A unified theory of strong coupling Bose polarons: From repulsive polarons to non-Gaussian many-body bound states

1) Clearly written.
2) Extensive review of current picture.
3) Methodology explained in details.

1) Explicit application limited to a simple case.
2) Importance of the findings somehow overstated.

The paper describes a possibly novel variational technique capable to treat non-linearities arising in the formation of Feshbach molecules and the occurrence of many-body bound states at intermediate energies between the attractive and repulsive polaron branches.

The paper is nicely written and interesting. I particularly appreciated the extensive review of the literature made by the authors. However, I found somehow a mismatch between the very general picture delineated in the introduction and the results section where the only case treated is the one of an impurity in a Bose gas.

For example the authors clearly state that the many-body bound states they discuss have already been studied in the literature and that their scope is to include effects of inter-boson repulsion. However, these effects seem to give no relevant physical effects. The physics discussed is rather straightforward and I could not find any reason why these effects are expected to be important.

In this perspective the authors state:

"It is crucial to retain the higher order terms H_{3} and H_{4} to describe essential strong coupling effects such as non-Gaussian correlations of Bose polaron..."

This sentence appears to be overly grandiose , but it is kind of a tautology. Including H_{3} and H_{4}, which are non-quadratic terms in the effective Hamiltonian expansion, leads to non-Gaussian correlations (it is obvious and, almost, tautological) but why are these ESSENTIAL? What physical phenomenon is enabled by non-quadratic terms and could not be captured by the standard mean field? Which experimental observation NEEDS non gaussian correlations to be justified and why is this relevant to the field?

None of these questions are discussed. As it stands the paper is just a methodological development, which although interesting, leads to no real finding. The fact that the methodology can IN PRINCIPLE be extended to include further effects or to describe Efimov states is not enogh for me to believe that the paper will "open a new pathway in an existing or a new research direction", because all mean-field approaches can be generalized to include multi-body correlations expanding around a saddle point solution. The problem is that these methods are numerically very challenging and the actual solutions of the equations beyond the quadratic order is often not possible.

In conclusion, the physical picture is rather straightforward and the many-body states described do not present particular surprises when beyond quadratic terms are included. The appearance of non-Gaussian correlation is obvious beyond quadratic order and does not bear "per se" any physical significance.

I share the 1st Referee opinion that the paper shall be published on SciPost core.

The paper can be published as it is on Scipost Core.

Accept in alternative Journal (see Report)

I thank the authors for addressing the comments in my first report. All in all, the manuscript has been substantially improved. Now, it became clear to me that the "main purpose in this work is to present a framework for modelling impurity-boson problems" with "a heavy (or static) impurity". This assumption appears in multiple places in the text; it is open to question which of the results hold also for a mobile impurity, arguably, the main motivation of this work. Based on this, I conclude that the manuscript does not meet the acceptance criteria of SciPost Physics. My recommendation is to resubmit to SciPost Physics Core, where this work can be published as is.

Accept in alternative Journal (see Report)