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Topological phase transition between the gap and the gapless superconductors
by Yuriy Yerin, Caterina Petrillo, A.A.Varlamov
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Submission summary
Authors (as registered SciPost users):  Yuriy Yerin 
Submission information  

Preprint Link:  scipost_202109_00018v2 (pdf) 
Date submitted:  20211107 08:43 
Submitted by:  Yerin, Yuriy 
Submitted to:  SciPost Physics 
Ontological classification  

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 of the disorder induced transition $s_{\pm}$$s_{++}$ states via gapless phase in multiband superconductors, and for better understanding of the gapless color superconductivity in quantum chromodynamics and the string theory.
Author comments upon resubmission
Thank you for your email message of October 14, 2021 and sending us the referee reports regarding the manuscript “Topological phase transition between the gap and gapless superconductors” by Yuriy Yerin, Caterina Petrillo, and A. A. Varlamov.
Please find enclosed the revised version of our manuscript, the answers to the Referee’s questions and comments, and the list of changes performed. For the sake of convenience to follow all changes have been made we also attached the pdf file where all improvements corresponding to the Referee questions and comments are highlighted by blue colour.
In view of recognition by the Referee of the hotness of the topic and originality of our findings we believe that the revised version will be approved by him and will be suitable for publication in SciPost.
List of changes
1. Additional arguments for the existence of the topological transition have been provided.
2. The supplemental material with details of the Euler characteristic calculation has been extended.
3. New parts of the manuscript are highlighted in blue.
Current status:
Reports on this Submission
Anonymous Report 1 on 2021126 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202109_00018v2, delivered 20211206, doi: 10.21468/SciPost.Report.4001
Report
In the new version of the manuscript, the Authors have added a number of clarifications regarding the scope of their calculations as well as the broader context of the study. However, I believe that the most important comment from my previous report ("Are any physical quantities of the system uniquely determined by this invariant?") has not been addressed properly. Let me elaborate on it to avoid misunderstanding.
While the Chern number of quantum Hall states or topological superconductors can be related to the Euler characteristic, the crucial property of these states is the existence of a physical observable (Hall or thermal Hall conductivity) that is determined by the topology only (and the universal constants), i.e. "quantized". A recent example, where Euler characteristic has been related to physical observables can be found in arXiv:2108.05870. The authors suggest ``the number of gaps in a system under consideration" as such a quantity  however, no way of actually measuring it is suggested. One can think of measuring DOS at $\omega=0$ via STM; however, it is known that in actual experiments (see e.g., Phys. Rev. 137, A557 (1965); Phys. Rev. Lett. 53, 2437 (1984)) a nonzero density of states at $\omega=0$ appears routinely even where it is not expected from an AbrikosovGor'kov theory standpoint. While the origin of this behavior is not fully clear (see e.g., Phys. Rev. Lett. 105, 026803 (2010), Phys. Rev. B 94, 144508 (2016)), it effectively rules out STM. Spectroscopic measurements, such as optical conductivity, would be contaminated at finite temperatures by quasiparticles. Taking into account in addition that the gapless superconductivity is expected to occur in an extremely narrow parameter regime, it does not appear that the topological characteristic can manifest itself robustly in experiments, severely limiting the impact of the present work.
The calculation of the thermoelectric coefficient, the second major result of the work, as the Authors acknowledge in their Reply, has been considered previously for a more general case: "the more general consideration in Ref. 34". Taking the points above into account, I can not recommend the current manuscript for publication in SciPost Physics, given that the criterion for this journal is to ``provide details on groundbreaking results". However, as the paper does explore interesting theoretical connections and provide guidelines for developments in thermoelectric effect, I believe that this paper is appropriate for publication in SciPost Core.
Additionally, there is a technical comment I would like to make. I believe that the statement that "at $\omega=0$, $\zeta=1$ the Gaussian curvature is not diverged and equal to zero" is not, strictly speaking, correct, when Eq. (17) for DOS is used. While $K=0$ when the limit $\zeta\to0$ is taken before $\omega \to0$, this is not so if the order of limits is reversed. Indeed for $\zeta\to 1,\omega=0$ one gets $\frac{\partial N}{\partial \omega},\frac{\partial^2 N}{\partial \omega \partial\zeta} = 0$, $\frac{\partial N}{\partial \zeta} \propto (\zeta1)^{1/2}$, $\frac{\partial^2 N}{\partial \zeta^2} \propto (\zeta1)^{3/2}$, $\frac{\partial^2 N}{\partial \omega^2} \propto (\zeta1)^{5/2}$. Using these expressions in (18) one gets $K\propto (\zeta1)^{2}$ that does diverge as $\zeta\to1$. While it is likely that the integral over both $\zeta$ and $\omega$ still converges near $\omega=0$, $\zeta=1$, I believe that the details of the analysis leading to this conclusions should be presented (and possible issues with the order of limits also considered for the $\omega=\Delta_g$, $\zeta\to 0$ case).
Author: Yuriy Yerin on 20211210 [id 2022]
(in reply to Report 1 on 20211206)Please find our response in the file attached
Anonymous on 20211211 [id 2025]
The authors present a perspective on the disorderinduced transition between gapped and gapless superconducting states as a Lifshitz transition. Lifshitz transition is wellknown to occur in metals when the Fermi surface changes its topology. The authors try to argue that the gappedgapless transition in superconductors also "has a topological nature".
While the similarity between this transition and Fermi surface Lifshitz transition is seen in the nonanalyticity in free energy (2 and 1/2 type), I do not see a solid basis for the claim that the former has a topological nature. Nowhere in the manuscript is a relevant topological invariant defined.
Apart from the claimed connection with topology, the work itself seems solid and deepens the understanding about disorder induced transition in superconductors. Therefore I believe it deserves publication in Scipost after the claim about topology in this context is softened.
Author: Yuriy Yerin on 20211228 [id 2057]
(in reply to Anonymous Comment on 20211211 [id 2025])We are grateful to the Referee for his/her overall positive evaluation of our work. Following the recommendations , we eliminated from the paper the mentioning of the possibility of characterization of the gapgapless transition by means of of the Euler characteristic. We removed the word “topological” also from the title of the paper and correspondingly we changed the title of the paper.
We stressed out on the emergence of the cuspidal edge at the density of states surface $N(\omega,\Delta_0)$ ($\Delta_0$ is the value of the superconducting order parameter in the absence of magnetic impurities) and the occurrence of the catastrophe phenomenon at the transition point.
We believe that our paper deserves publication in the present form.