Accurate projective two-band description of topological superfluidity in spin-orbit-coupled Fermi gases

1- New, simplified approach to superconductors with spin-orbit and Zeeman couplings in the topological phase. 2x2 matrix instead of 4x4, several analytic expressions result.

2- Writing and figures are clear.

1- Motivation for and payoff of the 2x2 formulation could be clearer.

Brand et al.'s paper study attractive fermions with Zeeman + spin-orbit couplings, a well-studied situation due to its importance for topological superfluidity. They eliminate the spin-down degrees of freedom that are far off in energy. The formalism they use is a formally exact Feshbach projection technique, together with an approximation that approximates a certain operator appearing in these expressions by its large-k limit. This reduces finding eigenstates, which normally require diagonalizing 4x4 matrices, to diagonalizing 2x2 matrices. Analyzing this, and occasionally employing other approximations, they arrive at simple analytic expressions to describe the topological superfluid in this system. They show the validity of the approach by comparing the exact and approximate calculations.

The problem of topological superfluidity is important, their results appear technically sound and new, and their approximate expressions are at least somewhat simpler to work with than with the solutions to the 4x4 matrix equations. I am not entirely convinced that this formulation sheds substantial new light on the problem, or that the rather modest simplifications justify introducing (sometimes uncontrolled) approximations. Nevertheless, the techniques should be made available to the community, in case they are useful, and so I believe that with revision this paper is suitable for publication.

I have a few comments and suggestions I request the authors to consider, listed below.

1- Is there a limit where the key decoupling approximation, just below Eq 5, can be shown to be accurate? E.g if Delta/h<<1 or something similar? This issue of where, at least in principle, we can expect the approximation to be valid (independent of numerical verification) should be addressed.

2- Comparisons of the 2x2 and 4x4 theories are shown for a rather limited set of parameter values. It would be valuable if, at least for the types of comparisons shown in Fig 1, comparisons at multiple other values of system parameters were shown. This should include other values where the approximation is accurate, but also where it breaks down, giving an indication of what the region of applicability of the present theory is.

3- How does this approximation compare to doing 2nd order degenerate PT in lambda (for example through a Schrieffer-Wolff transform)? Is it equivalent? Does the present approach capture things that this doesn't?

4- On page 7, the authors refer to "some ambiguity in the prefactor." Is this just due to ambiguities in conventions to define a_2D? If so, the authors should state so. If not, the authors should explain what they're referring to.

5- On page 8, the authors find that the sum in 17b diverges logarithmically at large k. This is a little surprising since the approximation used to simplify the Feshbach-projected equations was exact for large k. What is the reason for this divergence, and why is the remedy used appropriate?

6- Figure 2 dicusses self-consistent vs non-self-consistent results. In the non-self-consistent results, exactly what calculation was done?

7- For the chemical potential, is there insight as to why21-23 are so much more accurate that the other 2x2 theory?

8- Perhaps most essentially, I would like to see stronger arguments and examples for results or insights that are significantly easier to obtain in the 2x2 theory than the 4x4 theory. After all, plotting eigenvalues of a 4x4 matrix as a function of some parameter is extremely easy. Some of the analytic results they have provided are clear examples, but if there are other examples, it would help convince readers of the utility of the approach and help them use it for themselves.