Paper 30: One-Dimensional Anyon Exchange Statistics from the Spatial Elastic Field Theory: Quantitative Derivation and Experimental Tests within the Huaxia Space-Energy Totality Conservation Theory
One-dimensional anyons—whose exchange statistics are continuously tunable between bosons and fermions—constitute one of the most important frontier discoveries in condensed matter physics and ultracold atom research in 2025 and 2026. We provide a physical explanation from first principles for anyon exchange statistics within the H-SET framework. In H-SET, particles are helical dislocation loops in space. In a one-dimensional channel, two dislocation loops cannot bypass each other—they must directly penetrate. The penetration operation leaves irreversible entanglement in the space elastic field, and the degree of entanglement determines the value of the exchange factor. We rigorously derive the quantitative relationship between the exchange factor and the effective space elastic modulus from the fundamental formulas of dislocation elasticity theory. Since the effective elastic modulus can be continuously tuned via Feshbach resonance in ultracold atom experiments, the exchange factor is naturally continuously tunable between the bosonic limit and the fermionic limit. We further provide four quantitative predictions that can be independently tested by experiment.
One-dimensional anyons—whose exchange statistics are continuously tunable between bosons and fermions—constitute one of the most important frontier discoveries in condensed matter physics and ultracold atom research in 2025 and 2026. We provide a physical explanation from first principles for anyon exchange statistics within the H-SET framework. In H-SET, particles are helical dislocation loops in space. In a one-dimensional channel, two dislocation loops cannot bypass each other—they must directly penetrate. The penetration operation leaves irreversible entanglement in the space elastic field, and the degree of entanglement determines the value of the exchange factor. We rigorously derive the quantitative relationship between the exchange factor and the effective space elastic modulus from the fundamental formulas of dislocation elasticity theory. Since the effective elastic modulus can be continuously tuned via Feshbach resonance in ultracold atom experiments, the exchange factor is naturally continuously tunable between the bosonic limit and the fermionic limit. We further provide four quantitative predictions that can be independently tested by experiment.
One-dimensional anyons, whose exchange statistics can be continuously tuned between bosonic and fermionic limits, are among the most significant frontier discoveries in condensed matter and ultracold atomic physics. Existing theories describe the statistical behavior of anyons at the level of mathematical structures or effective models, but none addresses the fundamental question of the physical origin of the exchange factor. This work provides a first-principles physical explanation for anyon exchange statistics within the Huaxia Spacetime Total Conservation Theory. In H-SET, particles are helical dislocation loops in the space medium. In a one-dimensional channel, two dislocation loops cannot bypass each other and must directly penetrate. The penetration operation leaves an irreversible twist in the spatial elastic field, and the degree of twisting determines the exchange factor. Starting from basic formulas of dislocation elasticity theory, we rigorously derive the quantitative relationship between the exchange factor and the effective spatial elastic modulus. Because the effective elastic modulus can be continuously tuned via Feshbach resonance, the exchange factor is naturally and continuously tunable between the bosonic and fermionic limits, in quantitative agreement with the core finding of a landmark experiment. We further provide four quantitative predictions that can be tested by independent experiments, including the quantitative mapping between the exchange factor and the scattering length, the momentum-distribution asymmetry ratio, the linear constraint on the Haldane statistical parameter, and the braid-group locking mechanism for the two-dimensional braiding phase. All derivations proceed from the established constants of H-SET, without introducing any free parameters specifically adjusted for the anyon phenomenon.
In February 2026, Hidalgo-Sacoto et al. (OIST/Oklahoma) demonstrated that anyons — particles with exchange statistics intermediate between bosonic and fermionic — can exist in one-dimensional systems, with a continuously tunable exchange factor controlled by interaction strength. This volume proves that this result is a direct consequence of the algebraic framework of Project T. We establish the Anyon Theorem: the commutation deficit C(K) = K(K−3)/2 determines whether a rank-K system admits discrete or continuous exchange statistics. Specifically: (1) C(K) = 0 (uniquely K = 3) forces the exchange factor to be exactly ±1, yielding only bosons and fermions; (2) C(K) > 0 (K ≥ 4) opens escape routes through commuting pairs, allowing the exchange factor to vary continuously — producing anyons; (3) the continuous tunability observed by OIST maps precisely onto the irrational coupling constants of collapsed ranks K ≥ 4. The framework makes a falsifiable prediction: in any system where the effective algebraic rank can be experimentally tuned, the transition from discrete to continuous exchange statistics should occur at the critical point where C(K) first becomes nonzero. The OIST experiment, performed in ultracold atomic gases with tunable interactions, represents the first independent confirmation of the dimensional collapse mechanism predicted by Volume VI. Keywords: anyons, exchange statistics, Yang-Baxter equation, commutation deficit, dimensional collapse, 0-Hecke monoid, ultracold atoms, braiding, Lean 4
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions---applicable to quasiparticles---constitute an intriguing topic from both the fundamental and practical points of view. This work develops a scattering framework for two identical one-dimensional (1D) bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, in both free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon---fermionic-anyon mapping (R. Hidalgo-Sacoto et al., Phys. Rev. A 112, 063309 (2025)), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ for two harmonically confined anyons. Nonuniversal contributions at order ${k}^{\ensuremath{-}4}$ are discussed.
Recent work has demonstrated that identical particles in one spatial dimension can exhibit continuously tunable exchange statistics, extending the conventional boson–fermion dichotomy. These one–dimensional (1D) anyons are described within rigorously defined many–body models and display universal high–momentum tails whose leading coefficients depend on the statistical parameter. In this article, we present a conceptual and geometric interpretation of these results within the Helix–Light–Vortex (HLV) framework. We do not propose a competing microscopic theory and we do not modify established Hamiltonians. Instead, we provide a mapping in which the tunable exchange phase can be interpreted as a memory–modulated effective statistics parameter inspired by the triadic spiral–time structure of HLV. The manuscript is intentionally positioned as an interpretive bridge between established 1D anyon theory and a geometric spacetime language. All HLV–related elements are explicitly marked as hypotheses. No experimental validation, benchmarking, or performance claims are made.
Low-dimensional quantum systems can host anyons, particles with exchange statistics that are neither bosonic nor fermionic. However, the physics of anyons in one dimension remains largely unexplored. In this work, we realize Abelian anyons in one dimension with arbitrary exchange statistics using ultracold atoms in an optical lattice, where we engineer the statistical phase through a density-dependent Peierls phase. We explore the dynamical behavior of two anyons undergoing quantum walks and observe the anyonic Hanbury Brown-Twiss effect as well as the formation of bound states without on-site interactions. Once interactions are introduced, we observe spatially asymmetric transport in contrast to the symmetric dynamics of bosons and fermions. Our work forms the foundation for exploring the many-body behavior of one-dimensional anyons.
Low-dimensional quantum systems can host anyons, particles with exchange statistics that are neither bosonic nor fermionic. However, the physics of one-dimensional (1D) anyons remains largely unexplored. Using ultracold bosonic Rb-87 atoms in an optical lattice, we realize and explore the anyon-Hubbard model, which hosts 1D anyons and features a rich phase diagram. We implement the model using techniques from Floquet engineering, realizing the statistical phase in the form of a density-dependent Peierls phase. Using our high-resolution imaging system, we explore both the dynamical and ground state behavior in the anyon-Hubbard model. In the first set of experiments, we probe 1D anyons via two-particle quantum walks, observing the anyonic Hanbury Brown-Twiss effect, as well as the formation of bound states without on-site interactions. Once interactions are introduced, we observe spatially asymmetric transport in contrast to the symmetric dynamics of bosons and fermions. In the next set of experiments, we adiabatically prepare and probe ground states in the anyon-Hubbard model. First, we observe continuous fermionization in the density profile with increasing statistical phase. Then, we probe the chirality of the bound states in the anyon-Hubbard model, drawing similarities to a continuum model of 1D anyons. This work constitutes the first experimental study of the anyon-Hubbard model, and lays the foundation for exploration in the strongly correlated many-body regime.