Boson-Anyon-Fermion Mapping and Anyon Construction in One Dimension

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 anti-symmetric 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 have demonstrated the inadequacy of contact interaction model for anyons in continuum and lattices, and meanwhile proposed a convenient route for engineering fractional phases in the platform of ultracold atoms.