Non-Hermitian spectra and Anderson localization

TL;DR

This paper derives an exact formula linking non-Hermitian spectra to Anderson localization lengths using duality and Jensen's identities, with initial numerical insights in low dimensions.

Contribution

It introduces a novel exact formula for the spectrum of Anderson's model involving non-Hermitian boundary conditions and averaging over a Bloch phase.

Findings

Exact spectrum formula involving eigenvalues with non-Hermitian boundary conditions

Preliminary numerical results in 1D and 2D cases

Insights into the smallest localization exponent

Abstract

The spectrum of exponents of the transfer matrix provides the localization lengths of Anderson's model for a particle in a lattice with disordered potential. I show that a duality identity for determinants and Jensen's identity for subharmonic functions, give a formula for the spectrum in terms of eigenvalues of the Hamiltonian with non-Hermitian boundary conditions. The formula is exact; it involves an average over a Bloch phase, rather than disorder. A preliminary investigation of non-Hermitian spectra of Anderson's model in D=1,2 and on the smallest exponent is presented.