Harmonic Analysis in Operator Algebras and its Applications to Index Theory and Topological Solid State Systems

Invariants of disordered semimetals via the spectral localizer

“…Let us also note that the kernel of H typically lies in one chiral sector so that the left-hand side of ( 10) is simply ± T (Ker( H)) with a sign pending on which chiral sector is occupied. Finally let us stress that (10) also holds for chiral Hamiltonians in arbitrary dimension [43]. Furthermore similar identities linking weak Chern numbers to surface current densities in d = 3 Weyl semimetals can be derived [15].…”

“…and Ch {2} (A) = 0. This shows that the term invariant is to be taken with a grain of salt, but at least Ch {j} (A) are known to vary continuously in the parameters of the Hamiltonian [43].…”

“…Then there are two weak winding numbers given by Let us stress that A is not invertible, but the inverse as unbounded operator can have sufficiently regularity for this formula to have a well-defined mathematical meaning. This is the case [43] if the Fermi level either lies in a pseudogap or at least in a region of Anderson localization (in technical terms: the Aizenman-Molcanov bounds [44] hold). Whether these conditions actually hold for a generic disorder is a heavily disputed question [19,24,26,28].…”

“…(10) Here α is the (possibly irrational) cutting angle of the half-space, T is the trace per surface along the boundary and finally σ 3 is the Pauli matrix in the grading of the chiral Hamiltonian as in (5). The robust equality (10) holds under the above conditions assuring the existence of Ch {j} (A) [43] and thus for all parameters of the Hamiltonian. Let us note that it is an extension to semimetals and to arbitrary α of similar identities for topological insulators [7].…”