One-dimensional Fermi liquids

TLDR

In one dimension, conventional Fermi liquid theory fails due to Peierls divergence, leading to anomalous dimensions and charge‑spin separation, and the low‑energy physics is captured by the Luttinger model, defining a universal class of gapless 1D systems. The paper reviews a decade of progress in one‑dimensional Fermi liquid theory. The authors analyze the low‑energy behavior of gapless 1D systems using the exactly solvable Luttinger model, whose correlation functions can be computed analytically. Mapping to the Luttinger model yields an asymptotically exact solution of the 1D many‑body problem, with the 1D Hubbard, t‑J, extended Hubbard, electron‑phonon systems, impurity‑doped metals, and quantum Hall edge states all falling into this class.

Abstract

We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-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, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.