Topological phases: isomorphism, homotopy and $K$-theory

Abstract

Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modelled by vector bundles over Brillouin tori, physically relevant equivalences include isomorphism, homotopy, and $K$-theory, which are inequivalent but closely related. We discuss an important subtlety which arises in the chiral Class AIII systems, where the winding number invariant is shown to be relative rather than absolute as is usually assumed. These issues are then analyzed and reconciled in the language of $K$-theory.

14 pages, revised with new references

References in corpus (3)

• Experimental realisation of the topological Haldane model
• Bott periodicity for $Z_2$ symmetric ground states of gapped free-fermion systems
• Parallel Transport and Band Theory in Crystals

Cited by in corpus (10)

• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Phase rotation symmetry and the topology of oriented scattering networks
• A noncommutative framework for topological insulators
• T-duality simplifies bulk-boundary correspondence
• T-duality simplifies bulk-boundary correspondence: some higher dimensional cases
• Bulk-interface correspondences for one dimensional topological materials with inversion symmetry
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices
• Application of semifinite index theory to weak topological phases
• Topological edge states of 1D chains and index theory