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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