1D Anyonic Statistics from Confined Hopf Solitons

Title: 1D Anyonic Statistics from Confined Hopf Solitons

Author: Alexander Novickis (alex.novickis@gmail.com)

Hidalgo-Sacoto, Busch, and Blume (2025, Phys. Rev. A) recently demonstrated 1D identical particles obey generalised exchange statistics parameterised by $\alpha \in [0, 1]$. We show a Hopf soliton confined to a 1D quantum wire has two programme-derivable exchange phases applying in distinct confinement regimes. B-Lite (Wilczek-Zee + Finkelstein-Rubinstein spin-statistics chain): in weak confinement ($\sigma \gtrsim 0.3\,\mu$m), $\alpha_H^{\rm B\text{-}Lite} = H \bmod 2$ — pinned at endpoints $\{0, 1\}$. B-Full-B (SU(2)$_{k=H}$ Chern-Simons descent): in strong confinement ($\sigma \in [0.05, 0.1]\,\mu$m), the F$_2$ Hopf invariant identifies with level-$k=H$ SU(2) CS (Faddeev-Niemi 1997), descends via Witten's CS-WZW correspondence (Witten 1989) to SU(2)$_{k=H}$ WZW $j = 1/2$ primary, giving the closed-form $\boxed{\alpha_H^{\rm B\text{-}Full\text{-}B} = 3 / [2(H+2)]}$. For $H = 1, 2, 3, 4$: $1/2, 3/8, 3/10, 1/4$ — non-endpoint values in the interior of the Hidalgo-Sacoto range. Both predictions are pinned across the scattering length $a_{\rm any}$ in their regimes. Protocol E2b cold-atom experiments at tunable $H$ discriminate three scenarios — Hidalgo-Sacoto generic, B-Lite, B-Full-B — providing a uniquely diagnostic signature of the F$_2$-Hopf-CS structure.

Key results include:

• Hidalgo-Sacoto, Busch, Blume (2025) introduce 1D anyons with continuous exchange statistics $\alpha \in [0, 1]$ governed by short-range interactions.
• A 3D Hopf soliton of charge $H$ confined to a 1D quantum wire inherits its 3D exchange phase via the spin-statistics chain $H = 2s$ (Wilczek-Zee 1983) + Finkelstein-Rubinstein topology, giving $\alpha_H = H \bmod 2$ — i.e., pinned at the endpoints $\{0, 1\}$ of the Hidalgo-Sacoto continuous range, independent of the scattering length.
• Discriminator from the generic framework: scan the scattering length on a sample with known Hopf charge. Hopf programme: α stays pinned. Generic Hidalgo-Sacoto: α varies continuously.
• The test extends Paper CXXVI Protocol E2 (BEC hopfion magnetism) to 1D cold-atom settings.
• The deeper question of whether the F₂ fiber structure or an effective SU(2)_1 Chern-Simons theory in 1D could give non-endpoint $\alpha = 1/2$ (semion) predictions is flagged for v0.3 (deferred research direction).

Series: Paper CXXXVII in the Hopf Soliton Programme