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

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.

Non-Hermitian skin effect as an impurity problem

“…Natural questions which have been investigated are related to whether non-Hermiticity disrupts topological properties [22,23], whether new topological invariants can be introduced [24][25][26], and whether BBC holds true and in which sense [27][28][29][30]. A major issue regarding the restoration a non-Hermitian BBC is that non-Hermitian 1D tight-binding Hamiltonians with point gapped spectrum under periodic boundary conditions (PBCs) always yield the non-Hermitian skin effect [31][32][33][34][35][36][37][38][39][40], that is the unusual accumulation of bulk eigenstates at the ends of the same lattice under open 9). In all panels we plot one representative eigenstate (black), as all of them have the same qualitative behavior.…”

Interplay of Disorder and Point-Gap Topology: Chiral Modes, Localization and Non-Hermitian Anderson Skin Effect in One Dimension

“…The notion of non-Hermiticity and its interplay with disorder has been a recent subject of wide ranging interest. The effect of disorder on the inherently non-Hermitian Hatano Nelson model [61][62][63][64][65][66] has been studied recently, where disorder has been introduced in the hopping to obtain interesting physics, such as disorder driven phase transitions from an extended phase to a bulk localized phase 67 and an non-Hermitian Anderson skin effect, where disorder gives rise to a skin effect 68 , and more. Some studies of the non-Hermitian Su-Schrieffer-Heeger (SSH) model [69][70][71][72][73][74][75] have also been performed, where disorder has been introduced in the hopping to find a non-Hermitian topological Anderson insulating phase 76 .…”

Bulk-Bulk Correspondence in Disordered Non-Hermitian Systems

“…Very recently, the study on disorder effect in non-Hermitian systems has also drawn extensive attentions [80][81][82][83][84][85][86][87][88][89][90][91][92][93]. Importantly, the NHSE could still exist in disordered samples [12][13][14][15][16][17] and leads to the breakdown of BBC for translational-symmetry-broken samples. Although has achieved great successes in clean systems, the applicability of GBZ theory in disordered systems is still not fully understood because this theory is heavily based on the translational symmetry [52][53][54].…”