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

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.