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

TL;DR

This paper introduces a $Z_2$-topological index for classifying free-fermion systems on disordered lattices using infinite dimensional self-dual CAR $C^*$-algebras, with implications for understanding ground states and system stability.

Contribution

It develops a $Z_2$-index based on self-dual CAR $C^*$-algebras for free-fermion systems on disordered lattices, incorporating Combes-Thomas estimates and topological analysis.

Findings

The $Z_2$-index is shown to be uniform across system sizes.

The structure of ground states is fully characterized.

Paths connecting different ground state sets are analyzed using weak$^*$ topology.

Abstract

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.