A $\mathbb{Z}_{2}$-Topological Index for Quasi-Free Fermions

We use infinite dimensional self-dual $\mathrm{CAR}$ $C^{*}$-algebras to study a $\mathbb{Z}_{2}$-index, which classifies free-fermion systems embedded on $\mathbb{Z}^{d}$ disordered lattices. Combes-Thomas estimates are pivotal to show that the $\mathbb{Z}_{2}$-index is uniform with respect to the size of the system. We additionally deal with the set of ground states to completely describe the mathematical structure of the underlying system. Furthermore, the weak$^{*}$-topology of the set of linear functionals is used to analyze paths connecting different sets of ground states.