The superfluid transition temperature
Tc of a unitary Fermi gas on a three-dimensional isotropic lattice with an attractive on-site interaction is investigated as a function of density
n, from half filling down to $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ per unit cell, using a pairing fluctuation theory. We show that except at very low densities (
n1/3<0.2), where
Tc/EF is linear in
n1/3,
Tc/EF exhibits a significant higher-order nonlinear dependence on
n1/3. This calls for extra caution against possible significant error in the zero-density-limit value of
Tc/EF from typical quantum Monte Carlo (QMC) simulations, obtained by linearly extrapolating data points at intermediate and high densities. Our result,
Tc/EF=0.256, at the
n=0 limit, is close to, and should be compared with, the maximum
Tc/EF near the Bose-Einstein condensation regime obtained from QMC calculations.
We study the bosonic atoms with a wide Feshbach resonance at zero temperature in terms of the renormalization group. We indicate that this system will always collapse in the dilute limit. On the side with a positive scattering length, the atomic superfluid is an unstable local minimum in the dilute limit and it determines the thermodynamics of this system within its lifetime. We calculate the equilibrium properties at zero temperature in the unitary regime. They exhibit universal scaling forms in the dilute limit due to the presence of a nontrivial zero temperature, zero-density fixed point. Moreover, we find that the
T=0 thermodynamics of this system in the unitary limit is exactly identical to the one for an ideal Fermi gas.