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

We derive analytical results on energy spectral phase transitions and deformations in the simplest model of a one-dimensional lattice displaying the non-Hermitian skin effect, namely, the Hatano-Nelson model with unidirectional hopping under on-site potential uncorrelated disorder on the complex energy plane. Although the energy spectrum under open boundary conditions (OBCs) 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 on the complex energy plane it is shown that in the delocalized phase the energy spectrum under PBC is locked and fully insensitive to disorder, whereas transition to the bulk localized phase is signaled by the change in a topological winding number. When the disorder is described by a discrete distribution, the bulk localization transition never occurs, whereas topological phase transitions associated with PBC energy spectral splittings can be observed.