Topological characterization of multifold band degeneracies in Altland-Zirnbauer symmetry classes
Abstract
Topological degeneracies of energy bands in crystalline matter are conventionally characterized by invariants computed on an enclosing sphere over which the spectrum remains gapped, a program completed for minimal degeneracies in all ten Altland-Zirnbauer (AZ) symmetry classes. Higher-order degeneracies have instead been studied almost exclusively under crystalline-symmetry protection. Here, we characterize generic n-fold degeneracies stabilized solely by AZ symmetries acting locally in momentum space. Their codimension grows quadratically with n, placing multifold nodes in parameter spaces combining momenta with tuning parameters or synthetic dimensions. Crucially, the enclosing-sphere paradigm faces a fundamental obstruction: two (n − 1)-fold degeneracy loci emanating from the n-fold node necessarily pierce every choice of enclosing sphere, leaving no uniform spectral gap and thus no standard homotopy classification. We elevate this obstruction into the diagnostic itself. Namely, on the two nodal manifolds where the loci cross the sphere, complementary spectral gaps are restored, admitting conventional band invariants (Chern numbers, Stiefel-Whitney classes, and winding numbers). This observation establishes a general two-way correspondence: (1) the multifold node is topologically protected whenever the associated nodal manifolds are robustly linked, and (2) invariants on cycles of one manifold encode their linking numbers with cycles of the other. Carrying out this program for minimal models of all ten AZ classes, we recast multifold band topology as the topology of linked nodal manifolds and lay the foundation for characterizing multifold nodes in models with arbitrarily many bands.
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BibTeX
@misc{iliasov2026topological,
title = {{Topological characterization of multifold band degeneracies in Altland-Zirnbauer symmetry classes}},
author = {Askar Iliasov and Zoltán Guba and Tsuneya Yoshida and Apoorv Tiwari and Tomáš Bzdušek},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.13016},
eprint = {2607.13016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13016},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Topological characterization of multifold band degeneracies in Altland-Zirnbauer symmetry classes
AU - Askar Iliasov
AU - Zoltán Guba
AU - Tsuneya Yoshida
AU - Apoorv Tiwari
AU - Tomáš Bzdušek
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2607.13016
UR - https://arxiv.org/abs/2607.13016
AB - Topological degeneracies of energy bands in crystalline matter are conventionally characterized by invariants computed on an enclosing sphere over which the spectrum remains gapped, a program completed for minimal degeneracies in all ten Altland-Zirnbauer (AZ) symmetry classes. Higher-order degeneracies have instead been studied almost exclusively under crystalline-symmetry protection. Here, we characterize generic n-fold degeneracies stabilized solely by AZ symmetries acting locally in momentum space. Their codimension grows quadratically with n, placing multifold nodes in parameter spaces combining momenta with tuning parameters or synthetic dimensions. Crucially, the enclosing-sphere paradigm faces a fundamental obstruction: two (n − 1)-fold degeneracy loci emanating from the n-fold node necessarily pierce every choice of enclosing sphere, leaving no uniform spectral gap and thus no standard homotopy classification. We elevate this obstruction into the diagnostic itself. Namely, on the two nodal manifolds where the loci cross the sphere, complementary spectral gaps are restored, admitting conventional band invariants (Chern numbers, Stiefel-Whitney classes, and winding numbers). This observation establishes a general two-way correspondence: (1) the multifold node is topologically protected whenever the associated nodal manifolds are robustly linked, and (2) invariants on cycles of one manifold encode their linking numbers with cycles of the other. Carrying out this program for minimal models of all ten AZ classes, we recast multifold band topology as the topology of linked nodal manifolds and lay the foundation for characterizing multifold nodes in models with arbitrarily many bands.
ER -