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

\overline{ϵ}=d\ensuremath{-}2

DOI: 10.1103/physreva.75.063617

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ϵ=4\ensuremath{-}d

, with

d

ϵ

\overline{ϵ}=d\ensuremath{-}2

We consider a polarized Fermi gas in the unitarity limit. Results are calculated analytically up to next-to-leading order in an expansion about

d=4

spatial dimensions. We find a first order transition from superfluid to normal phase. The critical chemical potential asymmetry for this phase transition is $\ensuremath{\delta}{\ensuremath{\mu}}_{c}=\frac{2\ensuremath{\mu}}{\ensuremath{\epsilon}}(1\ensuremath{-}0.467\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$ is the expansion parameter and $\ensuremath{\mu}$ is the average chemical potential of the two fermion species. Stability of the superfluid phase in the presence of supercurrents is also studied.

d

is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be $\ensuremath{\mu}/{\ensuremath{\epsilon}}_{F}=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\mathrm{ln}ϵ\ensuremath{-}0.0246{ϵ}^{5/2}+\ensuremath{\cdots}$ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be $\ensuremath{\Delta}/\ensuremath{\mu}=2{ϵ}^{\ensuremath{-}1}\ensuremath{-}0.691+\ensuremath{\cdots}$. The minimum of the fermion dispersion curve is located at $|\mathbit{p}|=(2m{\ensuremath{\epsilon}}_{0}{)}^{1/2}$, where ${\ensuremath{\epsilon}}_{0}/\ensuremath{\mu}=2+O(ϵ)$. Extrapolation to

d=3

gives results consistent with Monte Carlo simulations.

We analyze strongly interacting Fermi gases in the unitary regime by considering the generalization to an arbitrary number

N

of spin-

1∕2

fermion flavors with

\mathrm{Sp}(2N)

symmetry. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$ this problem is exactly solved by the Bardeen-Cooper-Schrieffer--Bose-Einstein condensate mean-field theory, with corrections small in the parameter

1∕N

. The large-

N

expansion provides a systematic way to determine corrections to mean-field predictions, allowing the calculation of a variety of thermodynamic quantities at (and in proximity to) unitarity, including the energy, the pairing gap, and the upper-critical polarization (in the case of a polarized gas) for the normal to superfluid instability. For the physical case of

N=1

, among other quantities, we predict in the unitarity regime, the energy of the gas to be $\ensuremath{\xi}=0.28$ times that for the noninteracting gas and the pairing gap to be 0.52 times the Fermi energy.

We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the

p

-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is,

p

-wave scattering length and effective range. At the unitarity limit where the

p

-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact

s

-wave interaction in three dimensions. It is in contrast to a Fermi gas with the

p

-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary

p

-wave Fermi gas within the many-body

T

-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary

p

-wave Fermi gas.