While such computations may be carried out explicitly for some models, typically by means of spectral flows, 3,6,26,41,50 they remain notoriously difficult for more complex systems. It remains to define and compute topological invariants that characterize such an asymmetry. We apply this procedure to evaluate the topological charge of several classical examples of (standard and higher-order) topological insulators and superconductors in one, two, and three spatial dimensions.
This result generalizes to higher dimensions and higher-order topological insulators, the bulk-edge correspondence of two-dimensional materials. We also prove topological charge conservation by stating that the two aforementioned indices agree. We prove that the edge conductivity is quantized and given by the index of a second Fredholm operator of the Toeplitz type.
This asymmetry is captured by the edge conductivity, a physical observable of the system. A practically important property of topological insulators is the asymmetric transport observed along one-dimensional lines generated by the domain walls. For Hamiltonians admitting an appropriate decomposition in a Clifford algebra, the index is given by the easily computable topological degree of a naturally associated map. The index is computed explicitly in terms of the symbol of the Hamiltonian by a Fedosov–Hörmander formula, which implements in Euclidean spaces an Atiyah–Singer index theorem. Augmenting a given Hamiltonian by one or several domain walls results in confinement that naturally yields a Fredholm operator, whose index is taken as the topological charge of the system. This paper proposes a classification of elliptic (pseudo-)differential Hamiltonians describing topological insulators and superconductors in Euclidean space by means of domain walls.
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