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).[1] Such a model is necessary as the commonly used Fermi liquid model breaks down for one dimension.

The Tomonaga–Luttinger's liquid was first proposed by Sin-Itiro Tomonaga in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963, J.M. 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 over the range :

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).

Physical systems

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:

See also



  1. ^ Blumenstein, C.; Schäfer, J.; Mietke, S.; Meyer, S.; Dollinger, A.; Lochner, M.; Cui, X. Y.; Patthey, L.; Matzdorf, R.; Claessen, R. (October 2011). "Atomically controlled quantum chains hosting a Tomonaga–Luttinger liquid". Nature Physics. 7 (10): 776–780. Bibcode:2011NatPh...7..776B. doi:10.1038/nphys2051. ISSN 1745-2473.
  2. ^ Mattis, Daniel C.; Lieb, Elliot H. (February 1965). Exact solution of a many-fermion system and its associated boson field. Vol. 6. pp. 98–106. Bibcode:1994boso.book...98M. doi:10.1142/9789812812650_0008. ISBN 978-981-02-1847-8. ((cite book)): |journal= ignored (help)
  3. ^ Ishii, H; Kataura, H; Shiozawa, H; Yoshioka, H; Otsubo, H; Takayama, Y; Miyahara, T; Suzuki, S; Achiba, Y; Nakatake, M; Narimura, T; Higashiguchi, M; Shimada, K; Namatame, H; Taniguchi, M (4 December 2003). "Direct observation of Tomonaga–Luttinger-liquid state in carbon nanotubes at low temperatures". Nature. 426 (6966): 540–544. Bibcode:2003Natur.426..540I. doi:10.1038/nature02074. PMID 14654836. S2CID 4395337.
  4. ^ Chudzinski, P.; Jarlborg, T.; Giamarchi, T. (2012). "Luttinger-liquid theory of purple bronze Li
    in the charge regime"
    . Physical Review B. 86 (7): 075147. arXiv:1205.0239. doi:10.1103/PhysRevB.86.075147. S2CID 53396531.