We develop and describe new approaches to the problem of interacting fermions in spatial dimensions greater than 1. 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 processes neglected in our model (but does not, as yet, address questions of stability). The charge sector is equivalent to an exactly solvable Gaussian quantum field theory; the spin sector, however, must be solved semiclassically. Using the Luther-Haldane approach we recover the collective mode equation of Fermi-liquid theory and in three dimensions reproduce the ${\mathit{T}}^3$ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.

• Received 26 October 1992

DOI:https://doi.org/10.1103/PhysRevB.48.7790