The Hopf invariants of 3D topological insulators are identified with Dixmier-Douady invariants of tensor Berry connections, which quantize magnetoelectric and shift-current responses.
Non- Abelian Hopf-Euler insulators,
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Probing Tensor Monopoles and Gerbe Invariants in Three-Dimensional Topological Matter
The Hopf invariants of 3D topological insulators are identified with Dixmier-Douady invariants of tensor Berry connections, which quantize magnetoelectric and shift-current responses.