A $\mathbb{Z}_{2}$-Topological Index as a $\mathbb{Z}_{2}$-State Index

TL;DR

This paper introduces a new $$-topological index for fermionic states in infinite-dimensional self-dual CAR algebras, reformulating existing indices in terms of states and GNS representations to better understand topological phases.

Contribution

It reformulates the $$-index for fermionic states in terms of states and GNS representations, extending the index's applicability beyond basis projections.

Findings

Reformulation of the $$-index in terms of states.

Connection between GNS representation equivalences and index parity.

Extension of topological index concepts to infinite-dimensional fermionic systems.

Abstract

Within the setting of infinite dimensional self-dual $\mathrm{CAR}$ $C^{*}$-algebras describing fermions in the $\mathbb{Z}^{d}$-lattice, we depart from the well-known Araki-Evans $\sigma(P_{1},P_{2})$ $\mathbb{Z}_{2}$-index for quasi-free fermion states and rewrite it in terms of states, rather than in terms of basis projections. Furthermore, we reformulate results which relate equivalences of Fock representations with the index parity into results which relate equivalences of GNS representations and the associated index parity.