Fermi gas near unitarity around four and two spatial dimensions

We construct systematic expansions around four and two spatial dimensions for a Fermi gas near the unitarity limit. Near four spatial dimensions such a Fermi gas can be understood as a weakly interacting system of fermionic and bosonic degrees of freedom. To the leading and next-to-leading orders in the expansion over ε=4-d, with d being the dimensionality of space, we calculate the thermodynamic quantities and the fermion quasiparticle spectrum, both at unitarity and around the unitarity point. Then the phase structure of the polarized Fermi gas in the unitary regime is studied in the ε expansion. At unitarity the unpolarized superfluid state and a fully polarized normal state are separated by a first order phase transition. However, on the BEC side of the unitarity point, in a certain range of the two-body binding energy, these two phases are separated in the phase diagram by more exotic phases, including gapless superfluid phases with one or two Fermi surfaces and a superfluid phase with a spatially varying condensate. We also show that the unitary Fermi gas near two spatial dimensions is weakly interacting, and calculate the thermodynamic quantities and the fermion quasiparticle spectrum in the expansion over ε=d-2