Why does bulk boundary correspondence fail in some non-hermitian topological models

Abstract

The bulk-boundary correspondence is crucial to topological insulators. It associates the existence of boundary states (with zero energy and possessing chiral or helical properties) with the topological numbers defined in bulk. In recent years, topology has been extended to non-hermitian systems, opening a new research area called non-hermitian topological insulator. In this paper, however, we will illustrate that the bulk-boundary correspondence does not hold in these new models. This is because a prerequisite condition: 'the boundaries cannot alter most of the bulk states, so as to the topological numbers defined on them' does not hold any longer. This cuts out the correspondence between the topological numbers and the boundary states. We will illustrate that, as approaching the open boundary condition by eliminating the strength of the hopping between the two ends of a chain, a new series of exceptional points must be passed through and the topological structure of the spectrum in the complex plane has been changed. This makes the spectrum topology different for the chains with and without boundaries. We also discuss that such exotic behavior does not emerge when the open boundary is replaced by a domain-wall. So the index theorem can be applied to the systems with domain-walls but cannot be further used to those with open boundaries.

Perspective on topological states of non-Hermitian lattices

“…Finally, we comment on an extreme case of defectiveness found in non-Hermitian lattices: a pristine (without disorder) finite system may become devoid of extended states having all eigenstates localized close to a boundary [47,48]; the same systems typically show an extreme sensitivity of the spectrum to a change in the boundary conditions (from open to periodic) [47]. This anomalous localization was attributed to the proximity (in parameter space) to exceptional points with an order scaling with the system size in [48], an effect which was interpreted as an environment-mediated interaction effect.…”

“…Recently, Lee and Thomale [66] presented a characterization of these peculiar boundary modes and provided conditions for their existence. The non-Hermitian skin effect presents a conceptual obstacle for building a bulk-boundary correspondence as typically the topological invariants are based on the bulk eigenstates of the system, which in these cases may have a completely different character than the eigenstates of the finite system, no matter how large [47]. We note that, after submission of this article, different experiments have put forward realizations of this anomalous localization in topoelectrical circuits [106,107], quantum walks [108] mechanical systems [109] and magnons [110].…”

“…Together with Floquet systems, non-Hermitian lattices represent our first strong steps on the land of nonequilibrium, dynamical, topological phases. The departure from the Hermitian paradigm brings many surprises, like the fact that because of the extreme sensitivity to boundary conditions, a pristine non-Hermitian lattice may become devoid of extended states [47,48], an effect termed the non-Hermitian skin effect [49,50].…”

“…On the other hand, there is an intense activity in the search for a consistent classification [51][52][53][54][55], the definition of the topological invariants [56][57][58], and the search for a bulk-boundary correspondence [47,48,59,60].…”

PT symmetry protected non-Hermitian topological systems

“…There is no general framework to understand topological phase in the presence of gain and loss. There are still many issues that has not been understood fully such as bulk-boundary correspondence 25 , topological invariants 26 and classification of topological systems with symmetries 27,28 in non-Hermitian systems. Two topologically distinct gapped systems in the same symmetry class can be continuously deformed each other without closing the band gap.…”