A. Houghton

Variational approximations for renormalization group transformations

Approximate recursion relations which give upper and lower bounds on the free energy are described. Optimal calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, a particularly simple lower bound approximation which preserves the symmetry of the Hamiltonian (the one-hypercube approximation) is described. The approximation is applied to both the Ising model and the Wilson-Fisher model. At the fixed point a parameter is set variationally and critical indices are calculated. For the Ising model the agreement with the exact results atd = 2 is surprisingly good, 0.1%, and is good atd=3 and evend=4. For the Wilson-Fisher model the recursion relation is reduced to a one-dimensional integral equation which can be solved numerically givingv=0.652 atd=3, or by ɛ expansion in agreement with the results of Wilson and Fisher to leading order in ɛ. The method is also used to calculate thermodynamic functions for thed = 2 Ising model; excellent agreement with the Onsager solution is found.

Bosonization and fermion liquids in dimensions greater than one

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

Stability and single-particle properties of bosonized Fermi liquids

We study the stability and single-particle properties of Fermi liquids in spatial dimensions greater than one via bosonization. For smooth nonsingular Fermi-liquid interactions we obtain Shankars renormalization-group flows to second order in the BCS coupling and reproduce well-known results for quasiparticle lifetimes. We demonstrate by explicit calculation that spin-charge separation does not occur when the Fermi-liquid interactions are regular. We also explore the relationship between quantized bosonic excitations and zero-sound modes and present a concise derivation of both the spin and the charge collective-mode equations. Finally we discuss some aspects of singular Fermi-liquid interactions.

Theory of fermion liquids

We develop a general theory of fermion liquids in spatial dimensions greater than one. The principal method, bosonization, is applied to the cases of short and long range longitudinal interactions, and to transverse gauge interactions. All the correlation functions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau Fermi fixed point. Novel fixed points are found, however, in the cases of a super-long range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the 2+1-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction

Violation of the Wiedemann-Franz law in a large- N solution of the t − J model

V({\bf q}) \propto |{\bf q}|^{y-1}

QUASICLASSICAL APPROACH TO TRANSPORT IN THE VORTEX STATE AND THE HALL EFFECT

which controls the density, and hence gauge, fluctuations. For

Approximate renormalization group calculation of cubic crossover

y 0

Localization: The effect of a weak magnetic field

the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case

Transport properties of the A phase of superfluid 3He at low temperatures

y = 0

On the viscosity of superfluid 3He

(Coulomb interaction) which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator which is of a marginal Fermi liquid form. Using Ward Identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion, nor vertex corrections, alters our results: the fixed point is accessible by bosonization. As the two-point fermion Greens function is not gauge invariant, we also investigate the gauge-invariant density response function. Near momentum