Knot topology of exceptional point and non-Hermitian no-go theorem

TL;DR

This paper develops a topological classification of exceptional points in non-Hermitian systems using homotopy theory, revealing their knot structures and introducing a non-Hermitian no-go theorem that constrains EP configurations.

Contribution

It introduces a homotopy-based topological classification of EPs, linking eigenenergy braids to knot theory, and proposes a non-Hermitian no-go theorem for EP arrangements.

Findings

EPs characterized by braid groups and knot invariants

Quantized discriminant as knot writhe

A non-Hermitian no-go theorem constraining EP configurations

Abstract

Exceptional points (EPs) are peculiar band singularities and play a vital role in a rich array of unusual optical phenomena and non-Hermitian band theory. In this paper, we provide a topological classification of isolated EPs based on homotopy theory. In particular, the classification indicates that an $n$-th order EP in two dimensions is fully characterized by the braid group B$_n$, with its eigenenergies tied up into a geometric knot along a closed path enclosing the EP. The quantized discriminant invariant of the EP is the writhe of the knot. The knot crossing number gives the number of bulk Fermi arcs emanating from each EP. Furthermore, we put forward a non-Hermitian no-go theorem, which governs the possible configurations of EPs and their splitting rules on a two-dimensional lattice and goes beyond the previous fermion doubling theorem. We present a simple algorithm generating the non-Hermitian Hamiltonian with a prescribed knot. Our framework constitutes a systematic topological classification of the EPs and paves the way towards exploring the intriguing phenomena related to the enigmatic non-Hermitian band degeneracy.