A brief introduction to Luttinger liquids

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 a very dilute gas of excitations where interactions are sufficiently harmless so as to preserve the correspondence to the free-electron excitations. Three key elements are: (i) The elementary excitations of the Fermi liquid are quasi-particles. They lead to a pole structure (with residue Z – the overlap of a Fermi surface electron with free electrons) in the electronic Green’s function which can be – and has been – observed by photoemission spectroscopy [2]. (ii) Transport is described by the Boltzmann equation which, in favorable cases, can be quantitatively linked to the photoemission response [2]. (iii) The low-energy physics is parameterized by a set of Landau parameters F l s,a which contain the residual interaction effects in the angular momentum charge and spin channels. The correlations in the electron system are weak, although the interactions may be very strong. Fermi liquid theory breaks down for one-dimensional (1D) metals. Technically, this happens because some vertices Fermi liquid theory assumes finite (those involving a 2kF momentum transfer) actually diverge because of the Peierls effect. An equivalent intuitive argument is that in 1D, perturbation theory never can work even for arbitrarily small but finite interactions: when degenerate perturbation theory is applied to the coupling of the all-important electron states at the Fermi points ±kF , it will split them and therefore remove the entire Fermi surface! A free-electron-like metal will therefore not be stable in 1D. The underlying physical picture is that the coupling of quasi-particles to collective excitations is small in 3D but large in 1D, no matter how small the interaction. Correlations are strong even for weak interactions! 1D metals are described as Luttinger liquids [3,4]. A Luttinger liquid is a paramagnetic one-dimensional metal without Landau quasi-particle excitations. “Paramagnetic” and “metal” require that the spin and charge excitations are gapless, more precisely with dispersions ων ≈ vν |q| (ν = ρ, σ for charge and spin). Only when this requirement is fulfilled, a Luttinger liquid can form. The charge and spin modes (holons and spinons) possess different excitation energies vρ 6= vσ and are bosons. This leads to the separation of charge and spin of an electron (or hole) added to the Fermi sea, in space-time, or q − ω-space. Charge-spin separation prohibits quasiparticles: The pole structure of the Green’s function is changed to branch cuts, and therefore the quasi-particle residue Z is zero. Charge-spin separation in space-time can be nicely observed in computer simulations [5]. The bosonic nature of charge and spin excitations, together with the reduced dimensionality leads to a peculiar kind of short-range order at T = 0. The system is at a (quantum) critical point, with power-law correlations, and the scaling relations between the exponents of its correlation functions are parameterized by renormalized coupling constants Kν . The individual exponents are non-universal, i.e. depend on the interactions. For Luttinger liquids, Kν is the equivalent of the Landau parameters. As an example, the momentum distribution function n(k) ∼ (kF −k) α for k ≈ kF with α = (Kρ + K −1 ρ − 2)/4. This directly illustrates the absence of quasi-particles: In a Fermi liquid, n(k) has a jump at kF with amplitude Z. BOSONIZATION, OR HOW TO SOLVE THE 1D MANYBODY PROBLEM BY HARMONIC OSCILLATORS The appearance of charge and spin modes as stable, elementary excitations in 1D fermion systems can be rationalized from the spectrum of allowed particle-hole excitations. In 1D, low-energy particle-hole pairs with momenta between 0 and 2kF are not allowed, and for q → 0, the range of allowed excitations shrinks to a one-parameter spectrum ων ≈ vν |q|, indicating stable particles (cf. Fig. 1). True bosons are then obtained as linear combinations of these particle-hole excitations with a definite momentum q. Most importantly, we now can rewrite any interacting fermion Hamiltonian, provided its charge and spin excitations are gapless, as a harmonic oscillator and find an operator identity allowing to express fermion operators as functions of these bosons. This is the complete bosonization program. For free fermions, the Hamiltonian describing the excitations out of the ground state (the Fermi sea), a can be expressed as a bilinear in the bosons,