Magnetic impurity in a one-dimensional few-fermion system

Dear Referees,

thank you for taking the time to review our manuscript, for highlighting weak and strong points of our work. This definitely helped us to improve our manuscript. Below, we address all comments from the reports. [For convenience, we quote the original reports.]

We hope that the revised version is now ready for publication in SciPost Physics.

Sincerely,

The Authors

Reply to Referee 1.

Referee Report: This is a carefully constructed manuscript presenting the results of an exact-diagonalization study of a few-body system. The model, methods, and the results are clearly presented. The only improvement I may suggest is to attempt a comparison of the "phase transition" line in Fig. 4 with the phase transition line known from the Yu-Shiba-Rusinov model. It may require some work though to recast the parameters used in that model into the language (of g and G) utilized in the manuscript.

Our reply: We thank the Referee for this very positive evaluation of our work. Following the recommendation of the Referee “Try to compare the results with those of the Yu-Shiba-Rusinov model”, we added a comparison of our results to the corresponding many-body theory in Appendix D. We argue that our results are in a reasonable agreement with what is known, see an illustration in the new Fig. 12. The difference can be attributed to finite-size effects and to a somewhat different formulation of a problem. In our case, we add the Fermi energy when we add a spin-down particle. This effectively makes the gap between sectors of different parities larger than in the Yu-Shiba-Rusinov model. To account for this, we use the empirical Eq. (20).

Changes the manuscript: We have added Appendix D.

Reply to Referee 2.

Referee Report: This manuscript studies a few-body Hamiltonian for fermions interacting via a contact interaction in the presence of a magnetic impurity. The aim is to analyze the system as a precursor of a quantum phase transition caused by the interplay between the interaction and impurity coupling strength. Such a theoretical study is relevant for ultracold atom systems, where the number of particles is well below the typically assumed number of particles in an actual many-body system. The manuscript is very well written and the numerical results seem solid to me. It also has a clear relevance for the experimental community working on ultracold atoms. Despite all the positive aspects of this manuscript, I have as a reviewer to strictly adhere to the acceptance criteria of SciPost Physics. The closest match is criterion 4, "Provide a novel and synergetic link between different research areas", with criteria 1-3 not being fulfilled. I don't see criterion 4 being fulfilled either, but the authors could of course try to make their case better in the reply and revised version. Unfortunately, mere correctness and good quality writing is not enough to fulfill (at least one) of the four acceptance criteria of SciPost Physics. On the other hand, I would strongly recommend the publication of the current version of the manuscript in SciPost Physics Core.

Our reply: We thank the Referee for taking the time to review our manuscript, and for a suggestion to make our case better in the reply and the revised version. Below, we argue that our work fulfills two acceptance criteria of SciPost Physics.

– First of all, we believe that our paper “opens a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work”. Indeed, our work introduces a new system for investigation in the context of few- to many-body-physics crossover. This research direction has seen a number of important experimental discoveries in the recent past, see, e.g., Science 342, 457 (2013), Nature 587, 583 (2020), Nature 606 , 287 (2022), which motivate a wave of theoretical studies. Our work is clearly different from the previous studies as it focuses on the interplay between local ‘magnetic’ perturbation and pairing, which is a one of the key topics in superconductivity. To the best of our knowledge, this interplay has never been studied before with few-body systems, partially due to the fact that the task is numerically highly demanding. Here, we show a pathway that allows us to go to relatively large system sizes. In the outlook, we briefly discuss follow-up work, which includes systems with many impurities, time dynamics, etc. Besides, future research will focus on studies of a magnetic impurity in two- and three-dimensional systems of attractive fermions; mobile magnetic impurities; transport properties, etc.

– Second, our work “provides a novel and synergetic link between different research areas”. Indeed, our work provides a link between the Yu-Shiba-Rusinov model and few-body physics. This link is useful to understand the emergence of many-body physics from few-body building blocks. In particular, we argue that the phase diagram of the Yu-Shiba-Rusinov model can be understood already by considering few-body systems. This sheds new light onto the properties of the magnetic impurity in the vicinity of an s-wave superconductor. In the revised version, we elaborate on the connection between our results and the corresponding many-body model, see Appendix D.

Reply to Referee 3.

Referee Report: The manuscript discusses a few (2 to maximum 9) attractively interacting spinfull fermions trapped by an external potential. The external potential is superimposed on another narrow potential where a single fermion is trapped. The authors consider a zero-temperature behavior of the system depending on the interplay of two interactions: g - between the fermions themselves, and G - between the fermions and the so-called impurity in the narrow potential. The main result of their numerical (exact diagonalisation) calculation is the phase diagram which delineates between magnetic and nonmagnetic ground state. The authors find, for instance, a critical value of g=2 below which the system remains non-magnetic independent of the strength of G-interaction. This is an interesting result. The authors also consider density-density correlations to understand their findings better. All in all, I find the results valid and new, however, I do not quite agree with their interpretation. First of all, there is no resemblance to the BCS mechanism the authors mentioned. Also discussion of any Shiba-like states is not applicable for the case the authors consider. Also phase transitions arise as a result of ODLRO, which is far from being present in the system under consideration. So I am afraid I do not agree with the interpretation of the results and their direct relation to any phase transitions. However, on their own right the results are interesting and when shortened (e.g. parts about infinite interactions moved to Appendices, and discussion related to BCS superconductivity removed) the manuscript could be resubmitted to Phys Rev A or maybe SciPost Physics Core.

Our reply: We thank the Referee for taking the time to review our paper, and for their frank comments. The Referee finds three weak points: 1. interpretation of the results 2. too many trivial details in the main text 3. does not satisfy the expectation criteria of SciPost Physics

Let us address these three points:

1. We respectfully disagree with the Referee that there “is no resemblance to the BCS mechanism”, which led them to conclude that our interpretation of the results is not correct. Indeed, BCS theory in its original formulation was applied to extended systems. However, pair correlations between attractively interacting fermions exist even at a few-body level, see e.g., Fig. 7 B. Therefore, it is common in the community of few-body physicists to use the language of BCS theory to study the emergence of these correlations, see, e.g., Nature 606 , 287 (2022). Note also that the superfluid order parameter can be strictly defined through the eigenfunction associated with the dominant eigenvalue of the two-body density matrix. This definition generalises the concept of ODLRO to finite and inhomogeneous systems, and provides a natural extension of BCS-like concepts to few-body systems. We also note that in the paper we avoid a direct discussion of Shiba-like states, because those are not the topic of our study. Finally we note that in the revised version we provide a direct comparison of our results to the Yu-Shiba-Rusinov model, see Appendix D. In spite of a number of approximations, we observe a good agreement, which further validates our interpretation of the results.

Changes to the manuscript – We have amended the text such that it is clear that we discuss here a few-body precursor of BCS theory. See the new discussion of this issue in the first paragraph of Sec. 2.1. We have also added Appendix D to further validate our interpretation.

2. Following the recommendation of the Referee, we have moved the corresponding discussion to Appendix A.

3. We believe that our work fulfills acceptance criteria of SciPost Physics. We address those in the Reply to Referee 2. For convenience of the Referee, we repeat them below:

– First of all, we believe that our paper “opens a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work”. Indeed, our work introduces a new system for studying in the context of few- to many-body-physics crossover. This research direction has seen a number of important experimental discoveries in the recent past, see, e.g., Science 342, 457 (2013), Nature 587, 583 (2020), Nature 606 , 287 (2022), which motivate a wave of theoretical studies. Our work is clearly different from the previous studies as it focuses on the interplay between local ‘magnetic’ perturbation and pairing, which is a one of the key topics in superconductivity. To the best of our knowledge, this interplay has never been studied before with few-body systems, partially due to the fact that the task is numerically highly demanding. Here, we show a pathway that allows us to go to relatively large system sizes. In the outlook, we briefly discuss follow-up work, which includes systems with many impurities, time dynamics, etc. Besides, future research will focus on studies of a magnetic impurity in two- and three-dimensional systems of attractive fermions; mobile magnetic impurities; transport properties, etc.

– Second, our work “provides a novel and synergetic link between different research areas”. Indeed, our work provides a link between the Yu-Shiba-Rusinov model and few-body physics. This link is useful to understand the emergence of many-body physics from few-body building blocks. In particular, we argue that the phase diagram of the Yu-Shiba-Rusinov model can be understood already by considering few-body systems. This sheds new light onto the properties of the magnetic impurity in the vicinity of an s-wave superconductor. In the revised version, we elaborate on the connection between our results and the corresponding many-body model, see Appendix D.