Unified multipole Bott indices for non-Hermitian skin effect in different orders

Non-Hermitian systems exhibit a distinctive phenomenon known as the non-Hermitian skin effect, where an extensive number of eigenstates become localized at the boundaries of a lattice with open boundaries. While the spectral winding number under periodic boundary conditions is a well-established topological indicator for predicting the skin effect in one-dimensional non-Hermitian systems, a suitable topological invariant to diagnose higher-order skin effects remains elusive. In this article, we propose a unified non-Hermitian multipole characterization framework that generalizes the concept of spectral winding to higher-order skin effects. Specifically, we develop a set of non-Hermitian multipole Bott indices capable of diagnosing skin effects of different orders. Our approach provides a comprehensive understanding of both first- and higher-order skin effects in non-Hermitian systems, offering new perspectives for exploring topological phenomena in non-Hermitian systems.