Brief Introduction to Luttinger Liquids

Kurt Schönhammer

Johannes Voit

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.

Sumathi Rao

We give a brief introduction to Luttinger liquids and to the phenomena of electronic transport or conductance in quantum wires. We explain why the subject of transport in Luttinger liquids is relevant and fascinating and review some important results on tunneling through barriers in a one-dimensional quantum wire and the phenomena of persistent currents in mesoscopic rings. We give a brief description of our own work on transport through doubly-crossed Luttinger liquids and transport in the Schulz–Shastry exactly solvable Luttinger-like model. With a view to making this article accessible to researchers in diverse fields such as nuclear physics, particle physics, formal field theory and condensed matter physics, we begin by first giving a brief introduction. In Sec. 1, we explain the words such as "Luttinger liquid" and "transport" in the title. We also motivate the study of this subject by showing that field is very important at the current time from the applications point of view and describe why it is theoretically fascinating. We then give a quick review of some of the important results in the field in Sec. 2, before we go on to describe some work that we have recently done and are doing in this field in Sec. 3. Finally, in Sec. 4, we conclude by re-emphasizing the importance of this area and drawing attention towards future possibilities.

Johannes Voit, Theoretische Physik, Universität Bayreuth +1 more

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