Bosonization and fermion liquids in dimensions greater than one

Abstract

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 processes neglected in our model (but does not, as yet, addressquestions 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 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.

Singular or non-Fermi liquids

“…This is known from calculations on ladder models [203]. Such states have also been proposed for copper-oxide compounds [148,128,58].…”

“…For more recent methods with the same conclusions, see [237,128]. Our emphasis will be on highlighting the assumptions in the theory so that in the next section we can summarize the routes by which the Fermi-liquid theory may break down.…”

“…(28), the renormalization group procedures are used to systematically generate successively lower energy and small momentum Hamiltonians with excitations of particle ever closer to the Fermi surface. The calculations are carried out either in terms of fermions [237] or newly developed bosonization methods in arbitrary dimensions [128]. The end result is equivalent to Eqs.…”

Stability and single-particle properties of bosonized Fermi liquids

“…This viewpoint has been elaborated on by two of us 3 . In the present paper we continue to develop this theory first by showing that Shankar's renormalization group result is obtained easily in the bosonized picture.…”

Spin-excitations of the quantum Hall ferromagnet of composite fermions

“…13 Although the bosonization method of fermionic systems is well established for one-dimensional electron systems, 14,15 only few generalizations to higher dimensions have been proposed. 16,17 One particular example is a recently developed bosonization scheme for the 2D electron gas (2DEG) in a magnetic field. 13,18 The extension presented in Ref.…”