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 determine the thermodynamic functions, the fermion quasiparticle spectrum, the critical polarizations, and the critical temperature as functions of the binding energy of the two-body state. We also show that the unitary Fermi gas near two spatial dimensions reduces to a weakly-interacting Fermi gas and calculate the thermodynamic functions and the fermion quasiparticle spectrum in the expansion over \barε=d-2. We discuss the matching of the two systematic expansions around four and two spatial dimensions in order to extract physical observables at d=3. We find good agreement of the results with those from recent Monte Carlo simulations.
We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $(\ensuremath{\xi})$ from the
ϵ expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO
ϵ expansions around four and two spatial dimensions with the use of Pad\'e approximants give $\ensuremath{\xi}\ensuremath{\approx}0.360\ifmmode\pm\else\textpm\fi{}0.020$ in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO
ϵ expansions $\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014$ in spite of the large NNLO corrections.