Editing Luttinger liquid (section)

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

<math display="block">H = \sum_{k} \epsilon_k c_k^{\dagger} c_k</math>

is separated into left and right moving electrons and undergoes linearization with the approximation <math>\epsilon_k \approx \pm v_{\rm F}(k-k_{\rm F})</math> over the range <math>\Lambda</math>:

<math display="block">H = \sum_{k = k_{\rm F} -\Lambda}^{ k_{\rm F} +\Lambda} v_{\rm F} k \left(c_k^{\mathrm R \dagger } c_k^{\mathrm R } - c_k^{\mathrm L \dagger }c_k^{\mathrm L}\right)</math>

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.