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Abelian spectral topology of multifold exceptional points

Marcus Stålhammar1,2,* and Lukas Rødland3,4

  • 1Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584CC Utrecht, The Netherlands
  • 2Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden
  • 3Department of Physics, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden
  • 4Max Planck Institute for the Science of Light, Staudtstrasse 2, 91058 Erlangen, Germany

  • *Contact author: marcus.backlund@physics.uu.se
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Phys. Rev. Research 7, 033246 – Published 12 September, 2025

DOI: https://doi.org/10.1103/8lyx-vtnw

Abstract

The advent of non-Hermitian physics has enriched the plethora of topological phases to include phenomena without Hermitian counterparts. Despite being among the most well-studied uniquely non-Hermitian features, the topological properties of multifold exceptional points, n-fold spectral degeneracies (EPns) at which also the corresponding eigenvectors coalesce, were only recently revealed in terms of topological resultant winding numbers and concomitant Abelian doubling theorems. Nevertheless, a more mathematically fundamental description of EPns and their topological nature has remained an open question. To fill this void, in this article, we revisit the topological classification of EPns in generic systems and systems with local symmetries, generalize it in terms of more mathematically tractable (local) similarity relations, and extend it to include all such similarities as well as nonlocal symmetries. Through the resultant vector, whose components are given in terms of the resultants between the corresponding characteristic polynomial and its derivatives, the topological nature of the resultant winding number is understood in several ways: in terms of (1) the tenfold classification of (Hermitian) topological matter, (2) the framework of Mayer-Vietoris sequence, and (3) the classification of vector bundles. The classification scheme further predicts the existence of topological bulk Fermi arcs protected by a Z2-invariant, induced by nonlocal symmetries, dubbed Z2-protected Fermi arcs. Our work reveals the mathematical foundations on which the topological nature of EPns resides, enriches the theoretical understanding of non-Hermitian spectral features, and will therefore find great use in modern experiments within both classical and quantum physics.

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Phys. Rev. Research 7, 033246– Published 12 September, 2025
  • Received 27 January 2025
  • Accepted 12 August 2025
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