ONE-DIMENSIONAL FERMI LIQUIDS

We review the progress in the theory of one-dimensional (1D) Fermi liq uids which has occurred over the past decade. The usual Fermi liquid t heory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, produc ing anomalous dimensions of operators, and because of charge-spin sepa ration. Both are related to the importance of scattering processes tra nsferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Lut tinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spec trum, parameterized by one renormalized velocity and one effective cou pling constant per degree of freedom, fully describe the physics of th is model. Other gapless 1D models share these properties in a low-ener gy subspace. The concept of a 'Luttinger liquid' implies that their lo w-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, a nd variants such as the t-J- or the extended Hubbard model. In additio n, 1D electron-phonon systems or metals with impurities can be Lutting er liquids, as well as the edge states in the quantum Hall effect. We discuss in detail various solutions of the Luttinger model which empha size different aspects of the physics of 1D Fermi liquids. Correlation functions are calculated in detail using bosonization, and the relati on of this method to other approaches is discussed. The correlation fu nctions decay as non-universal power laws, and scaling relations betwe en their exponents are parameterized by the effective coupling constan t. Charge-spin separation only shows up in dynamical correlations. The Luttinger liquid concept is developed from perturbations of the Lutti nger model. Mainly specializing to the 1D Hubbard model, we review a v ariety of mappings for complicated models of interacting electrons ont o Luttinger models, and thereby obtain their correlation functions. We also discuss the generic behaviour of systems not falling into the Lu ttinger liquid universality class because of gaps in their low-energy spectrum. The Mott transition provides an example for the transition f rom Luttinger to non-Luttinger behaviour, and recent results on this p roblem are summarized. Coupling chains by interactions or tunnelling a llows transverse coherence to establish in the single- or two-particle dynamics, and drives the systems away from a Luttinger liquid. We dis cuss the influence of charge-spin separation and of the anomalous dime nsions on the transverse dynamics of the electrons. The edge states in the quantum Hall effect provide a realization of a modified, chiral L uttinger liquid whose detailed properties differ from those of the sta ndard model. The review closes with a summary of experiments which can be interpreted in favour of Luttinger liquid correlations in the 'nor mal' state of quasi-1D organic conductors and superconductors, charge density wave systems, and semiconductors in the quantum Hall regime.