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
T3ln(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.
We show that, for ultracold neutral bosonic atoms held in a three-dimensional periodic potential or optical lattice, a Hubbard model with dominant, attractive three-body interactions can be generated. In fact, we derive that the effect of pairwise interactions can be made small or zero starting from the realization that collisions occur at the zero-point energy of an optical lattice site and the strength of the interactions is energy dependent from effective-range contributions. We determine the strength of the two- and three-body interactions for scattering from van der Waals potentials and near Fano-Feshbach resonances. For van der Waals potentials, which for example describe scattering of alkaline-earth atoms, we find that the pairwise interaction can only be turned off for species with a small negative scattering length, leaving the
88Sr isotope a possible candidate. Interestingly, for collisional magnetic Feshbach resonances this restriction does not apply and there often exist magnetic fields where the two-body interaction is small. We illustrate this result for several known narrow resonances between alkali-metal atoms as well as chromium atoms. Finally, we compare the size of the three-body interaction with hopping rates and describe limits due to three-body recombination.
Universal low-energy properties are studied for three identical bosons confined in two dimensions. The short-range pairwise interaction in the low-energy limit is described by means of the boundary condition model. The wave function is expanded in a set of eigenfunctions on the hypersphere, and the system of hyperradial equations is used to obtain analytical and numerical results. Within the framework of this method, exact analytical expressions are derived for the eigenpotentials and the coupling terms of hyperradial equations. The derivation of the coupling terms is generally applicable to a variety of three-body problems provided the interaction is described by the boundary condition model. The asymptotic form of the total wave function at a small and a large hyperradius $\ensuremath{\rho}$ is studied, and the universal logarithmic dependence $\ensuremath{\sim}{\mathrm{ln}}^{3}\phantom{\rule{0.2em}{0ex}}\ensuremath{\rho}$ in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the
(2+1)-scattering length are calculated.