Bosonization

Let us look closely at the effective equation of motion for the Fourier components of the electron density in the limit where plasma oscillations occur. We showed in the previous chapter that in the limit https://www.w3.org/1998/Math/MathML"> ω o 2 ⪢ k 2 v F 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429037245/cc3a6600-aaec-4350-ba37-6c8d1f17d018/content/inline-math288.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the equation of motion https://www.w3.org/1998/Math/MathML"> ρ ¨ k ⁢   + ⁢   ω p 2 ρ k = ⁢   0 ⁢                               (9.1) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429037245/cc3a6600-aaec-4350-ba37-6c8d1f17d018/content/math452.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> resembles that of a harmonic oscillator. However, harmonic excitations obey Bose statistics. This suggests that plasma oscillations in some way represent bosonic excitations of the interacting electron gas. Equivalently, plasma oscillations can be thought of as harmonic excitations of the electrons in the gas. That this state of affairs holds profound consequences for the electron gas was pointed out initially by Tomonaga [T1950], who gave an explicit prescription for constructing the sound wave spectrum in a dense Fermi system. Three years later, Bohm and Pines [BP1953] showed that there is a natural connection between the sound wave (Bose) spectrum and the random-phase approximation. Since then, the equivalence between long-wavelength excitations in a dense Fermi system and a collection of bosons has proven to be of fundamental importance in solid state physics as well as in relativistic field theories. In this chapter, we focus on how a collection of interacting electrons in Id obeying standard anticommutation relations can be described by an equivalent set of boson modes. The problem of constructing such a boson field theory for a collection of fermions is known as bosonization. In this context, we will first bosonize the Id Hubbard [H1964] model and, in so doing, obtain the Luttinger liquid [L1960]. Luttinger liquids form the general basis for analyzing the properties of interacting electrons in Id, in so far as such systems are dominated by short-range Coulomb interactions.