A brief introduction to Luttinger liquids

Abstract

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 K ρ , 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 ≪ E F ) 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 2k F 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

3D gauged supergravity from wrapped M5-branes with AdS/CMT applications.

“…A big future challenge for this geometry concerns whether it captures signatures of the Luttinger liquid model [55,56]. In contrast to higher dimensions, Fermi-Landau liquid theory is not applicable in one spatial dimension, and for theories with a gapless branch, the effective description of the physics may be given in terms of a Luttinger liquid.…”

From Maxwell-Chern-Simons theory in AdS 3 towards hydrodynamics in 1 + 1 dimensions

“…Roughly speaking, Luttinger liquid theory can be understood as equivalent to applying conformal field theory to a system of interacting fermions in 1 + 1 dimensions [32]. A Luttinger liquid is defined as a paramagnetic 1+1-dimensional metal without Landau quasiparticle excitations.…”

SET-based experiments for HTSC materials

“…For the benifit of the non-theoretical reader we now summarize and state some relevant definitions, formulas and properties of Luttinger Liquid [LL], for more details see [16,17]. Moreover, since Single-Walled Nanotube [SWNT] is involved in our suggested configurations, see section II, we give some relevant details of SWNT.…”