A theorem about scattering from slow to fast subsystems and application to resonantly Floquet-driven impurities

The manuscript seems to be mathematically rigorous. The derivation is rather detailed.

The main weakness is that the author merely derives a formal relation for a quite artificial limit. There is no physical insight beyond that.

The authors considers scattering between two subsystems with very different
time scales and proofs two theorems that hold in the limit in which the
ratio of these time scales tends to zero. The theorems concern formal
relations obeyed by intermediate expressions from scattering theory. The
result is also formulated for the corresponding expression for an ac-driven
impurity.

While the manuscript seems technically correct, it lacks physical
motivation and finally presents expressions that may or may not be useful
for solving any specific problem. Moreover, to me the limit considered
looks rather artificial and does not provide physical insight.

In conclusion, I cannot recommend the manuscript for publication, at least
not in the sections AMOP or Condensed Matter. I do not want to judge
whether it could fit to a more mathematically oriented journal or the
corresponding section of SciPost.

None

The author of this article derives theorems for systems composed of two subsystems that can scatter into each other where they can be described by the Lippmann-Schwinger equations. Furthermore, the theorems apply to the cases where one subsystem is much slower than the other. The interesting result is that for an initial incoming scattering state in the slow subsystem, its probability to scatter into the fast subsystem vanishes in the “extremely slow subsystem limit” . Furthermore, they apply their methodology to pair-breaking in a Fermi-Hubbard driven system and show analytical results.

I find the paper to be rather interesting and I am inclined to recommend it for publication. However, before that is done I would like the author to take into consideration the following comments and re-write the manuscript

1.) In the introduction, I think the author should cite more papers in order to relate the applicability of their methodology to the literature. For example, the author mentions that the method is especially useful in Floquet-Driven impurities which is good but a broader set of avenues must be thought of for its applicability (and cited). Please also cite the Floquet-theory papers, merely citing #[3] does not give the right credit so the author must do a thorough search of the literature.

2.) Related to available methodologies for solving driven-quantum systems, I think the authors should certainly mention/cite literature on the Floquet- Nonequilibirum Green Function techniques that have been covered in the plethora of papers like Phys. Rev. Lett. 113, 266801, Phys. Rev. B 87, 045402, Phys. Rev. Research 2, 033438, etc (& see cited papers within) that not only are nonperturbative and numerically exact but can also take into account effect of mixed states and quantum transport.

3.) Furthermore, I think the author may find their approach applicable to examples from the field of spintronics where there is distinction of the energy scales for classical or quantum localized spins that interact with electrons in a tight-binding chain. See for example Phys. Rev. Lett. 124, 197202
I think if the author can comment on points 1-3 in the introduction, manuscript will be stronger.

4.) Please add more examples from literature where this approach can be applicable so as to not stay restricted.
In introduction, “usual Schrieffer-Wolff regime” — please cite some reference so that the interested author can look into it. General readers may not know the “regime”. Furthermore, right after this, please introduce some story of what “pair-breaking” is before using it in the introduction (and put citations).
In the toy example, for the line “this means that particle on P will propagate slower by a factor of alpha ” — it is always better to describe that using a physical quantity, like momentum so that readers have something tangible to think of in terms of time-scale.

5.) For the equations, I recommend putting a comma or period after all of them as they are part of the manuscript writing as well as use Eq.(1) instead of (1).

6.) For the P and Q projection operators please write a mathematical expression in Section 2.2 so it is clear to the readers. Furthermore, what is the normalization constant for the state |phi> introduced in this section. Typically, that is \sqrt{N} where N is the “number of sites” in the TB chain.
Minor thing but for infinitesimal, it is better to use so as to avoid conflict with energy E = -2J cosk (they appear similar).

7.) Section 3 is written nicely where all the theorems are introduced, however for section 4.3, I encourage the authors to plot results in Eq. 59 and 60 and show what they look like, otherwise it does not give much insight. The plot should be able to show the importance of the methodology and provide interpretation as a function of k. Furthermore, please cut down on the equations in this section and move them to the appendix and focus only on the final result and what it teaches using your approach otherwise it is too difficult
to juggle with equations blocking the main message.

Lastly, I think the authors should also point out to other examples in the literature that can be solved using this approach as a future work. I would like to see the above changes implemented in the manuscript.

This manuscript deals with scattering properties between subsystems with well separated time/energy scales. By considering a slow and a fast subsystem, the author rigorously proves an effective decoupling in the limit where the ratio alpha=T_fast/T_slow tends to zero. This implies, for example, that scattering from the slow subsystem to the fast one is suppressed as alpha -> 0. This result is proved in general by two theorems: the first states that, under the alpha -> 0 limit, the projection of the wave function into the space where the effective potential is nonzero vanishes, implying an 'artificial boundary condition' for the incoming particle in the slow subsystem. The second theorem uses this result to conclude that in this limit the projection of the wave function in the fast subsystem also vanishes. The results obtained are then illustrated in a Fermi-Hubbard chain, where the dynamics of pairs and individual particles are well separated. Since the associated energies for the two subsystems are not the same, that is, the energy bands do not overlap, the author includes a time periodic external potential, so that it induces scattering processes between the subsystems. The application of the theorem then shows that, at the separation limit of the time scales, pair breaking is suppressed.

In my opinion, the results obtained in this work are interesting and can be applied to a wide variety of systems. The manuscript is well written and its claims are rigorously verified. I also find the application of the toy model to introduce the terminology and elements of the scattering problem quite pedagogical. However, from the point of view of the novelty of the work, I am a little concerned with the way in which it was presented. Mainly the introduction shows a lack of background on a topic that is already known. For example, in the toy model discussed, it is possible to renormalize the energy at the 0_P site due to the presence of the Q chain. In the limit alpha --> 0, the P chain can be seen as a homogeneous chain with an impurity at its center, whose energy scales as 1/alpha. This can be considered as a high barrier that separates the system P into two parts. Therefore, it is not so surprising that in this limit the wave function at the 0_P site vanishes, and with it the probability of transmission to the other side of the chain.

Regarding the general acceptance criteria for SciPost Physics, I cannot recommend the manuscript in its current form. A revision of the introduction, claiming relevance to this topic, and including related bibliography, should be given for its consideration.