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

Non-Analytic Crossover Behavior of SU($\mathcal{N}_c$) Fermi Liquid