The $\mathbb{Z}_2$-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors

Abstract:We define a relative $\mathbb{Z}_2$-index $\mathcal{N}(\omega_1,\omega_2)$ for locally-comparable, parity-invariant pure states on a unital $C^*$-algebra. We prove that the index is well-defined, multiplicative, invariant under parity-preserving automorphisms, and locally constant in the norm topology. For the one-dimensional self-dual CAR algebra, we apply this construction to the half-chain automorphism $\sigma$ and define the many-body Majorana number $\mathcal{N}\sigma(\omega):=\mathcal{N}(\omega,\omega\circ\sigma)$ for parity-invariant $\sigma$-local pure states. In the quasi-free Hilbert--Schmidt regime, this index agrees with the usual single-body Majorana number. We then introduce symmetric local automorphism paths and prove that $\mathcal{N}_\sigma$ completely classifies the resulting automorphic-path-components. This automorphic-path equivalence retains the bulk Majorana number but may forget a relative zero-dimensional parity obstruction.

arXiv:2608.28796 [math-ph]

(or arXiv:2608.28796v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2608.28796 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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