Topological charge conservation for continuous insulators

Abstract: This paper proposes a classification of elliptic (pseudo-)differential Hamiltonians describing topological insulators and superconductors in Euclidean space by means of domain walls. 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. A Fedosov-H\"ormander formula implementing in Euclidean spaces an Atiyah-Singer index theorem allows for an explicit computation of the index in terms of the symbol of the Fredholm operator. For Hamiltonians admitting an appropriate decomposition in a Clifford algebra, the index is given by the easily computable degree of a naturally associated map. A practically important property of topological insulators is the asymmetric transport observed along one-dimensional lines generated by the domain walls. This asymmetry is captured by a line conductivity, a physical observable of the system. We prove that the line conductivity is quantized and given by the index of a second Fredholm operator of Toeplitz type. We also prove a topological charge conservation stating that the two aforementioned indices agree. This result generalizes to higher dimensions and higher-order topological insulators the bulk-edge correspondence of two-dimensional materials. 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.