EXACT RESULTS FOR HIGHLY CORRELATED ELECTRON-SYSTEMS IN ONE-DIMENSION

One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The di scovery of the ceramic high-temperature superconductors has revitalize d the interest in low-dimensional charge and spin fluctuations of high ly correlated electron systems. Several mechanisms proposed to explain the high-Tc superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are rele vant. Numerous one-dimensional models for correlated electrons have be en studied with various approximate, asymptotically exact and exact me thods. These results lead to the concept of Luttinger liquid for inter acting electron gases without excitation gaps (metallic systems). Char acteristic of Luttinger liquids are the charge and spin separation, ma rginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularitie s in the one-particle spectral function and the related absence of a d iscontinuity in the momentum distribution at the Fermi level, the powe r-law decay of correlation functions for long times and large distance s, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact resul ts derived within the framework of Bethe's ansatz for integrable one-d imensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of t he N-component electron gas interacting via a a-function potential (re pulsive and attractive interaction), which is probably the simplest mo del of correlated electrons. Emphasis is given to the procedure to ext ract the groundstate properties, the classification of states, the exc itation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. T he method is then applied to numerous other models, e.g. (i) a two-ban d model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with at tractive and repulsive U, (iii) the degenerate Hubbard model with repu lsive U, which displays a metal-insulator transition at a finite U, (i v) a two-band Hubbard model with repulsive U, (v) the traditional supe rsymmetric t-J model (vi) a two-band supersymmetric t-J model with ban d-splitting and (vii) the N-component supersymmetric t-J model. Finall y, results for models with long-range interactions, in particular r(-2 ) and sinh(-2)(r) potentials, are briefly reviewed.