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PreprintCC BY 4.0arXiv

The β„€2-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors

  • Anna Mazhar
  • Jacob Shapiro
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Abstract

We define a relative β„€2-index 𝒩(Ο‰1,Ο‰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 Οƒ and define the many-body Majorana number 𝒩σ(Ο‰):=𝒩(Ο‰,Ο‰βˆ˜Οƒ) for parity-invariant Οƒ-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 𝒩σ 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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Cite

BibTeXRIS
BibTeX
@misc{mazhar2026β„€2index,
  title = {{The $\mathbbZ_2$-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors}},
  author = {Anna Mazhar and Jacob Shapiro},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28796},
  eprint = {2608.28796},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28796},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The β„€_2-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors
AU  - Anna Mazhar
AU  - Jacob Shapiro
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28796
UR  - https://arxiv.org/abs/2608.28796
AB  - We define a relative β„€_2-index 𝒩(Ο‰_1,Ο‰_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 Οƒ and define the many-body Majorana number 𝒩σ(Ο‰):=𝒩(Ο‰,Ο‰βˆ˜Οƒ) for parity-invariant Οƒ-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 𝒩_Οƒ 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.
ER  -