Persistent Current of SU(N) Fermions

1) the mechanism describing the emergence of spinon excitations
in the ground state is new (to my knowledge) and interesting

2) the properties of the model are found by applying
well-established theoretical and numerical techniques

1) Various unclear points or comments should be modified and
improved. These are mainly concentrated in the section "Methods" which plays a crucial role to get a better understanding of the
subsequent discussion and of physical results

2) The term "mesoscopic" is repeatedly used and seems to be an
important aspect in this paper but no comment is given in the text. To have an idea of what it concretely means one must evince this
information from the figure captions. A clear comment (in the text) on the values of L used in this paper for different regimes is
necessary.

3) the study of different regimes of the same system has been
made by using different lattice sizes

This paper investigates the properties of persistent currents in a
system of fermions trapped in a ring geometry. Fermions are
characterized by N-component spins and repulsive interactions.
The crucial parameter is the filling factor $\nu$, which implies non zero persistent currents for incommensurate values (and for any value of the interaction parameter). In this regime, spinon excitations
appear in the ground state as the effective magnetic flux is
increased. The presence of spinons causes the partial screening of
the external flux and the onset of characteristic oscillations of
persistent currents. For integer values of filling, a Mott transition is found if the interaction values are larger than certain finite critical
values (and if $N >2$).

The most part of the analysis in this paper concerns the currents at
incommensurate fillings which, as a matter of fact, corresponds to
the regime of very low filling (large number of sites). In this case
model (1) reduces to the GYS model and thus the analytical
approach related to the Bethe ansatz (BA) can be used to study the
formation of ground states including spinons. A shorter final part
based on (exact o DMRG) numerical calculations is devoted to the integer filling regime which highlights the effects (energy gap,
persistent-current behavior, maximal-current drop) induced by
the transition to the insulating regime.

Apart from the section "Methods" the paper is, in general, well
organized. In the remaining part, investigating the
incommensurate and commensurate-filling regimes, the
discussion is often very technical but, in general, sufficiently clear
except for the comments relevant to the undefined Bethe quantum numbers.

The properties of the model are found by applying the well-
established theoretical and numerical techniques such as the BA
based analysis, exact numerical diagonalization and the DMRG
approach. The results presented in this paper are technically sound. The only false note is the fact that the study of different regimes
(with incommensurate and commensurate filling) has been made
by using different lattice sizes for the same system.

The central result of this paper, the new (to my knowledge)
mechanism describing the emergence of spinon excitations in the
ground state, is nice and interesting. I expect that the analysis
developed in this paper should stimulate further theoretical work.

This paper must be revised to improve its readability. To this end
various unclear points or comments should be modified and
improved. These are mainly concentrated in the section
"Methods" which is essential to allow a better understanding
of the subsequent discussion and of physical results. This section
is very technical and various comments and formulas are
introduced implicitly assuming that any reader can understand
them. Unclear definitions must be supported by further
concise but clarifying comments (which cannot be relegated
in supplemental material).

Comments

1) Page 2 (LC). The long part from "For $N=2$ the BA ..." to "... for
finite value of $U/t$ [4,26]" apparently contains a considerable
amount of information. It supplies a list of properties and results
relevant to model (1). This list, however, is almost disjoint
from the subsequent discussion in the paper and its contribution to better understand the paper is almost null. The worst point
concerns the comment "The BA eigenstates are customarily
labeled by a certain set of quantum numbers $\{ I_a, J_{\beta} \}$" and the
subsequent comments. After noting that $I_a$ and $J_{ \beta_j }$ are undefined, the comment on 1) the zero-flux case and consecutive quantum
numbers and 2) on the non-vanishing $\phi$ case where $\{ I_a, J_{\beta} \}$ can
change, turn out to be incomprehensible. Unfortunately, one
discovers that these quantum numbers are repeatedly used in
various important formulas in the sequel (for example for defining $X$ and in eq. (2)). The authors must reorganize this part clarifying
how the many properties/results listed are related to the following part of the paper. In particular, a clear definition (not relegated to
the supplementary material) of the quantum numbers that
characterize su(N) must be given.

2) The authors discuss the approach to model (1) in the integrable
and non-integrable regimes. In the first case, in addition to the GYS-model regime, they consider the half-filling regime with large
interaction captured by the Sutherland model (see page 2, (RC)).
The latter, apparently, has nothing to do with the Lai-Sutherland
model with $n_j = 1$. This point should be clarified.

3) The title of the section entitled "Methods" seems quite
inappropriate: only a few lines in the section are devoted to
methods and techniques.

4) The term "mesoscopic" is repeatedly used and seems to be an
important aspect in this paper (but no comment is given in the text on this term). To have an idea of what it concretely means one must evince this information from the figure captions. The use of 30, 40
sites to get the results in figs. 1 and 2 is clearly related to the
reduction of model (1) to the GYS model and than to the application of the BA analysis. But 30 or 40 sites represents (in many papers on lattice systems) macroscopic systems. Since L= 6, ... 18 is used to
get the numerical results in figs. 3 and 4, the meaning of
"mesoscopic" is confused. This term must be carefully discussed in text and clarified. As an alternative, I do not exclude that the best
thing to do is to remove "mesoscopic" from the paper. In any case, a sufficiently extended comment on the values of L used in this
paper for different regimes is necessary.

5) The study of a system in regimes with different filling
should be made without changing the other model parameters.
The fact that the system properties are derived by considering
rather different sizes (see comment 4) of the lattice does not
increase the scientific rigor of this paper. This is a delicate
point that these authors have ignored but deserves some
careful comment.

see my reports

1) Very detailed study of spinon creation and its consequences to the periodicity of the persistent current in the SU(N) Hubbard model from many different angles, using Bethe ansatz together with exact diagonalization and DMRG techniques.

2) The results are novel and interesting as far as it concerns the periodicity of the persistent current with the total flux.

3) The results can be of relevant for future experiments in the emergent field of atomtronics.

1) The presentation is at times not very clear. Specific comments have been made in the enclosed report.

2) The discussion of the experimental consequences has is poor.

The authors describe the results of an investigation of the persistent current
in various limits of the SU(N) Hubbard model with periodic boundary conditions.
The results are obtained by studying various integrable limits of the model
using the Bethe Ansatz (BA). Some of the results are also obtained
using exact (numerical) diagonalisation and the density-matrix renormalization
group (DMRG). The authors focus on the small system, mesoscopic regime,
which can be relevant to experiments in “atomtronics”.

Some of the main results are the existence of persistent currents that result from
the creation of spinons when a “magnetic flux” is piercing the system. An interesting finding is that the periodicity of the persistent current is not only determined by the flux quantum as in non-interacting systems, but it also depends on the interaction strength as shown in Fig. 1. For commesurate fillings, a suppression of the persistent current is observed as the system develops a Mott gap, which can be used as a “detector” of the thermodynamic quantum phase transition.

I think the paper is of a sufficiently high technical quality and the results
are interesting enough to be granted publication. However, prior to acceptance,
I would ask the authors to address several (minor and perhaps not so minor)
issues, which are listed below:

1) On page 2, second column, near the bottom of the page,
the authors state that they only consider systems with spin-singlet states
where the total magnetization S^z = 0. They should either provide the definition of S^z operator for N > 2 or clarify this statement. I believe I understand what they mean by zero total magnetization, that is, the numbers of different species are all equal, but it is rather confusing in this context.

For instance, for the sequence N = 2F +1 where F is the hyperfine spin of the alkaline-earth atom, S^z = F^z, the nuclear spin projection. However, this is not the case for values of N not being to this sequence as it is the case of N = 3, for which there are several possibilities of trapping three components exists.

2) To be frank, I find Fig. 2 excessively crowded and not very clear. I think the authors should consider simplifying it, and if necessary split it into two separate figures. There is too much information on this figure that its main message is hard to grasp.

3) On page 4, in the conclusions section, “The reduction of the effective flux quantum indicates that a form of ’attraction from repulsion’ can occur in the system…” along with the following sentences seem to me rather unclear and citation of Refs. 9, 11, 37, 38 does not clarify the mechanism. If the authors have a clear physical picture they should explain it, otherwise they should remove this sentence(s).

4) Although the analysis of the spinon creation in the ground state with flux and its relevance to the persistent current is very detailed and carried out from many different angles, I find the discussion regarding the connection to experiments is insufficient. For instance, due to the additional difficulties to cool fermions (as compared to bosons), finite temperature effects are always important. This implies that, by means of over the barrier thermal activation, the persistent current could decay and it would be useful if the authors could give some discussion of such effects (ideally some estimates to guide the experimentalists and give them an idea of the relevant time scale would be most welcome as well).

6) Some remarks about the references: Personally, I do not like the current trend of
citing only review articles and ignoring original references. For instance,
Ref. [13] is a review article about the SU(N) symmetry exhibited by alkaline-earth gases. I think that, IN ADDITION to review articles, it would advisable to cite the
original references where the existence of the symmetry was first pointed out:

AV Gorshkov, M Hermele, V Gurarie, C Xu, PS Julienne,
J Ye, Zoller P, Demler E, MD Lukin and AM Rey Nat. Phys. 6 289 (2010)

MAC, A.F. Ho and M. Ueda New Journal of Physics 11, 103033 (2009)

The same remarks apply to Ref. 21 about DMRG and possibly others.

Finally, let me also point out that the journal name for Ref. 38 is missing.

The suggested changes are described in the enclosed report.