New topological invariants in non-Hermitian systems

TL;DR

This paper reviews recent advances in topological phases of non-Hermitian systems, highlighting new invariants, unique properties, and potential applications in photonics and beyond.

Contribution

It introduces novel topological invariants specific to non-Hermitian systems and discusses their physical implications and experimental realizations.

Findings

New topological invariants like complex winding numbers and half-integer Chern numbers.

Unique phenomena such as skin effect and topological displacement.

Extension of topological concepts to photonic 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.