Role of impurity statistics and medium constraints in polaron-polaron interactions

• Provides a unified view of a number of recent publications, one of the most beautiful and clear papers I have read in a while.
• Arguments discussed in adequate detail and length, without skipping steps, making this work highly accessible to a broad readership

• none, in my view

In their manuscript, Levinsen et al. present a thorough analysis of medium-induced interactions between two impurities, as relevant to Bose-Fermi / Bose-Bose or Fermi-Fermi mixtures. This is one of the clearest papers I have read in a while, and the authors make an important contribution to the entire field. In fact, several earlier works have discussed medium-induced / -mediated interactions in different specific settings before, including in experiments, and it has been noted that surprisingly many different scenarios can be realized (medium-induced repulsion vs. attraction, anomalous enhancement of interactions,...), to the point that a coherent overview has been lacking. In this paper, the authors really clean up the entire field: They point out that a subtle but important distinction must be made between a host medium with fixed density vs. fixed chemical potential. One of the most remarkable results of this manuscript is the derivation, from fundamental thermodynamic relations and thus without any weak-coupling approximations, of an exact relation between interactions at fixed density vs. fixed chemical potential. Another remarkably elegant achievement of this paper is the formulation of a Chevy-type two-impurity wavefunction as a tool to derive medium-induced interactions. This approach is exact at weak couplings, where the authors derive a series of exact results, but also constitutes a promising ansatz for analyzing stronger interactions in the future. The paper is very clearly and pedagogically written, with detailed calculations presented in several appendices. Given the high relevance of this work to ongoing experimental and theoretical studies in a range of systems (from cold atoms to semiconductors), and the importance of the obtained results, I most strongly recommend publication of this manuscript in SciPost Physics. It definitely constitutes a milestone piece of work that SciPost Physics aims to publish.

Before publication, the authors can take into account the following few remarks / comments:

1) Below Eq. (2), the additional kinetic terms appearing for fermions are described. The authors could elaborate a bit more on the physics of these terms (Pauli pressure?). Then they note that this term must be removed in the calculation of the interactions: it is not entirely obvious to me why this term "must be removed". It appears that taking the proper partial derivatives as in Eq. (2), and the limit n_\sigma -> 0 afterwards, these terms simply yield no contribution. In this case, the description "these terms must be removed" is confusing, since it appears as though one can and should simply keep the terms, since they don't affect the interactions. Or do the authors have something different in mind - in this case they should elaborate.

2) In Sec. 4.2. the authors define momentum dependent interactions, where p1 and p2 denote the momenta of the first and second particle. From a fundamental symmetries point of view, this is confusing, since p1 and p2 are not separately conserved, but scattering would lead to populations of entirely different momentum configurations (with the same p1+p2 of course). On the level of the 2-impurity Chevy-type wavefunction, I understand that p1 and p2 appear as emergent conserved quantities (by the nature of the approximation), and in this case everything is well of course. But I would find it useful if the authors could comment on the meaning of p1 and p2, and under which conditions they envision that the p1-p2-dependent interaction can be accessed (e.g., does one have to perform post-selection in scattering experiments, to states without momentum-exchange?).

3) Below Eq. (65), the interactions should by called RKKY (not RRKY), I suppose?

4) In the introduction the authors refer to recent reviews [43-47] on Bose and Fermi polarons. They could consider adding a reference to the recent review article on Bose polarons: [Grusdt, et al., Rep. Prog. Phys. 88 066401 (2025)].

Publish (surpasses expectations and criteria for this Journal; among top 10%)

1- maybe a bit too long for my taste

Publish (surpasses expectations and criteria for this Journal; among top 10%)

-Exhaustive study of polaron-polaron interactions from an overlooked perspective. - Detailed calculations - Controlled and transparent methodology -Exact thermodynamic relation between interaction definitions

• Some notation and discussions can be improved.

Dear authors,

Please find attached my questions in the attached pdf.

Publish (easily meets expectations and criteria for this Journal; among top 50%)