New topological invariants in non-Hermitian systems

Abstract

Both theoretical and experimental studies of topological phases in non-Hermitian systems have made a remarkable progress in the last few years of research. In this article, we review the key concepts pertaining to topological phases in non-Hermitian Hamiltonians with relevant examples and realistic model setups. Discussions are devoted to both the adaptations of topological invariants from Hermitian to non-Hermitian systems, as well as origins of new topological invariants in the latter setup. Unique properties such as exceptional points and complex energy landscapes lead to new topological invariants including winding number/vorticity defined solely in the complex energy plane, and half-integer winding/Chern numbers. New forms of Kramers degeneracy appear here rendering distinct topological invariants. Modifications of adiabatic theory, time-evolution operator, biorthogonal bulk-boundary correspondence lead to unique features such as topological displacement of particles, 'skin-effect', and edge-selective attenuated and amplified topological polarizations without chiral symmetry. Extension and realization of topological ideas in photonic systems are mentioned. We conclude with discussions on relevant future directions, and highlight potential applications of some of these unique topological features of the non-Hermitian Hamiltonians.

Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions

“…Indeed, a large portion of recent studies has focused on PT symmetry and the role of exceptional points [50], whose intriguing properties lead to unconventional phenomena such as unidirectional invisibility [44,51], single-mode lasers [52] and enhanced sensitivity to perturbations [53,54]. Understanding the topological properties of non-Hermitian systems has also been the focus of many research efforts [55][56][57][58][59]. Initial interest revolved around exceptional points exhibiting unique topological features with no counterparts in Hermitian systems, such as Weyl exceptional rings [60], bulk Fermi arcs and half-integer topological charges [61].…”

Perspective on topological states of non-Hermitian lattices

“…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].…”

Active topological photonics

“…It is also known that non-Hermiticity can explicitly disrupt the conventional bulk-boundary correspondence (BBC) held in Hermitian systems [119,130,[139][140][141][131][132][133][134][135][136][137][138]. Especially, asymmetric couplings make not only topological edge states but also non-topological bulk states localize around an either end to which the stronger hopping is directed (non-Hermitian skin effect) [ Fig.…”