Universal Upper Bound for the Entropy of Superconducting Vortices and the Quantum Nernst Effect

Round 1

Reviewer 1 Report

This is an interesting group-theory analysis of the Nernst effect in superconductors. The authors demonstrate the existence of the upper bound ln(2) in the entropy per vortex in a 2D superconducting layer that explains the universal maximum in the temperature dependence of the Nernst coefficient. They provide a simple explanation to their findings in terms of the information theory. The article can be published as is.

Author Response

We are most happy that the Reviewer highly evaluated our results and recommended an unconditional publication of our article.

Reviewer 2 Report

This paper is a theory try to explain the recent result of Nernst. I should say frankly, as a condensed matter researcher, that I do not understand those abstract methods which seems to me similar to string theory. But more or less I do understand that incompressibility might be the center point to explain the result by Phys. Rev. Lett. 2021, 126, 077001.

The authors explained the result by using the incompressibility in a system with superconducting gap. For me, it is quite difficult to understand that they associated the result to the gap. I think that, there are two different kinds of vortex follow, one is vortex liquid, and other is vortex solid. The maximum appears the phase transition line between two phases. The phase transition line is vortex melting transition line. Indeed the vortex solid is  incompressible.

In summary, I think that incompressibility is a correct idea, but authors did not correctly identify the source of the incompressibility in my opinion.  

I suggest that the authors should address this problem carefully.

Comments for author File: Comments.pdf

Author Response

Point-by-point reply to Reviewer 2

1. Reviewer: This paper is a theory try to explain the recent result of Nernst. I should say frankly, as a condensed matter researcher, that I do not understand those abstract methods which seems to me similar to string theory.

Reply: Our paper is mathematically rigorous proof that when the density of quantum vortices in a Cooper pair condensate achieves its highest possible value, the entropy per layer carried by each vortex assumes the universal value. The mathematical method employed is the algebra-based dynamical symmetry of fundamental excitations of the considered system. The possibility of using the algebra for constructing the proof follows from the incompressibility of the Cooper pair condensate, a fundamental physical object that we study. We highly appreciate that the Reviewer frankly admitted the lack of understanding of the employed mathematical methods.

2. But more or less I do understand that incompressibility might be the center point to explain the result by Phys. Rev. Lett. 2021, 126, 077001.

Reply: We admire the Reviewer’s intuition; as mentioned in quoted PRL, the consulted leading experts in the vortex physics and Nernst effect in superconductors mentioned in Phys. Rev. Lett. 2021, 126, 077001 appeared unable even to hypothesize the origin of the universal entropy carried by vortices near the superconducting transition temperature.

3. The authors explained the result by using the incompressibility in a system with a superconducting gap. For me, it is quite difficult to understand that they associated the result to the gap.

Reply: Compressibility of the Cooper pair would mean the possibility of changing the superfluid density of the Cooper pair condensate by external perturbation, i.e., the change in the magnitude of the superconducting gap. This change is impossible as long as the external action is not strong enough to destroy Cooper pairing (i.e., the superconducting gap); therefore the existence of the superconducting gap means incompressibility of the Cooper pair condensate. 

4. I think that, there are two different kinds of vortex follow, one is vortex liquid, and other is vortex solid.

Reply: This is correct; the vortex system can exist either in the form of the Abrikosov vortex lattice or in the form of the vortex liquid. The authors are aware of the existence of vortex solid and vortex liquid. Please note that one of the authors is a laureate of the 2003 John Bardeen Prize for his contributions to the theory of the vortex matter that have led to the prediction of novel experimental results in vortex physics and a laureate of the 2015 Abrikosov prize for pioneering contributions to vortex dynamics.

5. The maximum appears the phase transition line between two phases. The phase transition line is vortex melting transition line.

Reply: This is a very interesting hypothesis, but, unfortunately, it is not confirmed by the experimental results. The findings of the Phys. Rev. Lett. 2021, 126, 077001 indicate that the maximum occurs at the superconducting transition temperature.

6. Indeed the vortex solid is incompressible.

Reply: This is again an entirely correct statement. We only wish to add that at the magnetic field well above the field of the first penetration (which corresponds to the experimental situation that we investigate), the vortex liquid has precisely the same compression modulus as vortex solid, I. e., is equally incompressible. This property is described in two most famous reviews on vortex physics: (i) G. Blatter, M. V. Feigel'man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Vortices in high-temperature superconductors, Rev. Mod. Phys. 66, 1125 – 1388 (1994) and E. H. Brandt, The flux-line lattice in superconductors, Reports on Progress in Physics, 58, 1465 – 1594 (1995). Here we would like to thank the Reviewer for attracting our attention to this point, so for the readers’ benefit, we have added these references to the reference list of our manuscript.

7. In summary, I think that incompressibility is a correct idea, but authors did not correctly identify the source of the incompressibility in my opinion.  

Reply: We are happy that the Reviewer approves our idea. In the items above, we explained the origin of incompressibility in more detail. We introduced this clarification into the main text based on the Reviewer's comments. We would like to summarize here that the incompressibility of the Cooper pair condensate (and the following from this fact incompressibility of the vortex matter derived in the above-cited reviews), which is expressed by the existence of the superconducting gap, is not the finding of the submitted manuscript, but one of the fundamentals of the superconductivity. In this work, we demonstrate that the incompressibility of the Cooper pair condensate enables us to employ the algebra and use the corresponding dynamical symmetry technique to analyze fundamental quantum topological excitations and the entropy transport mediated by these excitations. We realize that this technique may not be common knowledge at this point, but this is precisely why we were capable of derivations beyond the reach of other experts in the field.

To repeatedly stress again, the superconducting gap defines the existence of the superconducting state and, therefore, the corresponding incompressibility of the Cooper pair condensate, which is a ground state of a superconductor. This incompressibility follows from the fact that perturbations with the energies below the gap cannot cause disturbance in the condensate density. Vortices are the topological excitations over this ground state, and the incompressibility of the vortex matter is derived from the incompressibility of the condensate.

8. I suggest that the authors should address this problem carefully.

Reply: The problem we addressed is formulated in the cited PRL as follows:

“In four superconductors belonging to different families, the maximum Nernst signal corresponds to an entropy per vortex per layer of ≈k_Bln2” in the abstract of the PRL paper and as

“To sum up, we find that in four superconductors with different normal states, pairing symmetries and critical temperatures, the Nernst transport entropy per vortex per
layer is of the order of k_B” in the conclusion of the paper. The experimental paper in question is marked as “Editors’ Suggestion” and “Featured in Physics,” which spotlights the significance and scientific appeal of this problem claimed to be a significant experimental result.

To address this problem, we used the fundamentals of superconductivity, stating the incompressibility of the Cooper pair condensate. This incompressibility enabled us to employ and apply the technique of the W_1+∞ algebra for the description of the dynamics of the topological excitations, vortices, and to derive the result observed in the experiment mathematically rigorously. To the best of our knowledge, so far, nobody else was able to derive this experimental result.