Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology

Abstract

The topological classification of gapped band structures depends on the particular definition of topological equivalence. For translation-invariant systems, stable equivalence is defined by a lack of restrictions on the numbers of occupied and unoccupied bands, while imposing restrictions on one or both leads to ``fragile'' and ``delicate'' topology, respectively. In this article, we describe a homotopic classification of band structures -- which captures the topology beyond the stable equivalence -- in the presence of additional lattice symmetries. As examples, we present complete homotopic classifications for spinless band structures with twofold rotation, fourfold rotation and fourfold dihedral symmetries, both in presence and absence of time-reversal symmetry. Whereas the rules of delicate and fragile topology do not admit a bulk-boundary correspondence, we identify a version of stable topology, which restricts the representations of bands, but not their numbers, which does allow for anomalous states at symmetry-preserving boundaries, which are associated with nontrivial bulk topology.

30+15 pages, 19 figures

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• Homotopy, Symmetry, and Non-Hermitian Band Topology
• Representation-protected topology of spin-singlet $s$-wave superconductors
• From quantum geometry to non-linear optics and gerbes: Recent advances in topological band theory
• Non-Hermitian Hopf insulators
• Euler band topology in superfluids and superconductors
• Three-dimensional spinless Euler insulators with rotational symmetry
• Gapped spinful phases obtained via Gutzwiller projections of Euler states
• Hinge modes of three-dimensional Euler insulators