This article reviews theoretical and experimental developments for one-dimensional Fermi gases. Specifically, the experimentally realized two-component delta-function interacting Fermi gasthe Gaudin-Yang model -and its generalisations to multi-component Fermi systems with larger spin symmetries. The exact results obtained for Bethe ansatz integrable models of this kind enable the study of the nature and microscopic origin of a wide range of quantum many-body phenomena driven by spin population imbalance, dynamical interactions and magnetic fields. This physics includes Bardeen-Cooper-Schrieffer-like pairing, Tomonaga-Luttinger liquids, spincharge separation, Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations, quantum criticality and scaling, polarons and the few-body physics of the trimer state (trions). The fascinating interplay between exactly solved models and experimental developments in one dimension promises to yield further insight into the exciting and fundamental physics of interacting Fermi systems.

The low-energy properties of the one-dimensional anyon gas with a -function interaction are discussed in the context of its Bethe ansatz solution. It is found that the anyonic statistical parameter and the dynamical coupling constant induce Haldane exclusion statistics interpolating between bosons and fermions. Moreover, the anyonic parameter may trigger statistics beyond Fermi statistics for which the exclusion parameter is greater than one. The Tonks-Girardeau and the weak coupling limits are discussed in detail. The results support the universal role of in the dispersion relations. Anyons, which are used to describe particles with generalized fractional statistics [1,2], are becoming of increasing importance in condensed matter physics [3] and quantum computation [4]. The concept of anyons provides a successful theory of the fractional quantum Hall (FQH) effect [5]. In particular, the signature of fractional statistics has recently been observed in experiments on the elementary excitations of a two-dimensional electron gas in the FQH regime [3]. These developments are seen as promising opportunities for further insight into the FQH effect, quantum computation, superconductivity, and other fundamental problems in quantum physics.In one dimension, collision is the only way to interchange two particles. Accordingly, interaction and statistics are inextricably related in 1D systems. The 1D Calogero-Sutherland model is seen to obey fractional exclusion statistics [6,7]. In the sense of Haldane exclusion statistics, the 1D interacting Bose gas is equivalent to the ideal gas with generalized fractional statistics [8,9]. We consider an integrable model of anyons with a -function interaction introduced and solved by Kundu [10]. Here we obtain the low-energy properties and Haldane exclusion statistics of this 1D anyon gas. We find that the low energies, dispersion relations, and the generalized exclusion statistics depend on both the anyonic statistical and the dynamical interaction parameters. The anyonic parameter not only interpolates between Bose and Fermi statistics, but can trigger statistics beyond Fermi statistics in a super Tonks-Girardeau (TG) gaslike phase.Bethe ansatz solution.-We consider N anyons with a -function interaction in one dimension with Hamiltonian [10]