We study one-dimensional anyons as a platform to explore the interplay between generalized exchange statistics and many-body correlations, with a view toward experimentally relevant implementations in engineered ultracold atomic systems.

We first analyze the momentum distribution of N-identical Lieb–Liniger anyons with mixed pseudopotential descriptions and derive their asymptotic high-momentum behavior, identifying universal coefficients governing the large-momentum tails relevant for time-of-flight measurements.

We then investigate internal correlations encoded in the one-body density matrix. Using the mapping between anyonic states and bosonic mean-field bright-soliton solutions as an analytical tool, we study the structure of the one-body density matrix and its natural orbitals. We show that the corresponding occupation numbers display a universal behavior, independent of the specific choice of underlying non interacting bosonic state. We further analyze the occupation spectrum and identify signatures consistent with odd-parity fractional fermionic pairing.

Finally, we connect these results to experimentally relevant implementations based on spin–orbit coupled ultracold atoms realizing two distinguishable bosons in one dimension in the Tonks Girardeau regime, where effective one-dimensional geometries and engineered interactions can be used to emulate anyonic physics. In particular, the anyonization mechanism can be understood as arising from a spin-dependent anyonic gauge field acting on the triplet sector, leading to controlled
exchange-statistics effects.

Overall, our results show how generalized exchange statistics govern universal correlation properties in both momentum-space and reduced-density-matrix observables, linking anyonic statistics, many-body coherence, and experimentally accessible signatures in one-dimensional quantum systems