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.
Nonrelativistic anyons in one dimension possess generalized exchange statistics in which the exchange of two identical anyons generates a phase that is governed by the spatial ordering of the particles and the statistical parameter $\ensuremath{\alpha}$. Working in the continuum, we demonstrate the existence of two distinct types of one-dimensional (1D) anyons, namely, bosonic anyons and fermionic anyons. We identify a many-body Hamiltonian with additive two-body zero-range interactions that supports bosonic and fermionic anyon eigenstates, which are, for arbitrary interaction strength, related through a generalized bosonic-anyon--fermionic-anyon mapping, an extension of the celebrated Bose-Fermi mapping for zero-range interacting 1D systems. The momentum distributions of bosonic and fermionic anyons are distinct: while both feature ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ tails, the associated prefactors differ. Our work reveals intricate connections between the generalized exchange statistics, the universal two- and three-body Tan contacts of systems consisting of
N identical particles, and the emergence of statistics-induced chiral symmetry breaking.
We study a one-dimensional system of interacting anyons with short-range interactions under external confinement. This model, which describes anyons with zero-range
s-wave and
p-wave scattering interactions in the strong interaction regime, continuously interpolates between spin-polarized fermions with
p-wave interactions and free bosons. At zero temperature, the correlation functions decay exponentially with distance, with oscillations governed by the statistics parameter. The decay rate is maximal for
p-wave fermions and decreases monotonically as the statistics parameter approaches the bosonic limit, where it vanishes. The momentum distribution is asymmetric, a hallmark of one-dimensional anyons, and takes the form of a shifted Lorentzian with universal power-law tails, ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{2}$. We prove analytically that, following release from a harmonic trap, the asymptotic momentum distribution converges to that of free bosons in the same trap, a phenomenon known as dynamical bosonization. We also establish the universality of the ground-state
n-particle reduced density matrices: their natural occupations are independent of the confining potential, while the associated natural
n-point functions for different confinements are related through a simple analytical transformation. In particular, for the one-particle reduced density matrix, we derive exact expressions for both the natural occupations and the natural orbitals at arbitrary particle number. These results extend and unify earlier partial findings for
p-wave fermions, and they provide a clear conceptual explanation of the double degeneracy observed in their spectrum.
We establish an exact mapping between identical particles in one dimension with arbitrary exchange statistics, including bosons, anyons, and fermions, provided they share the same scattering length. This boson-anyon-fermion mapping facilitates the construction of anyons from a linear superposition of spatially symmetric and antisymmetric states. This scheme is general and has been demonstrated in a spin-1/2 Fermi gas, where both s- and p-wave bound states can be supported by manipulating spin channels. With a suitable symmetry-breaking field, these bound states are hybridized to form a fractional-wave molecule. The condensation of these molecules in a many-body system leads to anyonic superfluidity, characterized by fractional statistics upon spin exchange within a Cooper pair. These anyonic states can be detected through asymmetric momentum distributions for each spin with a chiral k − 3 tail. Our results demonstrate the inadequacy of the contact interaction model for anyons in continuum and lattices, and have also suggested a convenient route for engineering fractional phases on the platform of ultracold atoms.
Anyons are low-dimensional quasiparticles that obey fractional statistics, hence interpolating between bosons and fermions. In two dimensions, they exist as elementary excitations of fractional quantum Hall states and they are believed to enable topological quantum computing. One-dimensional (1D) anyons have been theoretically proposed, but their experimental realization has proven to be difficult. Here, we observe anyonic correlations, which emerge through the phenomenon of spin-charge separation, in a 1D strongly-interacting quantum gas. The required spin degree of freedom is provided by a mobile impurity, whose effective anyonic correlations are associated with an experimentally tunable statistical angle. These anyonic correlations are measured by monitoring the impurity momentum distribution, whose asymmetric feature demonstrates the transmutation of bosons via anyons to fermions. Going beyond equilibrium conditions, we study the dynamical properties of the anyonic correlations via dynamical fermionization of the anyons. Our work opens up the door to the exploration of non-equilibrium anyonic phenomena in a highly controllable setting.
An elementary introduction is given to the concept of anyons—quantum particles whose ‘statistics’ interpolate smoothly between those of bosons and fermions. Such particles—or quasi-particle excitations—can occur in two-dimensional systems in which the particles experience strong short-range mutual repulsions. These circumstances effectively mean that one has to apply quantum mechanics to a configuration space in which some points are excluded (i.e. it has ‘holes’)—namely, those points where two particles would coincide. The Aharonov—Bohm effect provides the simplest physical example of quantum mechanics in such a non-simply connected space, and is also fundamental to a simple model for anyons. A qualitative introduction is given to two currently proposed physical applications of anyons: high temperature superconductivity, and the (fractional) quantum Hall effect.