 The next speaker is Andrei Varlamov, and he will present work about topological phase transition with Ingepp and Gappels supercontact in state. He is Andrei. Yeah, yeah, I understand. Now it's okay. Okay. Okay, so I would like to thank the organizers for this very nice traditional and a little bit nostalgic conference. For me, I mean that after all this pandemic and other events, it is so nice to see the people whom I know, 40 years, 30 years. And today I would like to speak about some I told about nostalgia. So I will speak about the synthesis of two fundamental seminal papers published in 1960 in Soviet jet. So this paper was accomplished together with Yuri Yeryan who is present here and with Katerina Petrilova who presents in three years in the quality of I don't know president of technological part. So the outline of my talk will be first reminder will be devoted to Gappels superconductivity. The second reminder will be devoted to the transition of the second and half order. Then I will ask myself what is the order of the gap Gappels transition in superconductor in superconductor. Then I will propose you the topological interpretation of gap Gappels transition. And then I will try to put in doubts the mean field treatment of this transition by the father's foundation foundation so for Bricose and Garkov. And naturally I will find that they were right. The account for fluctuations actually shows that this approach is very stable with respect to the non-homogeneity of distribution of paramagnetic impurities. And finally, I want to speak about the possible experimental detection of what consequences this treatment not just can have to experimental study of this transition and I will speak about the giant thermoelectric effect. So let's start from year 1960 when in 1959, Bricose and Garkov developed the well known for all of us cross diagram technique for account for impurities. And in 19 so they did it for normal metal and then they immediately applied it for superconducting alloys. Next year they considered the spin flip risk at rings and found very non-trivial thing. After 46 years of the question what is superconductivity Bardin Cooper and Schreiber proposed their outstanding theory where the superconductivity the phenomenon of supercurrent was strongly directly related with the gap in the spectrum. So what was the consequence of the discovery of Bricose and Garkov that it is possible to have gapless superconductor and still as a gapless regime and still to have supercurrent. So this was fundamental because we understood that gap and supercurrent are not necessarily the same. So still they left in the long range order in 1970 from the papers of Brazilians, we understood that even this is not necessary for two-dimensional system because you can have supercurrent even without long range order. So it was the first strong extension of BCS theory I would say. So what they did actually they wrote the standard BCS theory in Garkov formulation and took into account the spin flip scatterings. And they found that critical temperature is first of all that exists some critical concentration of paramagnetic impurities which kills superconductivity and C. But in the range between 0 between 91 percent of this critical to this critical there is an interval of concentrations where the superconductor is gapless and still you have supercurrent. Okay then they found how paramagnetic impurities shift critical temperature etc etc. Very good now I will go so now what does this mean for density of states? So in gap regime naturally you have this weird BCS picture then in the critical in the point 0 9 1 2 of critical concentration you have the cusp and then you have the some non-singular behavior of density of states. Now let's go to Haikov the city which is now every day in the newspapers. Ilja Lipschitz yes. The question appears in the chat which is irrelevant what happens to the superfluid stiffness in this regime? Sorry? What happens to the superfluid stiffness? Superfluid remains. So it's fine okay. Yes yes it's fine it's fine yes. So Ilja Lipschitz so every day during this conference we listen about Van Hauff's singularity. Van Hauff's singularity what do we intend? We intend the singularity in the density of states of what? I want to recall you that the paper of Van Hauff in 1953 was written for phonons but of course then we can apply this also for electrons. So in 1960 Ilja Lipschitz studied the free energy of the metal when the thermosurface changed the number of its components of topological connectivity. I mean the formation of new void or the break of the hyperboloid okay the break of the neck in let's say let's suppose that I have the brilliant zone and I have the thermosphere. I add electrons. At some moment thermosurface grows to touch the border of the brilliant zone and from the closed thermosurface you pass to open one due to periodicity. This is if you do vice versa it will be the break of the neck. So Ilja Lipschitz studied thermodynamic potential close to it and found that such changes results in appearance in thermodynamic potential of the power z in power five over two where z is a parameter which governs transition and this is epsilon or mu minus mu critical where mu critical is a value of chemical potential this critical point very good. I want to recall you that in 1933 and in first introduced classification of the phase transition and he identified the order of phase transition with the order of derivative which passes the break singularity. So first order phase transition the entropy breaks second order heat capacity breaks stir order transitions force order so he told that all this is possible so in terms of error and first classification the Lipschitz called this transition and the two and a half order phase transition why because appearance of z in power five over two means that after three derivations you will get one over square root of z and he called this two and a half phase transition the density of states exactly like in the case of Honol's in one-horse scenario under goes the casp square in a three-dimensional case. Okay then consequences. Indeed the paper was published in 1960 and then he spoke only about thermodynamics and actually nothing happened so people tried to apply to superconductivity those times and they did not find something dramatic. Dramatic event happened in 1981 where Valery Yegorov with Fyodorov they studied the thermoelectric effect in the alloy lithium X magnesium 1 minus 6 and they found that close to the point X equals 0.19 they found you see this is the behavior of thermoelectric effect as of under this curve is for room temperature something happens at nitrogen and you see evident peak at the helium temperatures so it was necessary to explain and I want to recall you the what is the difference between the zebra coefficient and conductivity so in conductivity just from Boltzmann equation you immediately have to do the formula with the derivative of fermi function integrating but if you will do the same with for thermoelectric coefficient you will have a beautiful zero why because of the oddness and evenness of the expression. So to get non-zero answer you need to take into account electron hole or something else the so next terms in v, nu, v, nu and tau so the fact that we observe the giant thermoelectric effect should be attributed to some changes in this the most simple what was done by Vox to say look guys nu has a cusp what formula says me that beta thermoelectric coefficient is a d log sigma over d mu so I will have derivative of nu and in denominator so this is the origin of giant effect this is wrong this is wrong because where from you take this square root from small void what is the fermi velocity there zero so v square kills so this explanation does not work and to get true explanation it is necessary so we did it many times years ago we just considered the microscopic theory of transport for topological transition so the model was taken as the hyperboloid of rotation and we wrote with Andrei Pansulya we wrote the green function at such spectrum with one negative mass and solve the problem for solve the problem for self-energy the self-consistent equation and what is important you see here that there are different scattering processes which is possible to consider electron scattering from the periphery of thermosurface to the periphery but there are processes from periphery to the small to the neck in this case and namely these processes where the neck appears as a trap for electrons gives you the main contribution in scattering time and the yes okay it is possible to do it is interesting that you can consider another model not one bent model with hyperboloid of rotation but you can consider the formation of new so I started oh my god okay so I will finish with this okay in any case it is possible to explain so conclusion electronic topological transition is strongly related to so it can be determined by the giant thermal electric effect okay now let's go I will ask my question what is the order of the gap gapless transition in superconductor you can open the parts book or the ambiguous car and grief in paper of 65 and find the expression for free energy at zero temperature quantum phase transition please derive it three times and you will see that in third derivative you have the exactly like in leafshitz transition square root so then you can get this curse which I already tell you but the question is that superconductor is not normal metal where in leafshitz approach all was clear in the interpretation of the appearance of new words the name change of the number of the components of connectivity here it was necessary to propose some topological treatment and uh yeah so we propose to find what happens from the point of your topology in this quantum phase transition we propose to study density of states not in two-dimensional space by the three-dimensional space we study the corresponding surfaces as a n density of states is a function of energy naturally and there's a function of the independent parameter so delta in the state without impurities and you see the in or the difference between these curves is a because of gecko parameter one over tau s delta and you see how changes the shape of this curse you see that something happens in the point that i equal one to understand this we should return to platonic solids and uh to recall you that exists like a layer one characteristic of it which is the uh sum and difference of number of vertices ages and phases for each platonic solid of which kepler wanted to construct all the universe uh the he is equal to but it for more complicated uh polygons it can be not two so we propose to characterize sorry so uh if there are two possibilities one possibility is formal we have density of states we can kill it the curvature curvature of the corresponding and calculate uh in the uh according to ailer bonnet theorem which relates the uh ailer uh number with integral of uh curvature we can calculate directly almost simple we can uh construct the polygons on these curves and just calculate uh in ailer bonk characteristics the answer will be the same what is important that for gap state is zero in the point of transition it changed to one now i want to do in a last minute i want to do the reverse consideration so i started from leaf sheets and then i applied it to uh superconductor i found new treatment because fermion surface did not mean too much and now i want to say but if leaf sheets would speak not about the change of number of components of connectivity but would speak about the ailer bonk characteristics what he wouldn't get you see now i make the parallel between superconductor and between the two types of leaf sheet transition and i see that in figure d the two sheet hyperboloid where each sheet is topologically equivalent to disk with he equal one so the ailer characteristic disjoint union of two disks will be two in figure e it will be one in figure f it will be zero and i can do the same for the void formation and i see again that the ailer characteristics change from four to two so now in one slide i'm finishing in one slide i can demonstrate you that why what is happening i don't understand yes in one slide i can demonstrate you that a bricoce of gerco mean field theory is very stable so i take the free energy from homogeneous superconductor and add kinetic term i rewrite in ginsburg-landau i rewrite this kinetic term as a derivative of the parameter over density of the concentration of the impurities and i see that actually for validity of mean field it is enough that logarithm on concentration slowly changed on the xi and final slide that if i will repeat the calculus of thermoelectric effect in superconductor or so for the quantization of flux so i will see that the fact is giant thank you very much i finished thank you very much and right so there are questions or comments here from the audience yeah just clarify so you say that there's global curvature or the characteristics whatever you call it enters into some physical effects so what is the physical observable this determined to express in terms of this uh we found only that uh actually it was already known that so what you have the uh so what is thermoelectricity in superconductor we know that there is no effect in bulk superconductor but you can construct the ring of two superconductors of different superconductors at different temperatures so instead of the quantization of flux integer quantization of flux you will have some non-integer contribution which is a demonstration of the thermoelectric effect which depends on thermoelectric coefficients of one and other superconductors in normal state here the effect will be giant and we can predict how it will depend on temperature and impunity consumption yeah yes so andrei um you emphasize the narrow region in concentration when there's gapless state but in this thermoelectric coefficients the width of your parameter that is much larger so it does seem that yes the anomaly starts way yes before so what really controls uh the width in the observable versus the width of the very good question yes so i want to gain two so this calculus was done absolutely following the ambiguous car griefing theory or the giant effect so i want to say first of all you're right that it is extended up to half of the the second that maximum as usually photopoly as in the case of leaf sheets is asymmetric and is belongs to the range of gap superconductor so its effect is epsilon fermi divided temperature so the the parameter of the in comparison with the background so it must be uh it should not be uh uh small i mean its range is large of course the same in the normal metal so maybe the confusion is that the range in density where you see the gapless superconductor the concentration where you see the gaps in the superconductor is very small yes is what you show at the beginning but excite here has nothing to do with that no no this is zeta uh it changes from one to zero but zeta is one over so transition happens at the point of one but this is not concentration zeta is one over tau s delta so all range from zero to one it corresponds to uh it corresponds in your in ns all above above zero nine one so this is just the range between zero and one between zero nine two one and one yeah yeah so this is gap stage uh state and this is gapless so it is large from one to infinity okay so i don't see other questions so we thank you again andrey