Spectral deformations in non-Hermitian lattices with disorder and skin effect: a solvable model

TL;DR

This paper analytically investigates spectral phase transitions and deformations in a non-Hermitian lattice model with disorder, revealing how spectral properties change with disorder strength and type, especially in relation to the skin effect and localization.

Contribution

It provides an exact analytical framework for understanding spectral deformations and phase transitions in the disordered Hatano-Nelson model with non-Hermitian skin effect.

Findings

PBC spectrum undergoes topological and spectral deformations with increasing disorder.

Disorder type influences the occurrence of bulk localization and topological transitions.

Spectral behavior under disorder is characterized by changes in spectral loops and winding numbers.

Abstract

We derive analytical results on energy spectral phase transitions and deformations in the simplest model of one-dimensional lattice displaying the non-Hermitian skin effect, namely the Hatano-Nelson model with unidirectional hopping, under on-site potential uncorrelated disorder in complex energy plane. While the energy spectrum under open boundary conditions (OBC) exactly reproduces the distribution of on-site potential disorder, the energy spectrum under periodic boundary conditions (PBC) undergoes spectral deformations, from one or more closed loops in the fully delocalized phase, with no overlap with the OBC spectrum, to a mixed spectrum (closed loops and some OBC energies) in the mobility edge phase, to a complete collapse toward the OBC spectrum in the bulk localized phase. Such transitions are observed as the strength of disorder is increased. Depending on the kind of disorder, different interesting behaviors are found. In particular, for continuous disorder with a radial distribution in complex energy plane it is shown that in the delocalized phase the energy spectrum under PBC is locked and fully insensitive to disorder, while transition to the bulk localized phase is signaled by the change of a topological winding number. When the disorder is described by a discrete distribution, the bulk localization transition never occurs, while topological phase transitions associated to PBC energy spectral splittings can be observed.