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

R:“I thank the authors for responding to my comments and for making corrections to the manuscript. The authors have brought it to a point, where I feel that the manuscript may be published. But not in SciPost Physics, rather in SciPost Physics Core.

Below, I explain the grounds for such a recommendation. I start with responding to some of the authors’ comments in their response letter. ”

1. R: The authors state: “we note that the persistent current, being a genuine mesoscopic quantity, is identically zero in the thermodynamic limit and therefore the ‘smooth continuum’ cannot be observed in that limit”. This is evidently wrong. The most obvious counterexample of a persistent current on macroscopic scales is given by superconducting levitation experiments: https://www.youtube.com/watch?v=AWojYBhvfjM. Even if one restricts the attention to cold atomic systems, Ref. 39 performed experiments with about 60 000 atoms. In which case, 1/Np < 10−4, leading to the possibility of a quasicontinuum to quite a good approximation.”

A: The confusion in the mind of the Referee arises since an erroneous understanding of what thermodynamic limit means: Number of particles & system size going to infinity in such a way that the ratio between the two quantities is finite.

Persistent currents result whenever the ring’s spatial scale is comparable to the particles’ coherence length. As it is well known in mesoscopic physics, the persistent current scales as L1 (see Equation 4.6 page 69 in the book “Introduction to mesoscopic physics” by Joe Imry). For the particular case of cold atoms mentioned by the Referee, persistent currents have been observed experimentally with systems having around 105 atoms in rings that have a size to the order of μm. Such system is mesoscopic. We remark that the fact that the persistent current vanishes in the thermodynamic limit can also be seen in the reference mentioned in the manuscript (see, for instance, the expression of the current Equation 2 in Ref 43.) where one can see that there is an inverse relationship with the length of the system.

We point out that the experiment in Ref. 39 was carried out for bosons and not fermions —- persistent currents of repulsive bosons have a corresponding angular momentum quantized to integer values (see Figure 3 in Ref. 39.).

1. R: “The authors state: “If we understand correctly, the Referee makes an argument on the ability of the angular momentum per particle to acquire fractional values based on particle-hole excitations in free and weakly interacting systems. Their argument would imply that any weakly interacting system would display 1/Np angular momentum fractionalization. This is in clear contrast with the experimental findings (Ref. 39 in the manuscript) in which the angular momentum per particle in a quantum fluid is indeed quantized irrespective of the number of particles.”

My argument indeed states that having a fractional values of angular momentum per particle is not a wonder. Moreover, the lower limit on the fractional values that can occur in a system with Np particles is given by 1/Np. Why does then Ref. 39 not observe such fractional values? Note that the system is cooled to its ground state or close to it. So, the observed deviation from an integer value should be proportional to Nex/Np, where Nex is the number of particles excited above the non-interacting ground state (due to either imperfect cooling or the change in the ground state due to the weak interactions). If the number of excited particles is ¡ 10%, the deviation lies well within the error bars in Fig. 3. Further, for large Ω, when the cooling may break down, the angular momentum per particle varies continuously with Ω in qualitative agreement with my argument.

In other words, assuming perfect cooling to the ground state, I would expect integer quantization at vanishing interactions, and near-integer values for weak interactions. Which is what was observed in Ref. 39.

Appearance of robust fractional steps would indeed be surprising. However, the authors of the present manuscript do not make this claim. Angular momentum per particle in the interacting system is never calculated in the manuscript. The link between the calculated experimentally observable quantities and the angular momentum per particle is only implied by the text, and the implication is based on the intuition coming from non-interacting systems.

What the authors consider, is the dependence of some observable quantities on the flux threading the system. Which is why I repeat my opinion from the previous report: I strongly oppose the phrasing “orbital angular momentum ... quantized to fractional values” appearing in the abstract. Saying that interactions change the flux dependence of the observables in an unexpected way, would be a much clearer and a much more solid statement.”

A: The Referee’s whole argument hinges on the fact that in the experiment of Ref. 39, the persistent current does not display fractional steps reflecting the fractional quantization of the angular momentum. The problem of the argument may arise from the fact that he seems to apply a classical physics reasoning to a quantum fluid. On top of this, we reiterate that we are considering fermions and not bosons.

A very well known fact of quantum many-body systems is that the response of the ground-state to the rotation is called the angular momentum. For example, the Referee can check the Dalibard lectures in Varenna (see Equation 24 in http://www.phys.ens.fr/~dalibard/publications/2015_Varenna_JD.pdf ).

1. R: “The paper contains a number of results concerning the flux dependence of observables. The citation of Ref. 43 and Fig. 12 appearing in the paper’s appendix state that the flux periodicity of some observables should be inversely propotional to the number of particles, when the interactions are infinitely strong. This is what led to the above discussion concerning the thermodynamic limit. I acknowledge the author’s remark that systems with small number of particles, where the effect of steps should be visible, can, in principle, be studied.

Figure 3 in the main text says something different about a specific observable. Namely, the appearance of a dip in the density distribution is delayed by 1/2 − 1/(2Np) for strong interactions. 1/2 − 1/(2Np) tends to 1/2 in the thermodynamic limit and is thus observable for a large number of particles. In my opinion, this can be called fractionalization (not clear, though, of what). Unfortunately, this formula is only hinted at by the results for a small number of particles. Moreover, the physics behind this effect (the relation to the reduced periodicity, the relation to angular momentum per particle etc.) is not clear.

For the attractive interactions, the effects scale with the number of species, which makes them suitable to be observed in the thermodynamic limit. Figure 5 shows a reduced flux interval between jumps. And Fig. 4 shows a similar reduction of the flux intervals between features. Again, these are results for a small number of particles. Their generalizability and physical origin are not clear.”

A: The above argument is based on the wrong application of a formula that we provided. We specifically indicate that the delay for the appearances is given by φ_{H} + (N_{p}−1)/(2N_{p}), where φ_{H} is the flux at which a hole appears for the system with zero interactions and Np is the number of particles.

1. R:“One sees, that the paper does contain a number of interesting results. However, there are two problems:

• The language of angular momentum fractionalization may be highly misleading (may not be too – but the paper does not present enough evidence and explanations to make the judgement);

• The paper does not present physical insight into the origin of the discussed effects and thus will hardly be understandable to a broader audience. When properly interpreted, the results might indeed spark multi- pronged follow-up work in the professional subcommunity (three directions being theoretical clarification of what’s going on, and constructing experiments for small and large number of particles); but this proper interpretation is extremely hard to read in the paper.”

A: We strongly disagree with the Referee. We produce arguments and references that we apply the correct language in the manuscript. On top of that, our results are perfectly understandable with basic notions in mesoscopic physics.

1. R:“Therefore, I have two recommendations. I suggest the authors to avoid the language of “angular momentum fractionalization” in order not to mislead the reader. I suggest the editor to accept the paper to SciPost Physics Core, where it can serve as a reference to the professional subcommunity.”

A: In our opinion the paper is very suitable for SciPost Physics.