Dimensional crossover of class D topological invariants

The topological invariants corresponding to a given symmetry class in the Atland-Zirnbauer classification depend on the dimension of the Hamiltonian. In this talk, we will consider a model in class D, given by magnetic adatoms placed on a superconductor (Shiba lattices), and study how the topology of a 2D island of magnetic adatoms transitions into the topology of 1D chains. To this end, we employ the spectral localizer, a real-space local topological marker endowed with a local measure of the topological protection. In 2D, the signature of the spectral localizer predicts the Chern number, while in 1D, the sign of the determinant of the dimensionally-reduced spectral localizer predicts the Z2 invariant. Through the entire dimensional crossover process, we can quantitatively monitor the topological protection of both the 2D Chern number and the 1D topology. The effect of chemical potential disorder will also be discussed.