Abstract. I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems. WHAT IS A LUTTINGER LIQUID ANYWAY? Ordinary, three-dimensional metals are described by Fermi liquid theory. Fermi liquid theory is about the importance of electron-electron interactions in metals. It states that there is a 1:1-correspondence between the low-energy excitations of a free Fermi gas, and those of an interacting electron liquid which are termed “quasiparticles” [1]. Roughly speaking, the combination of the Pauli principle with low excitation energy (e.g. T ≪ EF) and the large phase space available in 3D, produces
The Hubbard model in the strong-coupling regime is mainly studied by Kondo-lattice theory or 1/d expansion theory, with d the spatial dimensionality. In two dimensions and higher, the ground state within the Hilbert subspace with no order parameter is a normal Fermi liquid except for n=1 and U/W=+infinity, with n the electron density per unit cell, U the on-site repulsion, and W the bandwidth; the cooperation between the Kondo effect, which favors a local singlet on each unit cell, and a resonating-valence-bond effect, which favors a local singlet on each pair of nearest-neighbor unit cells, stabilizes the Fermi liquid, whose ground state is a singlet as a whole, in the strong-coupling regime. In the whole Hilbert space with no restriction, the normal Fermi liquid is unstable at least against a magnetic or superconducting state. This analysis confirms an early Fermi-liquid theory of high-temperature superconductivity, F. J. Ohkawa, Jpn. J. Appl. Phys. 26, L652 (1987). The ground state for n=1 and U/W=+infinity is a Mott insulator. Actual metal-insulator transitions cannot be explained within the Hubbard model. In order to explain them, the electron-phonon interaction, multi-band or multi-orbital effects, and effects of disorder should be considered beyond the Hubbard model. The crossover between local-moment magnetism and itinerant-electron magnetism corresponds to that between a localized spin and a normal Fermi liquid in the Kondo effect and it is simply a Mott metal-insulator crossover.
Electrons in metals interact via the Coulomb interaction. The size of this effect is not negligible: it is of order e 2 / a where a is the mean distance between electrons. For a metal this ≈ 3 eV, that is, comparable to E F . An approach to the problem would be to treat the interaction as a perturbation. However, the perturbation is strong and long-ranged, and this gives rise to many difficulties, such as divergences in finite orders of perturbation theory. The problems can be overcome by an application of the methods of quantum field theory to the problem, and is treated in books on many-body theory. We will not pursue this approach here, but try to get some of the central conclusions with physical reasoning. At first glance, we might be tempted to conclude that the sharp Fermi surface that we have discussed, and which is readily observed, should be wiped out by the interaction. The much smaller energy k B T does lead to smearing of the Fermi distribution; it is natural to conclude that this is true for interactions as well. This natural conclusion is not correct in three dimensions. The situation in one dimension is complicated, and the Fermi surface does disappear, leading to a state called the Luttinger liquid. In three dimensions the sharp Fermi surface survives the presence of interactions. This is an important phenomenon which demands explanation.