disordered chain [N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996), cond-mat/9603165]. We compute the relation between the real and imaginary parts of the energy in the thermodynamic limit, as well as finite-size effects. The results are in good agreement with numerical simulations for weak disorder (mean free path large compared to the wavelength).