We investigate the superfluid phase transition in an $\mathrm{SU}\left(N\right)$-symmetric Fermi gas with $N$ distinct spin states using the functional renormalization group. To capture pairing phenomena beyond mean-field theory, we introduce an auxiliary bosonic field and employ the leading order of the derivative expansion of the partially bosonized effective average action. By discretizing the effective potential on a grid and numerically integrating the flow equations, we resolve the thermodynamic behavior near the transition. Our results reveal a fluctuation-induced first-order phase transition for $N\ge 4$, which is absent at the mean-field level. In the unitary regime, we provide quantitative predictions for the critical temperature as well as for the discontinuities in the superfluid gap and entropy density as functions of $N$. With increasing $N$, the critical temperature decreases, while the discontinuities become more pronounced, indicating a stronger first-order transition.

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• Fermi gases
• First order phase transitions
• Superfluidity
• Functional renormalization group
• SU(N) symmetries