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.
A quantity known as the contact is a fundamental thermodynamic property of quantum many-body systems with short-range interactions. Determination of the temperature dependence of the contact for the unitary Fermi gas of infinite scattering length has been a major challenge, with different calculations yielding qualitatively different results. Here we use finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods on the lattice within the canonical ensemble to calculate the temperature dependence of the contact for the homogeneous spin-balanced unitary Fermi gas. We extrapolate to the continuum limit for 40, 66, and 114 particles, eliminating systematic errors due to finite-range effects. We observe a dramatic decrease in the contact as the superfluid critical temperature is approached from below, followed by a gradual weak decrease as the temperature increases in the normal phase. Our theoretical results are in excellent agreement with the most recent precision ultracold atomic gas experiments. We also present results for the energy as a function of temperature in the continuum limit.