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Zero density limit extrapolation of the superfluid transition temperature in a unitary atomic Fermi gas on a lattice

The superfluid transition temperature $T_c$ 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 $5.0\times 10^{-7}$ per unit cell, using a pairing fluctuation theory. We show that except at very low densities ($n^{1/3} <0.2$), where $T_c/E_F$ is linear in $n^{1/3}$, $T_c/E_F$ exhibits significant higher order nonlinear dependence on $n^{1/3}$. Therefore, linear extrapolation using results at intermediate densities such as in typical quantum Monte Carlo simulations leads to a significant underestimate of the zero density limit of $T_c/E_F$. Our result, $T_c/E_F=0.256$, at $n=0$ is subject to reduction from particle-hole fluctuations and incoherent single particle self energy corrections.