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  "slug": "projective-braid-group-representations-from-a-microscopic-long-range-kitaev-chai",
  "title": "Projective Braid Group Representations from a Microscopic Long-Range Kitaev Chain",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "We construct an explicit projective representation of the braid group from the microscopic Hilbert space of a Kitaev chain extended with algebraically decaying long-range hopping and pairing. The construction proceeds in three steps. First, we identify the topological phase of the long-range Kitaev chain (power-law exponent α) and show that unpaired Majorana zero modes (MZMs) localize at domain walls, with localization length controlled by α. Second, we write the braid unitary U_ij = exp((π/4)γ_iγ_j) acting on the zero-mode Majorana operators γ_i and verify, by explicit matrix computation in a fixed-parity sector, that the braid relation σ₁σ₂σ₁ = σ₂σ₁σ₂ holds exactly while the relation σ⁴ = 1 is lifted: U⁴ = −𝟙 rather than +𝟙. This defines a genuine projective (2-cocycle-twisted) representation whose class is the Ising one, with per-exchange projective phase anchored at π/8. Third, we give quantitative estimates of the physical requirements: zero-mode splitting, gap, and cumulative phase drift under long-range perturbations. [...]",
  "abstract_markdown": "We construct an explicit projective representation of the braid group from the microscopic Hilbert space of a Kitaev chain extended with algebraically decaying long-range hopping and pairing. The construction proceeds in three steps. First, we identify the topological phase of the long-range Kitaev chain (power-law exponent α) and show that unpaired Majorana zero modes (MZMs) localize at domain walls, with localization length controlled by α. Second, we write the braid unitary U_ij = exp((π/4)γ_iγ_j) acting on the zero-mode Majorana operators γ_i and verify, by explicit matrix computation in a fixed-parity sector, that the braid relation σ₁σ₂σ₁ = σ₂σ₁σ₂ holds exactly while the relation σ⁴ = 1 is lifted: U⁴ = −𝟙 rather than +𝟙. This defines a genuine projective (2-cocycle-twisted) representation whose class is the Ising one, with per-exchange projective phase anchored at π/8. Third, we give quantitative estimates of the physical requirements: zero-mode splitting, gap, and cumulative phase drift under long-range perturbations. [...]",
  "date": "2026-10-03",
  "first_published": "2026-10-03",
  "last_revised": "2026-10-03T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/projective-braid-group-representations-from-a-microscopic-long-range-kitaev-chai",
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    "titles": [
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        "title": "Projective Braid Group Representations from a Microscopic Long-Range Kitaev Chain"
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        "description": "We construct an explicit projective representation of the braid group from the microscopic Hilbert space of a Kitaev chain extended with algebraically decaying long-range hopping and pairing. The construction proceeds in three steps. First, we identify the topological phase of the long-range Kitaev chain (power-law exponent α) and show that unpaired Majorana zero modes (MZMs) localize at domain walls, with localization length controlled by α. Second, we write the braid unitary U_ij = exp((π/4)γ_iγ_j) acting on the zero-mode Majorana operators γ_i and verify, by explicit matrix computation in a fixed-parity sector, that the braid relation σ₁σ₂σ₁ = σ₂σ₁σ₂ holds exactly while the relation σ⁴ = 1 is lifted: U⁴ = −𝟙 rather than +𝟙. This defines a genuine projective (2-cocycle-twisted) representation whose class is the Ising one, with per-exchange projective phase anchored at π/8. Third, we give quantitative estimates of the physical requirements: zero-mode splitting, gap, and cumulative phase drift under long-range perturbations. [...]",
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