Three-body correlations in nonlinear response of correlated quantum liquid

Behavior of quantum liquids is a fascinating topic in physics. Even in a strongly correlated case, the linear response of a given system to an external field is described by the fluctuation-dissipation relations based on the two-body correlations in the equilibrium. However, to explore nonlinear non-equilibrium behaviors of the system beyond this well-established regime, the role of higher order correlations starting from the three-body correlations must be revealed. In this work, we experimentally investigate a controllable quantum liquid realized in a Kondo-correlated quantum dot and prove the relevance of the three-body correlations in the nonlinear conductance at finite magnetic field, which validates the recent Fermi liquid theory extended to the non-equilibrium regime.

Dimensionless axis labels (e.g., eV/kBTK) are very appealing from a theoretical point of view but make a comparison of experimental data difficult. May I suggest adding the scaling factors (as in this particular case kBTk in eV) in the figure captions for readability?
As a side note, reaching the unitary limit does not require the nanotube to be particularly "clean". As the authors are certainly aware, it mostly means that it couples equally well to both contacts.
Finally, given that it is also an experimental work and extensively discusses few data curves of one(?) device, we should probably get to know the device a bit more. (This should be in the supplement, and referred to in the main text. Given the nice quality of the already shown data in combination with the universal behaviour of the Kondo effect, I dont expect this to have any impact on the statement and theoretical outcome, but it would be useful to place the findings into context.) I would expect at least * a brief discussion of device fabrication, materials and geometry * a "Coulomb diamonds" style plot of the differential conductance at the Kondo ridge of Fig. 2  If the device is identical to one shown in referenced publications (Ref. 33?) and this information is already shown elsewhere, please state so explicitly! Then again, space in the supplement is cheap, and it may be useful to have reference data in place.
In the following, we would like to answer his/her comments and questions. Please also see the list of the revisions below and the color parts in the main text. In order to follow the formatting instructions of Nature Communications, we have added section names and figure titles, made many minor corrections in all the sections, and moved several explanations to Methods. These changes are not highlighted to make the manuscript easy to read. (For clarity, your comments are shown in blue and italic.)

[Referee's comment (1)] In the Introduction, it is not clear what the authors mean exactly with two-body
correlations ¥chi_{¥sigma_1¥sigma_2}, the definition of this quantity should be given immediately.

[Answer]
As the referee suggested, we have added the definition of two-body correlation immediately after the explanation of χ σ 1 2 in the 3 rd paragraph. The two-body correlation is exactly conventional linear susceptibility.

[Answer]
As the referee pointed out, we have included the term, "the", at the 2 nd to last sentence of caption of Fig. 1c. [Answer] In order to avoid readers' misunderstanding of the 1 st sentence in the 7 th paragraph, we have changed the sentence to "Importantly, when the onsite Coulomb interaction ( ) in the QD is zero, a complete analytical form for Equation (1) is obtained by replacing B K with 2γ 0 (see Methods and Supplementary Note 1). Here, 2γ 0 virtually corresponds to the half width of a resonance peak." We have also shown the detail analysis procedure for = 0 case in the subsection "Analysis in the = 0" in Methods.
[Referee's comment (4) [Answer] We thank you for your nice comment, which is certainly an important point. Unfortunately, we do not access the regime at temperature of the order of K . The reason is that K~1 .6 K is higher than the highest temperature for stable operation of our dilution fridge (800mK), and systematic measurement around and above K was not easy. We have added comments on this issue in Discussion, as the referee suggested.
Finally, we thank the referee very much again for his/her fruitful comments. We strongly believe that the manuscript has been further improved thanks to them and would be very happy that he/she would find our manuscript now publishable. We thank the referee very much for recommending our work to be published in Nature Communications and noting that "the presented results are definitely worth publication." We also highly appreciate several precious comments to further improve our manuscript.
In the following, we would like to answer his/her comments and questions. Please also see the list of the revisions below and the color parts in the main text. In order to follow the formatting instructions of Nature Communications, we have added section names and

[Answer]
For readers to understand three-body correlations, we have added a schematic view of the correlation between the three electrons accounting for the Fermi liquid correction as Fig.   1a. We hope such a simplified picture would help readers to have some intuition on the three-body correlation. In addition, in the 6 th paragraph, we have mentioned that two independent parameters, ↑↑↑ and ↑↑↓ , characterize the nonlinear response for specific cases shown in the subsection "Properties of 1 2 3 " in Methods.
In addition, we take your comment "Unpacking the manuscript took some effort." seriously, and we have made several minor revisions to enhance the readability such as by adding mutual references to Figures and Equations in the manuscript. We thank you very much for your helpful comment.

[Referee's comment (2)]
A carbon nanotube, due to its inherent spin *and* valley degeneracy, can display both the SU(2) and the SU(4) Kondo effect. * The discussion does not state anywhere (or I can't find it) which case is discussed. The SU(2) case is implicit in that the theory talks only about a single spin-1/2 system, but this should in any case be clarified in the text and maybe even in the abstract / introductory paragraph.
* How do the authors justify that their data is described by the SU(2) Kondo effect, when also SU(4) can occur in carbon nanotubes? The theory obviously "fits" (and the data also looks like SU(2)), however, this is a gap in the argumentation. And in particular when the fourfold degeneracy is not strongly enough broken the additional states have a clear impact on the nonlinear conductance.

[Answer]
We thank you for pointing out this important point, which we failed to mention in the original manuscript. As you wrote, we discuss only the SU(2) Kondo state in this manuscript. We have revised the manuscript to explicitly mention that we are dealing with the SU(2) Kondo case. In the 9th paragraph, we write that the symmetry of the Kondo state was experimentally identified by the shape of Coulomb diamond and shot noise measurement as demonstrated before [Ferrier et al, Nat Phys. 12, 230 (2016);Phys. Rev. Lett. 118, 196803 (2017)]. The effective charge of the shot noise is clearly different between in the SU(2) case and in the SU(4) case and thus we are perfectly sure that the SU(2) Kondo state is realized in this manuscript.

[Referee's comment (3)]
Dimensionless axis labels (e.g., eV/kBTK) are very appealing from a theoretical point of view but make a comparison of experimental data difficult. May I suggest adding the scaling factors (as in this particular case kBTk in eV) in the figure captions for readability?

[Answer]
We thank you for this nice comment to increase the readability. As the referee suggested, we added the scaling factor, B K = 138 μeV and have mentioned that B / B K = 1.0 corresponds to = 2.4 T in the caption of Fig. 3b. [Referee's comment (4)] As a side note, reaching the unitary limit does not require the nanotube to be particularly "clean". As the authors are certainly aware, it mostly means that it couples equally well to both contacts.

[Answer]
As the referee pointed out, the cleanness is not required for achieving the unitary limit. We removed the term "clean".
[Referee's comment (5)] Finally, given that it is also an experimental work and extensively discusses few data curves of one(?) device, we should probably get to know the device a bit more. (This should be in the supplement, and referred to in the main text. Given the nice quality of the already shown data in combination with the universal behaviour of the Kondo effect, I dont expect this to have any impact on the statement and theoretical outcome, but it would be useful to place the findings into context.) I would expect at least * a brief discussion of device fabrication, materials and geometry * a "Coulomb diamonds" style plot of the differential conductance at the Kondo ridge of Fig. 2 and Fig. 3 and its surroundings * possibly similar data in a magnetic field as point of reference for Fig. 3 If the device is identical to one shown in referenced publications (Ref. 33?) and this information is already shown elsewhere, please state so explicitly! Then again, space in the supplement is cheap, and it may be useful to have reference data in place.

[Answer]
As the referee pointed out, the device is identical to the one studied in Ref. [35,36,37], which we explicitly mentioned in the 9 th paragraph. We also added a section to show the fabrication process and supplemental data in the Supplementary Note 4.
Finally, we thank the referee very much again for his/her fruitful comments. We strongly believe that the manuscript has been further improved thanks to them and would be very happy that he/she would find our manuscript now publishable.