  We consider the thermodynamic potential of a dilute Fermi gas with a contact
interaction, at both finite temperature $T$ and non-zero effective magnetic
fields $\mathbf{H}$, and derive the equation of state analytically using second
order perturbation theory. Special attention is paid to the non-analytic
dependence of $\Omega$ on temperature $T$ and (effective) magnetic field
$\mathbf{H}$, which exhibits a crossover behavior as the ratio of the two is
continuously varied. This non-analyticity is due to the particle-hole pair
excitation being always gapless and long-ranged. The non-analytic crossover
found in this paper can therefore be understood as an analog of the
Ginzberg-Landau critical scaling, albeit only at the sub-leading order. We
extend our results to an $\mathcal{N}_c$- component Fermi gas with an
$\mathrm{SU}(\mathcal{N}_c)$-symmetric interaction, and point out possible
enhancement of the crossover behavior by a large $\mathcal{N}_c$.
