A Luttinger liquid, or Tomonaga–Luttinger liquid, is a theoretical model describing interacting electrons (or other fermions) in a one-dimensional conductor (e.g. quantum wires such as carbon nanotubes).
Luttinger reformulated the theory in terms of Bloch sound waves and showed that the constraints proposed by Tomonaga were unnecessary in order to treat the second-order perturbations as bosons.
But his solution of the model was incorrect; the correct solution was given by Daniel C. Mattis [de] and Elliot H. Lieb 1965.
[2] Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons.
Starting with the free electron Hamiltonian:
is separated into left and right moving electrons and undergoes linearization with the approximation
Expressions for bosons in terms of fermions are used to represent the Hamiltonian as a product of two boson operators in a Bogoliubov transformation.
The completed bosonization can then be used to predict spin-charge separation.
Electron-electron interactions can be treated to calculate correlation functions.
Among the hallmark features of a Luttinger liquid are the following: The Luttinger model is thought to describe the universal low-frequency/long-wavelength behaviour of any one-dimensional system of interacting fermions (that has not undergone a phase transition into some other state).
Attempts to demonstrate Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in condensed matter physics.
Among the physical systems believed to be described by the Luttinger model are: