Luttinger liquid

A Tomonaga-Luttinger liquid, more often referred to as simply a 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). Such a model is necessary as the commonly used Fermi liquid model breaks down for one dimension.

The Tomonaga-Luttinger liquid was first proposed by Tomonaga in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963, 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 Mattis and Lieb 1965.

Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons. Starting with the free electron Hamiltonian:

${\displaystyle H=\sum _{k}\epsilon _{k}c_{k}^{\dagger }c_{k}}$

is separated into left and right moving electrons and undergoes linearization with the approximation ${\displaystyle \epsilon _{k}\approx \pm v_{F}(k-k_{F})}$ over the range ${\displaystyle \Lambda }$:

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