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

• URL: http://arxiv.org/abs/2608.28796v1
• Date: Fri, 28 Aug 2026 19:03:39 GMT
• Title: The $\mathbb{Z}_2$-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors
• Abstract summary: We prove that the index is well-defined, multiplicative, invariant under parity-preserving automorphisms.<n>We then introduce symmetric local automorphism paths and prove that $mathcalN_$ completely classifies the resulting automorphic-path-components.
• Score: 0.5013248430919224
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We define a relative $\mathbb{Z}_2$-index $\mathcal{N}(ω_1,ω_2)$ 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 $σ$ and define the many-body Majorana number $\mathcal{N}σ(ω):=\mathcal{N}(ω,ω\circσ)$ for parity-invariant $σ$-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 $\mathcal{N}_σ$ 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.

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