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Topological phase transition between the gap and the gapless superconductors
by Yuriy Yerin, Caterina Petrillo, A.A.Varlamov
Submission summary
As Contributors:  Yuriy Yerin 
Preprint link:  scipost_202109_00018v1 
Date submitted:  20210915 11:09 
Submitted by:  Yerin, Yuriy 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
It is demonstrated that the known for a long time transition between the gap and gapless superconducting states in the AbrikosovGor'kov theory of superconducting alloy with paramagnetic impurities is of the Lifshitz type, i.e. of the $2\frac12$ order phase transition. We prove that this phase transition has a topological nature and is characterized by the corresponding change of the topological invariant, namely the Euler characteristic. We study the stability of such a transition with respect to the spatial fluctuations of the magnetic impurities critical concentration $n_s$ and show that the requirement for validity of its mean field description is unobtrusive: $\nabla \left( {\ln {n_s}} \right) \ll \xi^{1} $ (here $\xi$ is the superconducting coherence length) Finally, we show that, similarly to the Lifshitz point, the $2\frac12$ order phase transition should be accompanied by the corresponding singularities, for instance, the superconducting thermoelectric effect has a giant peak exceeding the normal value of the Seebeck coefficient by the ratio of the Fermi energy and the superconducting gap. The concept of the experiment for the confirmation of $2\frac12$ order topological phase transition is proposed. Obtained theoretical results can be applied for the explanation of recent experiments with lightwavedriven gapless superconductivity, for the new interpretation disorder induced transition $s_{\pm}$$s_{++}$ states via gapless phase in multiband superconductors, and for the better understanding of gapless color superconductivity in quantum chromodynamics and the string theory.
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Reports on this Submission
Report 2 by Carlo Beenakker on 20211014 (Invited Report)
Report
The key message of the paper is that the density of states exhibits a topological transition when the gap closes, signified by a change in the Euler characteristic χ of the surface N(ω,ζ) for ζ<1 and ζ>1.
The claim is not substantiated in any way, it is merely stated that χ=2 for ζ<1 and χ=1 for ζ>1. Since it is a central claim, there should be no ambiguity. For starters, I am not even sure that the Euler characteristic is well defined on the surface N(ω,ζ), because the cuspidal edge at ζ=1 has a divergent Gaussian curvature. The poles at ζ=0 are also worrisome. If these poles can be regularised, then I would think that both the surfaces at ζ<1 and ζ>1 can be contracted to a point, which would imply that χ=1 for both surfaces and there is no topological transition.
Anonymous Report 1 on 20211012 (Invited Report)
Strengths
Clear presentation
Weaknesses
Results are not sufficiently novel or do not impact physical observables in a major way
Report
In the present manuscript the Authors studied theoretically the transition between gapped and gapless states in swave superconductors driven by magnetic impurities. Two main result are presented: a reinterpretation of the gappedgapless transition as a topological transition and the prediction of the enhancement of the quasiparticle thermoelectric effect related to this transition. The results are presented in a clear way and appear absent of noticeable error. However, I am not convinced that the results constitute a significant advance in the field. The topological invariant the Authors propose does not reflect in any observable quantized properties, and does not indicate the presence of a topological state of matter (unlike, e.g., Chern number in a p+ip 2D superconductor). The transition to the gapless phase, as pointed out by the Authors, does not reflect a new universality class and belongs to the same class as the Lifshitz transition. Moreover, thermoelectric effect has been investigated in superconductors with magnetic impurities previously (Refs. 32,32) using various approximations; the Authors do not make it clear how their approach is different and if their results offer anything new except for the discussion. Finally, there are also questions regarding the generality of results (for example, whether the same results will hold if the gap is not isotropic) and the approximations used (in particular, the Born approximation appears to be implied).
Consequently, I do not recommend the manuscript for publication in SciPost Physics. Below I expand on the points mentioned above and provide questions and suggestions to be addressed in the manuscript.
Requested changes
(1) The Authors offer a reinterpretation of this transition in terms of a topological invariant  the Euler characteristic calculated for the DOS surface. Are any physical quantities of the system uniquely determined by this invariant?
$\bullet$ Is there any interplay of the Euler characteristic with the known cases of topology in superconductors (e.g. p+ip state in 2D)? Being a property of DOS, I believe that it will not distinguish between trivial (swave) and topological (p+ip in 2D) states.
(2) All calculations were performed for the case of an ideally isotropic gap. How will gap anisotropy affect the results? In particular, for strong anisotropy with deep minima, can the behavior of DOS and free energy near the transition change qualitatively?
(3) The Authors refer to AbrikosovGor'kov theory for the disordered superconductivity throughout the text  does that imply that Born approximation for scattering is used? In particular, does the justification for the stability of the meanfield description given in "Smearing of the transition due to spacial fluctuations of the magnetic impurities concentration", rely on Born approximation? How will rare region effects and the presence of impurity bound states affect the argument?
(4) The relation to previous works on thermoelectric coefficients in superconductors with magnetic impurities has to be discussed in more detail. In Ref. 33, multiple scattering effects were considered beyond the Born approximation, so it appears that the results of Ref. 33 are more general. In Ref. 32, on the other hand, results in Born approximation were reported and Eq. (11) and (12) there are indeed quite similar to Eq. (13) and (8) of the current work. However, the denominators in Eq. (13) and Eq. (11) of Ref. 32 appear different  could the Authors explain this difference?
$\bullet$ Since results at finite $T$ are reported in Fig. 3, was $\Delta(\zeta)$ calculated selfconsistently for finite $T$ or were the zerotemperature expressions used? If the latter is true, this has to be mentioned and justified.
Minor comments/suggestions:
 Fig. 1 misses a color scale; the drastic change from purple to blue seems to suggest a jump, which can be confusing to readers, since the transition is actually continuous.
 Mentions of the applications to $s_{++}/s_{\pm}$ transition and color superconductivity in QCD and string theory are not really substantiated or discussed. The Authors should either provide a discussion of what new physics can their approach reveal in those systems or refrain from stating the connection (at least in the abstract and conclusion).
 Some links in citations are not working (e.g. 17,18); Ref. 17 links to the same URL as Ref. 18; Ref. 18 is missing the journal information.
Anonymous on 20210921 [id 1771]
A question which the authors may want to address, to help me out of my confusion. Figures 2c and 2d show a topological change in the Fermi surface before and after the gap closing transition. It is argued that an equivalent topological transition appears in the density of states as a function of frequency , where figure 2c corresponds to figure 2a and figure 2d to 2b. I am unsure how this comparison works.
Figure 2c is at a fixed value of the parameter $\zeta<1$, say $\zeta=1/2$, while figure 2d is for a fixed $\zeta>1$, say $\zeta=3/2$ and these two figures are indeed topologically distinct. Now to compare with figures 2a and 2b I would look at the curves $N(\omega)$ versus $\omega$ at $\zeta=1/2$ and $\zeta=3/2$. These two curves are topologically identical, both show a cusp at $\omega=0$. So in what sense does the topological distinction of figures 2c,2d carry over to figures 2a,2b?
I ask because this topological correspondence is the central point of the paper. Basically I don't see how the 2D surfaces of figures 2c,2d correspond to 1D cuts in figures 2a,2b.
I notice that in an earlier version of this manuscript on arXiv:2105.01934v1 there was indeed a comparison of surfaces at fixed $\zeta$. In this version figures 2a,b contain $\zeta$ on one of the axes, so the comparison is to a surface cut and then I don't see a topological distinction.
Author: Yuriy Yerin on 20210923 [id 1779]
(in reply to Anonymous Comment on 20210921 [id 1771])Thank you so much for the valuable comment. You have raised a crucial question. We provided a detailed explanation in the file attached.
Attachment:
reply_comment1.pdf
Anonymous on 20211013 [id 1844]
(in reply to Yuriy Yerin on 20210923 [id 1779])Thank you for the helpful answer. A smaller question. Near figure 2 I read " the Euler characteristic changes from χ = 2 (gap state) to χ = 1 (gapless state)." At the end of the paper, instead, it says "the Euler characteristic changes from χ = 0 (gap state) to χ = 1 (gapless state)."
This is contradictory, and moreover I think both statements are incorrect.
If I look at figures 2c and 2d, and take into account that I can translate the Fermi surface by a reciprocal lattice vector, I see a surface that is topologically equivalent to a sphere in figure 2c and to a torus in figure 2d, so the Euler characteristic would change from χ = 2 (gapped state) to χ = 0 (gapless state).
Author: Yuriy Yerin on 20211013 [id 1846]
(in reply to Anonymous Comment on 20211013 [id 1844])Many thanks for the careful reading of the manuscript.
The statement at the end of the manuscript is a footprint of the first revision https://arxiv.org/pdf/2105.01934v1.pdf
We agree that is incorrect. Thank you again.
However, we partially agree that the Euler characteristic would change from χ = 2 (gapped state) to χ = 0 (gapless state). According to our numerical calculations, the Euler characteristic, which unambiguously changes during the transition from the gapped state to the gapless state, in the latter case has exactly 1 and not 0 due to the singularity in the form of the cuspidal edge in Figure 2b.