Bosonization and Fermion Liquids in Dimensions Greater Than One

TL;DR

This paper extends bosonization techniques to higher dimensions, developing simplified models to study interacting Fermions and their instabilities, revealing behaviors like Luttinger liquids and Fermi surface effects.

Contribution

It introduces a new approach to bosonization in dimensions greater than one, using simplified models with Fermi points to analyze Fermion liquids.

Findings

Model exhibits Luttinger liquid behavior away from half-filling

Instabilities occur at half-filling due to nesting

Simplified Fermi surface models are effective for studying higher-dimensional Fermions

Abstract

(Revised, with postscript figures appended, corrections and added comments.) We develop and describe new approaches to the problem of interacting Fermions in spatial dimensions greater than one. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting Fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing Fermion liquids. The complicated functional renormalization group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the Bosonization of higher dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase space and small-angle scattering .... (7 figures, appended as a postscript file at the end of the TeX file).