This is a very nicely written paper discussing the emergence of spontaneous symmetry breaking in a version of the Curie-Weiss model. It includes a thorough treatment of many aspects of spontaneous symmetry breaking, in a mathematically rigorous fashion, which will be useful to any readers interested in the details of spontaneous symmetry breaking in general, and in the Curie-Weiss model in particular.
Beyond being a nice, clear, and precise description of the effects of a symmetry breaking field for finite N and hbar in a setting where it has -to the best of my knowledge- not been discussed in these terms before, the paper unfortunately does not offer much new insight. The idea that a finite symmetry-breaking field is required to break the symmetry of a finite system is completely standard. Similarly, the realisation that the strength of this field may be taken to infinitesimal values in the thermodynamic limit has been at heart of the way we teach spontaneous symmetry breaking for a long time.
I appreciate the present submission as an explicit and pedagogical example of spontaneous symmetry breaking. Examples that can be treated in an accessible, yet mathematically rigorous, way are rare, and it is valuable to have more such examples. Beyond that, I don't see any new physics in this manscript.
I would therefore urge the authors to rewrite their introduction in such a way that they make clear that this manuscript presents a new and pedagogical example of known physics, rather than a novel research result.
Besides this general remarks, I also noticed some other issues in the manuscript:
1- In the introduction, and in various places throughout the manuscript, the author fail to acknowledge, and even misrepresent, past results in the field. The realisation that finite symmetry breaking fields are needed to break the symmetry of finite systems has been around at least since the works of PW Anderson. The effect of finite perturbations and the need for suitable convergence in the thermodynamic limit has been discussed in detail for magnetic systems by various authors throughout the 1980's, and has been generalised to many other types of systems since.
The authors fail to cite much of the previous work in this direction, and instead present their own papers from the past few years as if these are the first results in the field.
1b - The failure to acknowledge previous work is made worse by the fact that the authors use their own personal terminology for concepts that have been discussed in the literature for many years under other names. For example, the "flea in Schrodinger's cat" is nothing other than the "symmetry breaking field" in any other discussion of spontaneous symmetry breaking.
2 -Although I agree with the idea of the discussion in section 4.1, I am not convinced the perturbation shown in figure 9 can be regarded as a proper choice for a symmetry breaking field. As the authors show in their manuscript, the two nearly degenerate ground states become arbitrarily well-localised in the limit of large N. Taking the limit of N to infinity while keeping the perturbation finite thus means that the ground state wavefunction will have zero overlap with any alterations of the potential away from the minima of the double well. Only perturbations that make the two minima in the potential non-degenerate can then serve as proper symmetry-breaking fields in the limit of large N.
In other words: I suspect that for the perturbation shown in figure 9, the energy difference between the nearly degenerate ground states scales as lambda/N, implying that N can always be chosen large enough for the perturbation chosen by the authors to be practically irrelevant. This is precisely opposite to the effect of a proper symmetry breaking field, which causes an energy splitting proportional to lambda*N, and which becomes more relevant with increasing N.
1 - On the bottom of page 9, and again in various places further on, it is confusing to talk about "new ground states". Two states simply become degenerate up to numerical precision, and any superposition of them is a (numerically) allowed ground state. If you always find a particular "new" ground state in the numerics, such as a localised state, that must mean there is a symmetry breaking field in your numerical implementation of the calculation. Without such a field, you ought to find random superpositions of the two degenerate states.
This should really be explained already on page 9.
On a related note, the number 80 in N=80 is arbitrary, and must depend on the machine precision of the computer used to do the calculations. The authors really ought to mention that.
2 - in the first line of the introduction, the authors choose ferromagnets as their example for the archetype of spontaneous symmetry breaking. I suspect the authors are aware that this is actually a very bad example. Ferromagnets are very special in the sense that their symmetry-broken state is an exact eigenstate of their symmetric Hamiltonian for any N. This is not at all generic, and in fact severely affects the physics associated with symmetry breaking in ferromagnets (see for example 10.1016/j.aop.2015.07.008 and references therein).
3 - It seems like the matrix elements discussed in theorem 2.1 are included in those of appendix 2 in 10.1103/PhysRevB.74.094430 ?