Luttinger's Theorem and Bosonization of the Fermi Surface

TL;DR

This paper explores the algebraic structures underlying bosonization of the Fermi surface in higher dimensions, extending one-dimensional techniques and clarifying the mathematical framework for Fermi liquids.

Contribution

It introduces the algebraic structure necessary for bosonization in two- and three-dimensional Fermi surfaces, generalizing Kac-Moody algebra and deriving the Gaussian reduction of Fermi liquids.

Findings

Generalization of Kac-Moody algebra to higher dimensions

Derivation of Gaussian reduction of Fermi liquid to harmonic oscillators

Review of 1D bosonization and spin-charge separation

Abstract

A course of four lectures given at the International School of Physics "Enrico Fermi", Varenna, Italy, July 1992, in which the underlying algebraic structure needed for bosonization of the Fermi surface in two- or three-dimensions was first described. This is an unchanged 1993 preprint version of a published but hard-to-find (and often mis-cited) 1994 article in the Varenna Summer School proceedings. The d > 1 dimensional generalization of the Kac-Moody algebra on the Fermi surface is presented, and the Gaussian reduction of the Fermi liquid to harmonic oscilator modes is derived. One-dimensional bosonization and the symmetries of spin-charge separation are also reviewed.