Winding Topology of Multifold Exceptional Points

TL;DR

This paper develops a topological classification framework for multifold exceptional points in non-Hermitian systems, introducing winding numbers that characterize their stability and distribution across parameter spaces.

Contribution

It introduces a systematic topological classification of generic and symmetry-protected multifold exceptional points using resultant winding numbers, revealing their stability and fundamental properties.

Findings

Topological invariants called resultant winding numbers are introduced.

EP$n$s are stable due to the topology of a map to a sphere defined by resultants.

Fundamental doubling theorems for EP$n$s in n-band models are established.

Abstract

Despite their ubiquity, a systematic classification of multifold exceptional points, $n$-fold spectral degeneracies (EP$n$s), remains a significant unsolved problem. In this article, we characterize the Abelian eigenvalue topology of generic EP$n$s and symmetry-protected EP$n$s for arbitrary $n$. The former and the latter emerge in a $(2n-2)$- and $(n-1)$-dimensional parameter space, respectively. By introducing topological invariants called resultant winding numbers, we elucidate that these EP$n$s are stable due to topology of a map from a base space (momentum or parameter space) to a sphere defined by resultants. In a $D$-dimensional parameter space ($D\geq c$), the resultant winding number topologically characterize a $(D-c)$-dimensional manifold of generic [symmetry-protected] EP$n$s whose codimension is $c=2n-2$ [$c=n-1$]. Our framework implies fundamental doubling theorems for both generic EP$n$s and symmetry-protected EP$n$s in $n$-band models.