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  • Letter
  • Open Access

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

Haiping Hu1,*, Shikang Sun1,2, and Shu Chen1,2,3

  • 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • 3Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China

  • *hhu@iphy.ac.cn
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Phys. Rev. Research 4, L022064 – Published 27 June, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L022064

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 Letter, we provide a topological classification of isolated EPs based on homotopy theory. In particular, the classification indicates that an nth order EP in two dimensions is fully characterized by the braid group Bn, 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.

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Phys. Rev. Research 4, L022064– Published 27 June, 2022
  • Received 28 November 2021
  • Revised 18 May 2022
  • Accepted 30 May 2022
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