Singularity Selector: Topological Chirality via Non-Abelian Loops around Exceptional Points

TL;DR

This paper introduces a topological invariant called topological chirality for non-Hermitian systems with exceptional points, enabling the classification of EP encirclements and extending to complex multi-EP scenarios, with experimental validation.

Contribution

It defines a novel topological invariant based on non-Abelian loops around exceptional points, providing a universal framework for classifying EP encirclements in various physical systems.

Findings

Confirmed binary rule for EP encirclement in optical microcavity

Validated topological chirality in non-Hermitian topological band

Extended invariant to multi-EP systems with higher-order topology

Abstract

Chirality is more than a geometric curiosity; it governs measurable asymmetries across nature, from enantiomer-selective drugs and left-handed fermions in particle physics to handed charge transport in Weyl semimetals. We extend this universal concept to non-Hermitian systems by defining topological chirality, an invariant that emerges whenever an exceptional-points (EP) pair is present. Built from the non-commutative fundamental group and its braid representation, topological chirality acts as a singularity selector: clockwise EP loops occupy a homotopy class that avoids EPs, whereas counter-clockwise mirrors are equivalent only if they cross the EPs themselves. We confirm this binary rule in an optical microcavity and a non-Hermitian topological band. The same two-sheeted topology governs EP pairs in spin systems, photonic crystals and hybrid light-matter structures, where EP encirclements have already been demonstrated, so the framework transfers without alteration and confirms its experimental viability. Our findings lay the cornerstone for interpreting loop-sensitive observables such as spectral vorticity, the complex Berry phase and the non-Abelian holonomy. Finally, a gluing-of-planes construction extends the invariant to an n-sheeted surface hosting 2m EPs, unifying higher-order EP pairs.