Beyond universality in repulsive SU(N) Fermi gases

1- The article provides a comprehensive perturbative description of the energy of a fermionic gas of interacting particles with spin $S=N/2$. Each term is thoroughly described, and the analytical description is general enough to be applicable to softcore and hardcore particles.
2- The results of the article are carefully benchmarked to numerical results of Monte-Carlo simulations, displaying a very sound agreement. Furthermore, discrepancies between results are also discussed.
4- The predictions formulated by the authors on the critical behavior of the ferromagnetic transition in $SU(N)$ Fermi gases can be readily investigated in existing experimental set-ups.

1- The article is ambiguous regarding the numerical method, or the parameter regimes used to extract some results.
2- The type of transitions and the phases that can be encountered as a function of $S$ are treated in a rather superficial manner.

In this article, the authors investigate itinerant ferromagnetism in a dilute Fermi gas. To this end, they derive an analytical expression for the energy per particle via a perturbative diagrammatic expansion up to the third order in the parameters of the problem, namely the $s-$ and $p-$wave scattering lengths and the $s-$wave effective range. In particular, the spin degrees of freedom are considered for an arbitrary spin length. Then, based on the derived expression, the authors determine the magnetization of the ground state and observe as a function of the gas parameter $x$ different types of transitions towards a ferromagnetic state, depending on the length $S$ of the spin. Finally, the authors characterize the emergence of itinerant ferromagnetism via the magnetization, the magnetic susceptibility and Tan's constant.

During the last decade or so, efforts made in experimental and theoretical physics converged in order to shed light on the physics of $SU(N)$ system, and this work is in line with this joint effort. In my opinion, its relevance for the description of available experiments on Yb and Sr gases as well as its good degree of agreement with Monte-Carlo simulation make this article suitable for publication in SciPost Physics.

However, before fully agreeing to a publication, I would like the authors to address two points:

-Firstly, the authors do not specify how they determine the magnetization displayed on Fig. 5. If it is evaluated by using the value of the order parameter $P$ that minimizes the energy $E$ for a given value of $x$, as shown on Fig. 3, then it should be explicitly stated in the manuscript. Similarly, the authors do not provide the values chosen for the parameters $a_1$ and $r_0$ on the softcore results shown on Fig. 10.

-Secondly, based on the behavior of the magnetization on Fig 5 and the susceptibility on Fig 7, it seems that for $S=7/2$ and $S=9/2$ the Fermi gas undergoes two phase transitions, namely a second-order phase transition from an unpolarized system and then a first-order phase transition towards a polarized phase. This interpretation of the results, that is also further supported by the two singularities observed in the magnetic susceptibility, hints at the existence of an intermediate phase with some sort of magnetic ordering. Indeed, several examples can be found in the literature of $SU(N)$ systems featuring rich phase diagrams, such as in Nucl. Phys. B 996 (2023) 116353. Could the authors comment on this remark?

I invite the authors to correct the following typos
1- Top of right column of p2 : "require of a combination" -> "require a combination".
2- Left column on p3: "$s=1/2$" -> "$S=1/2$".
3- Right column on p6: "$k_F a_0=0,85$"-> "$k_F a_0=0.85$" and "$k_F a_0=0,9$"-> "$k_F a_0=0.9$".

4-On p6, the authors claim that the use plane-waves Slater determinant in Diffusion Monte Carlo describe more efficiently polarized Fermi gases. I would ask the authors to add a reference to support this statement.
5- On top of the right column of p7, could the authors clarify what they mean by "partial discontinuous transitions"?

Ask for minor revision

The authors have exhaustively addressed the comments I raised in my previous report. In particular, they clarified the definition of quasi-continuous transition, they provided additional information on the adaptive Monte Carlo integration, and, interestingly, they further discussed the effects of intra-species interactions. I recommend publication of the manuscript in SciPost Physics.