An Extension of the Atiyah-Singer Index Theorem C; Skin Indicator for Open and Non-reciprocal Non-Hermitian Systems

In recent years, non-Hermitian physics has revealed numerous novel phenomena
 in open quantum systems, such as the non-Hermitian skin effect, which challenges
 the conventional bulk-boundary correspondence. This paper proposes a general
ized index theorem that unifies the description of topological characteristics in non
Hermitian systems by introducing biorthogonal Fredholm modules and a skin effect
 correction term. The core formula is the skin indicator, which combines the topo
logical invariants of the classical Atiyah-Singer index theorem with a Jump term
 that quantifies the breakdown of the bulk-boundary relationship. This framework
 is applied to chiral edge currents in the quantum Hall effect and impedance quan
tization in non-Hermitian topological circuits, revealing the intrinsic mechanisms
 of non-reciprocal topological phase transitions. The work presented here provides
 new theoretical tools for the topological analysis of open systems