Interference dynamics of matter-waves of SU($N$) fermions

1. Comprehensive study.

1. Mixing results and their explanations.
2. Not enough information in the main text for a non-specialist to comfortably read the paper.

In the present work, the authors investigate ultracold fermionic gases in toroidal traps. They investigate both non-interacting and interacting (both attractive and repulsive) gases. The authors derive predictions for interference experiments. Namely, for homodyne interference which gives access to the momentum distribution of the gas particles and heterodyne interference, which may produce spiral interference patterns, linked to the total angular momentum of the gas.

The main novelty of the consideration is that the authors study the gas in which fermions possess an SU(N) internal degree of freedom (as opposed to SU(2) studied previously). For homodyne detection, the authors focus on investigating the appearance of the dip in the center of the interferogram. For heterodyne detection, the authors focus on the appearance of glitches in the spiral. Both as a function of the flux penetrating the center of the system.

This is a comprehensive study, which may be useful for analyzing experiments in ultracold gas systems and verifying exotic properties of strongly-interacting systems. At the same time, the scope of the results corresponds rather to SciPost Physics Core (PRB-level) rather than SciPost (PRL-level) results.

Further, the paper in its present form is hard to understand unless one is very familiar with ultracold atoms and with strongly-interacting 1D systems. It is very hard to disentangle the results from attempts to explain them on the go. I strongly recommend the authors to work on making the paper more reader-friendly. I mention below a few important aspects of how this can be done.

1. In the abstract, the authors mention fractional quantization of the angular momentum. I understand that this is some sort of slang. The wave function of each particle should be single-valued as a function of the polar angle phi, leading to integer quantization of the total angular momentum – fractional quantization of the angular momentum seems unphysical from this point of view. On the other hand, if the authors prefer considering twisted boundary conditions due to the inserted flux, the angular momentum of such a “twisted” wave function can be quantized to fractional values even for non-interacting systems. If the statement is crucial to the authors, it should be explained early on in the paper. On the contrary, if the statement is not essential to the main message of the paper, it should not appear in the abstract – where it raises doubts about the validity of the work.

2. Early in section III.A, the authors refer to n, quantum numbers of the levels occupied by particles. While this is a free particle system and one can guess that n corresponds to momentum quantization along the chain, this is not spelled out explicitly. In connection to total angular momentum l appearing in the same paragraph, this makes the reading somewhat confusing. Further, the authors refer to parabolas of different angular momentum – it would be most reader-friendly to provide a figure of such parabolas and how changing the flux makes the system move from one parabola to another. The authors actually provide the figures and the explanations in Appendix A.1. I find that having them in the main text would make the paper significantly easier to read.

3. In the last paragraph of III.A, the authors write about W, the ratio of the number of particles in the system to the number of fermionic flavours. In particular, the authors discuss the difference between “equal and commensurate value” of W and “the case when W is commensurate but not equal for different SU(N)”. I can guess that what is meant here is the number of fermions occupying each flavour being an integer multiple of the chain length, yet possibly not the same for different flavours. However, providing a slightly more extended discussion, where the reader would be given a proper definition, would be beneficial. It would also be nice to briefly discuss why equilibration within each flavour is assumed, but no equilibration between flavours is considered possible.

4. In section III.B.1, when discussing repulsive interactions, it would be beneficial to discuss the origin and the “fractionalized” parabolas in the main text, and not refer the reader to the Appendix.

An alternative approach to presenting the paper results could be highlighting the key novel results without explaining their origin. And then derive the results/provide their microscopic explanations later in the paper/in the Appendix. The present way of presenting, when the result explanation is expected to be understandable, but there is not enough information to actually understand it, makes the text very uncomfortable to read.

A few smaller remarks.

In the introduction, the authors write: “ultracold atoms feature robust coherence properties withour cryogenics”. According to some definitions (e.g., https://en.wikipedia.org/wiki/Cryogenics), cryogenics is defined by studying low temperatures – not necessarily by using cooling liquids to achieve those. The authors may want to reformulate the phrase.

In the paragraph after the one containing Eq. (1), the authors write: “for strong attractive interactions (U << t)”. I presume U < 0, |U| >> t is meant.

In Eqs. (A14-A17), a discrete sum formula is connected to the Bessel functions represented through a continuous integral (A15, by the way, the differential is missing from the integral). Is the connection valid for any chain length L or only in the thermodynamic limit L->infty?

1. Improve the presentation of the paper following one of the routes suggested in the report.

1- The subject is timely
2- The study of SU(N) well complements corresponding studies for bosons and for 2-component fermioni c gases in rings
3- The study of momentum distribution and interferograms is well integrated
4- The article is overall clear

1- The main weakness in my opinion is that the study of attractive interactions is only partially performed
2- The summary of literature on attractive SU(N) fermions is too compact, and it would be useful together with a more detailed study of the attractive interactions case

The authors study the interference dynamics of SU(N) potentials in ring potentials. The paper is well motivated and clearly written, with the results well explained. My main remark is anyway that, given the clear motivation of the problem, the attractive interactions case should have been more expanded:

-) In the section II it is written

"For strong attractive interactions (U<<t) SU(N ) fermions are able to form bound states of different types and nature, which in turn causes part of the particles to localize together, while still adhering to the Pauli exclusion principle."

While this is true, I think a more detailed description of the phase diagram(s) of attractive SU(N) fermions is needed, see the point below. Notice that in the previous sentence the parenthesis U<<t should read |U|>>t if I am not wrong.

2) When in section III attractive interactions are considered, plots are discussed, but U is not varied. I think a discussion of different values of U would be important in section 3 [probably also in section 4, but that I can understand it could be defered to a subsequent publication], also in connection with the previous point 1).

3) Morever, for the two-component case the transition to the Tonks-Giraredeau gas has been discussed in literature, as in Fuchs, Recati and Zwerger, PRL (2004) and other papers, that helps to understand the quasi-1d case. A similar discussion for the SU(N) case with N>2 should be improved/provided, at least qualitatively.

If the previous remarks are taken into account, the readability and significance would improve and the paper could be published in my opinion.

1- Better discuss the results for SU(N) with attractive interactions in literature in the 1d and quasi-1d cases.

2- Improve the discussions of momentum distribution (and possibly of spiral interferograms) as a function of |U|.