Topological characterization of multifold band degeneracies in Altland-Zirnbauer symmetry classes

Abstract

Topological band degeneracies are conventionally characterized by invariants defined on enclosing spheres over which the energy spectrum remains gapped. This program has been completed for minimal degeneracies in all ten Altland-Zirnbauer (AZ) symmetry classes, whereas higher-order degeneracies have been studied almost exclusively under crystalline-symmetry protection. In this work, we characterize generic n-fold band degeneracies whose stability derives solely from AZ symmetries acting locally in momentum space. We find that their codimension grows quadratically with n, placing such multifold nodes in parameter spaces that combine physical momenta with tuning parameters or synthetic dimensions. However, the enclosing sphere paradigm faces a fundamental obstruction: two (n-1)-fold degeneracy loci intersecting at the n-fold band node pierce every enclosing sphere, implying that no uniform spectral gap (and thus no standard homotopy classification) exists. Here, we turn this obstruction into the diagnostic itself. On the nodal manifolds where the two loci intersect the enclosing sphere, complementary spectral gaps are restored, allowing us to characterize each with conventional band invariants. This enables us to establish a two-way correspondence: (1)~the multifold node is topologically protected whenever the nodal manifolds are robustly linked on the enclosing sphere, and (2)~band invariants on cycles of one nodal manifold encode their linking numbers with cycles of the other. We carry out this characterization for minimal models of all ten AZ classes, computing the band invariants wherever an explicit parametrization is available. Our results recast multifold band topology as the topology of linked nodal manifolds in momentum space, while our methods provide the foundation for a general characterization of multifold band degeneracies in models with arbitrarily many bands.