Inverse Design of Winding Tuple for Non-Hermitian Topological Edge Modes

TL;DR

This paper develops an inverse design method for topological edge modes in non-Hermitian systems by deriving their wave functions and constructing winding invariants, enabling better control and application of these modes.

Contribution

It introduces a novel inverse design approach for topological invariants in non-Hermitian systems, including a spectral winding number, to characterize and manipulate edge modes.

Findings

Derived wave functions for edge modes in a non-Hermitian SSH model.

Constructed a winding tuple to characterize edge mode existence and distribution.

Defined a spectral winding number linked to band energies.

Abstract

The interplay between topological localization and non-Hermiticity localization in non-Hermitian crystal systems results in a diversity of shapes of topological edge modes (EMs), offering opportunities to manipulate these modes for potential topological applications. The characterization of the domain of EMs and the engineering of these EMs require detailed information about their wave functions, which conventional calculation of topological invariants cannot provide. In this Letter, by recognizing EMs as specified solutions of eigenequation, we derive their wave functions in an extended non-Hermitian Su-Schrieffer-Heeger model. We then inversely construct a winding tuple $\left \{ w_{\scriptscriptstyle GBZ},w_{\scriptscriptstyle BZ}\right \} $ that characterizes the existence of EMs and their spatial distribution. Moreover, we define a novel spectral winding number equivalent to $w_{\scriptscriptstyle BZ}$, which is determined by the product of energies of different bands. The inverse design of topological invariants allows us to categorize the localized nature of EMs even in systems lacking sublattice symmetry, which can facilitate the manipulation and utilization of EMs in the development of novel quantum materials and devices.