On the K-theory of magnetic algebras: Iwatsuka case

Abstract

In the tight-binding approximation, an Iwatsuka magnetic field is modeled by a function on $\mathbb{Z}^2$ with constant, but distinct values in the two parts of the lattice separated by a straight line of slope $\alpha\in [-\infty,\infty]$. In this paper, the $K$-theory of the magnetic $C^*$-algebras generated by an Iwatsuka magnetic field for any possible $\alpha$ is computed. One interesting aspect concerns the analysis of the behavior of the system in the transition from rational to irrational $\alpha$. It turns out that when $\alpha$ is irrational, the magnetic hull associated with the flux operator forms a Cantor set. On the other hand, for rational $\alpha$ this set coincides with the two-point compactification of $\mathbb{Z}$. This characterization, along with the use of the Pimsner-Voiculescu exact sequence, is the main ingredient for the computation of the $K$-theory. Once the $K$-theory is known, with the use of the index theory one can deduce the bulk-interface correspondence for tight-binding Hamiltonians subjected to an Iwatsuka magnetic field. Notably, it occurs that the topological quantization of the interface currents remains independent of the slope $\alpha$.