 It's a under somehow. Yeah. I mean, it's. Yeah. Okay. Should I wear. Yes. Microphone. Okay. Okay. So, so the second talk of the session is given by Professor Lenny Blasman from Yale. Good morning. It's great to be here in person. We had another time. So, I'll talk about. Andre reflection. Okay. In scanning. So it's again, Andrea. And it's somewhat ready to the previous talk at least in terminology. I'll try to try to. Yeah. Come to it. So, here's the. The auto and I'll try to follow. So I'll first I'll motivate. And then I'll do the long introduction actually. So I'll discuss only conductance and how the. The density of states is recovered from experiments. In my serious understanding. Oh, I see. Okay, just a second. Let's try to. Must be pulling there. Yeah. Okay. Okay, so, and then I'll. Discuss how to go beyond a totally approximation. And to do the non perturbative relation of conductance. And I mentioned three known pass. You'll see which pass. I mean, which are well developed first wave superconductivity. And we'll see how to generalize one of this ways. To, to look at beyond the S wave. Now, then I'll just present the final analytical result. And we'll discuss what can be learned out of it in terms of what are the characteristics of the order parameter, how it affects the spectrum from Victor strong tunneling. I'll talk a little bit about experimental applications, not too much. And conclude. So, as the motivation. The motivation is hunting for unusual. So, then a counter to the S wave superconductivity and goes back on memory memory memory, at least to high TC. Here are two figures of the face diagram. From two different reviews. One kind of artists like another one is more steam like. The diagrams are similar and we see that there is competition between phases. Well, everything is controlled by competition between various kind of quasi-insulating phases, quasi or magnetic phases and superconductivity as interplay actually leads to unusual ways of pairs of pairing electrons in the superconducting state, which some people will know. And there are manifestations of non-space superconductivity in de-wave materials that go back to wonderful curly experiments with seeing a trapped flux or just the ground state flux in a specially prepared ring of ITC and to ARPIS. So here is ARPIS examples. And basically in ARPIS one can see that the gap in some direction turns zero. So basically by now it's well established that superconductivity which basically lives in AV planes, there are kind of four loops of de-wave structure with the perimeter, it's too positive to negative and there are some directions where the gap is vanishing and that's what actually this graph illustrates. It's back in mid-nitis. Now another way to see the gap structure is in STM, actually in STM there were two different directions of studies, so-called quantum quasi-particle interference effects, which I will not discuss at all and just more direct or straightforward, I would say, way to look at it at the properties of the spectroscopy. So the principle of STM is shown on this figure. So basically there is a needly tip that goes close to the surface of the material and actually by the exponential virtue of the decay of the wave functions, the sensitivity of this instrument is much higher than the geometry may tell you because basically the trajectories, the imaginary times, the trajectories of electrons, the tunnel from a tip into the surface basically form a very narrow cone and exponentially decay if you try to make this cone wider. So that actually leads to remarkable sensitivity space-wise but also one may just park this thing at one point and measure every characteristics and then DIDV actually tells you about the nature of the density of states as we will see at a given point. And here's an example. So actually that's from Oisin Fischer et al. Ravey article and this kind of things is that what Oisin Fischer was doing for years. If one can look, for example, at this conventional spec inductor in magnetic field and if one parks tip at the part of the vortex, then one sees more or less normal density of, normal IV characteristic and concludes the density of states is constant. And if you go away, then it's BCS and there is a gap. Now, so here is kind of a various cases of superconductivity. Starting actually, I found the graph in Giver's Nobel lecture and it is a nice graph in the sense that actually it shows DIDV normalize to the normal DIDV and there are numbers, one, two, three and so on. Quite often people use something like arbitrary units which actually may be somewhat deceptive. But what's important is that this is a conventional superconductor, it's lead in contact with manganese through a insulating barrier and there is a well-defined gap. And this is an obium, also a very well-defined gap. Now, even in more exotic materials like boron carbide, which is actually anti-framagnetic, still as wave superconductivity is alive and well and there's actually angles telling yesterday, anti-framagnetism is not so harmful because there are some still symmetry you can flip things and shift. So it's not very surprising. Now, in high-TC, citation is different and there is what is called pronounced this shape of the DIDV. Now, in some experiments, actually this shape goes way down and almost touches zero. So this color graph is from Jenny Hoffman's lab if I remember right. And basically what she measured DIDV at different points of a sample and sample is inhomogeneous. So the curves are different or somewhat different, but the commonality is that it's V-shape everywhere. Now, actually this is BISCA 2201. I think it's, it's a Fisher-Magioprili experiment and it's again V-shape, but it doesn't go to zero. And I'm not sure if it's resolved to why that or not, still. Now, again, samples may be homogeneous. So inhomogeneous, this is an example actually, this is Magioprili of a very uniform ethium Ipka, one, two, three. And it's remarkably uniform and this V-shape is more or less unchanged from point to point. So that was in the times in high days of high-TC. Now, what we've also tried at that time was looking at what I would call strong tunneling. Into cuprace. So one may make a junction at that time, it was mostly not STM junctions, but just spattered or whatever junctions which have a large conductance and the area may be small or large. It's actually important because what may matter is not just the conductance, but the transmission coefficient of a barrier. And these are two different things because conductance just scales with the area of the junction whereas transmission coefficient is independent of the area, right? So you can get the same value of conductance by making area smaller, but transmission larger. And some quantity scale not linearly with transmission but it's some higher powers and then it matters. So strong tunneling, I would mean to know it's large transmission coefficients rather than just large conductance of the junction. So if one tries to make a stronger contact into AB planes, so just kind of tunneling from the top, basically there is no clear effect happening if you increase the transmission coefficient. But if one tunnels into another phase, especially into AB plane, 110, sorry, 110, then actually there is an effect. And that's from the review of the torture. Also experiments were done in mid 90s. So as you see for tunneling into the side, so basically if you have AB plane, it's tunneling in plane tunneling across a line which has direction 110. And that corresponds to tunneling as shown on this sketch in the direction that corresponds to a nodal point. Then there is a clear zero bias peak in DIDV, again, a storm less conductance, I apologize, but that's what experimentalists did. Once again, so here's a geometry and as you see, this contact is in a very uncomfortable direction. The layers go vertically and one tunnels, one has a contact on the kind of on the side. So that was this high TC. Now there is one more thing which kind of goes together with the previous slide. And actually I didn't know about the experiments before yesterday, so one may try to go with a tip on a grain of high DC, in this case it was YBCO. And so the grain is in natural direction, so the AB planes are horizontal, but one may tunnel into the middle of the grain or at the edges. And skipping details, there are two very characteristic points somewhere far away from the edges, of the grain and then it's this proverbial shape. And, but once you tune to a point that corresponds to the 110 phase, which is just a point here because it's cutting or it's in different, it's along 100 and 010 directions, the real ages. So if you go to the corner, then, voila, there is a peak. So that again goes together with the previous slide where the tunneling was done on purpose into the side phase of the crystal. So now new stuff. So things kind of, I would say there was a flashback and this field exploded again with TBG twisted by layer graphene discovery. And I think this diagram is in memory of every person now. So basically, again, it looks like ADC and there is superconductivity and correlate insulator, all kinds of phases. Again, it's because of the narrow bends and strong interactions. And there are competing phases, conjecture of pneumatic superconductivity. There is experiment and some series. So again, it's a transport experiment. So no tunneling. And again, the conclusions were kind of cautious. There was no, definitely statement about, nodal superconductivity, D wave, but the possibility of having something like D or P wave superconductivity. And conveniently there is a theory, which actually, it's one of the understandable papers, where just having pairings that couple various one-half singularity, one-half saddle points, one can develop various scenarios for getting to spaghetti state. And there are possibilities of having the states which break spatial symmetry, but keep time reversal intact or break both spatial and time reversal symmetries. So this A2, E1 and E2 are representations, just the one-dimensional, two-dimensional representation and two components of two-dimensional representation of C3 group that characterizes the actually D C3 group that characterizes the material. So now what about tunneling? Once again, of course, people tried it. So now there are no layers, it's just single layer, right? So no tunneling into the plane, at least on purpose. So just from the top. And on TBG, there is a recent work, which actually prompted us to do this study theoretically that shows V shape in the conventional regime of weak tunneling. And actually this is a good example. It does tell the numbers. So 30 nanosemons, and I'll tell what nanosemons is in a second. So this is, and trust me, it's a weak junction. It's a very large resistance. Or if you remember what nanosemons is, now in trial layer graphene, there is the work from Caltech and they think of the possibility of having a transition of some kind between the nodulus and nodal gaps at differing fielding factors. It's all in whole doping between minus two, minus three, if I remember right, as well as the twisted by layer. Now, I'm not sure I understood the theory arguments in the paper about the transition, but meanwhile I learned about an older work of Goray Redzhovsky and Andreyov, which do have a discussion, actually that's about the possible BCSBC transition as per resonance. So basically, my understanding is very simple, that you may, okay, so if you have almost three particles, you can construct by some interaction a P wave or a USB connector, which will be gap full and will not break the reversal. But then if you increase the interaction, eventually the two telephones must form some kind of a molecule and there's some kind of energy to pull it apart. So basically, there should be a gap in a single particle spectrum, although at the macroscopic level, it's still this perglon dow and you may have d wave order parameter. And that would correspond actually to d wave, but it must be gap full just because of this simple energy considerations. And that's, again, a way to understand in my view what these people see and tell. Now, yep, yep, right, yeah, right, yeah. That's excellent question. It's a perfect segue to what I'll do later. So jumping ahead, it's because of Andrea reflection. So we're seeing you that Andrea reflection, as I understand it, it's a two electron tunneling. There's nothing is reflected, it's just two electrons a tunnel in a single quantum act, okay? So you take one electron tunnel through the barrier, it sits in a virtual state, wait for a counter part. Another one comes and they form a copper pair, right? So it's like one coherent act in which you tunnel two particles. So you have a product of tunneling amplitudes and this is still an amplitude of a process, okay? Now you have to calculate conductors, you have to square the amplitude and give the amplitude to the force power, right? So that's the difference. And that's why actually you don't, the Andrea conductors doesn't scale the same way as single particle conductors, right? So that's why this normalization to just normal conductance may be deceptive because it can make a transmission coefficient very small and increase the size of the barrier and have the same conductance, but you'll completely kill Andrea by making transmission small, does it answer? Okay, thanks. Okay, so now going further about numbers, so thanks for this question again. So the unit transmission for kind of a point contact where there is just one channel, one dimensional channel if you wish for electrons, your response to conductance, which is around 80 times 10 to minus four microseconds. So in tunneling, I assume is a point contact, especially on the scale of Murray pattern. So that's why I would call a certain nanosemons a big junction and say 150 microseconds, a strong junction where the conductance is close to the quantum unit, so meaning the transmission is order of one, okay? So this is the difference between weak and strong, all right? And again, in this respect, STM is kind of simple because it really is a point contact. So basically the kind of the smallest of the Fermi wavelengths electrons kind of seeps through some very small, yeah? Right, yeah, that's exactly what I'm telling now that I assume that, oh yeah, sure, yeah. So the question was, do I assume that there is a single channel in the tip? Now again, what is channel? You may imagine for a second a contact of some area and measure area units of the Fermi wavelengths, okay? So that would be the number of channels. And yes, I assume that the STM tip is a single channel contact with transmission between zero and one, okay? How good is this assumption? It's a bit of a question. I personally think that in the case of graphene of TBG especially, of TBG, where the Fermi wavelength is tiny in the material, I think it's a reasonable assumption. As long as the contact is less than the size of the moire unit cell, right? Right, absolutely, this is true. Nevertheless, the point of the tip, there may be in some special cases, maybe not a single channel because of the symmetry of the atom that sticks out. Right, yeah, but tip is very sharp. And actually, again, look, I mean, one should talk to experimenters, but they also kind of dip the tip into something and then take it out to make it sharper. So as far as I understand, usually it's a reasonable assumption, but it's a good question and you'll see that actually the assumption leads us into a ditch in this particular case. So yeah, so this is the strong tunneling data. And again, what I guess we thought incorrectly, but not probably, but what we took at face value, you know that, okay, it's interesting that basically in some experiment we go from V shape to a maximum in the conductance at stronger tunneling. And how does it happen kind of defied our imagination or at least our understanding. So now just a few words about the standard things. Now, the conventional scanning tunnel experiment measurements of in the BCS, conventional BCS regime, the standard way to do it is just to write tunnel Hamiltonian. So this is the tip three particles. This is to do material that we explore, but suppose it's just BCS superconductor. And then tunnel Hamiltonian that is in the case of point like tunneling has matrix elements that the products of the wave functions in the tip and in the sample at the same point, at the same point in plane, right? So R zero. So there is one parameter in this Hamiltonian and it's a model Hamiltonian. So this parameter is in the model, but one should get rid of it at the end if you want to express observable in terms of observable. So then the standard things, it's BCS or bugaloo representation of Fermi, electron Fermi operators, the conventional expression for the excitations in the superconductor with the spectrum excitations, which is conventional square root. And if it's not S waves, then delta depends on the direction on K vector on the Fermi surface. And one can just write the Fermi's golden rule. Now you sub K of R zero here is the wave function of electron in the 2D. So in principle, it's some block function at a given point R zero. And then it's standard manipulation. Basically one can divide out the normal state densities of states. And I'm sure that I screwed up the coefficient, but basically the normal conductance is proportional to this model parameter T mod squared times the density of states, products of density of states in the tip and in the 2D in the normal state, right? And as a result, one has this wonderful formula that DIDV in superconducting state equals, not proportional, but equals to the normal conductance divided by the normal density of states in the solid material times the BCS density of states at energy EV, actually at energy mode EV, things are particle symmetric. And maybe for some more conventional formula is basically the density that DIDV is proportional to the imaginary part of retired green function at a given point and given energy. So the important thing that in the two last expressions, the model parameter is chased out and DIDV is expressed in terms of the measurable quantity which is normal conductance, okay? So that's the essence of phenomenology that you may have a model, but better express everything in the model in terms of things that can be accessed. So what do we have in case of this lowest order tunneling? Okay, DIDV is proportional to the density of states in BCS and if it says wave, then there is a hard gap. So zero DIDV up to the gap value. And then there is one whole singularity which is one over square root textbook thing. Another thing that is perhaps almost as textbook that in case of nodal superconductivity, very simple analysis of the same integral over momenta gives you a mode EV behavior as small biases. This is the proverbial shape and a bit less known. So there is still one whole singularities but they are weaker, it's a logarithmic order peaks, okay? And again, it's very simple. I mean, it's similar to some formulas that Harris was showing yesterday in a more sophisticated setting. Pardon? Yeah, thanks, yeah. So thank you very much. 2D, I mean, in 3D there are some, actually even with STM, there are some more difficult, not difficult, but additional things because then there is some angular dependence. So let's discuss 2D, it's, yep. Okay, so now, wait, so now, but how to deal with arbitrary transmission, right? So I'll expose several ways to go. They're all kind of bitten paths. So it's, there is no any, anything unusual here. So the first path is what was done by Blunter-Chickelman-Clapevac, it's a famous BTK paper in 1982. And basically, okay, what can we do? We can just do one dimensional problem, one side normal method, another side, superconductor, BDG, Poglio-Digene equations, a barrier, and just, just barrel through just matching the wave functions, solve transmission problem, and write current, okay? And this was done. Now, the barrier that was chosen in this work was a delta function barrier with the stress characterized by parameter Z. And that's in experimental papers, they love it, just everything in terms of Z. Now, one can find the idea, for example, analytically, it's not very difficult. For example, at EV less than delta, one gets this kind of expression that depends of course on EV and on Z. Now, they did everything, they just looked at various values of Z, but what is important, they experimentally, so what is important that still this Z is directly related to the normal conductance, but in a very simple way. And it's very straightforward to re-express things so that Z vanishes and everything is pressed in terms of the observable conductance, which actually enters in various places, including the denominator of this expression, for example, okay, so that was BTK 1D. Now, if it uses what is called Andrea approximation and things work out only when delta over EF is small, now, if it's not small, then you can get anything and it will be extremely model dependent because in this case, basically even without a barrier, you have two media with different refraction coefficients, so to speak, so you'll best get anything. And if you don't believe me, you can look at this quoted paper and see what one may get if you abandon the Andrea approximation. So the peak actually, so this peak is the one whole singularity and it fix a delta, you cannot shift it. The thing that happens when you change Z, which is transmission coefficient, is that what you get, what is called sub-gap conductance. So here there is no tunneling, very big tunneling, you make something stronger, you start getting conductance below the gap and then it gets larger and at Z equals zero, which actually you get to full transmission at the flat top. Okay, so that was 1D. Now, one can generalize 1D to a flat boundary, of course, because you can separate variables. And most interesting thing was done by Tanaka in this respect, and this peak for 110 phase was actually obtained directly by him. He capitalized on the earlier papers that discovered the mid-gap surface state in D-wave TRS preserving superconductor by chair and who. And one thing that is not parallel conservation is crucial for this bus and there is no way to generalize it on a point like tunneling except if it's a wave. Now I have to confess. So this building is worn and that's because it's here for a long time. So there was a conference held here in 1997, 25 years ago, actually. And it turns out that I was one of his organizers. Now, and actually Tanaka was speaking in this very room. And I still remember this talk, not that I understood much of what he was telling, but this is written by Den Ralph and Vinay Ambigayakar and they master language much better than me. So there was a wonderful personal, as I discovered yesterday, personal reflection. I think it's by Vinay about the way to summarize the presenters of theory. And then here you give a theory to students, they solve some differential equations and before long they come back with the results. So, which is perfect segue for me to discuss pass two. So one starts with Tanaka-Miltonian and then just both the user green functions try to evaluate conductance to all orders. Well, you will try to do it. And that was actually attempted by this paper and it's non-equilibrium, it's a waste of scattering, but non-equilibrium, so it's Keldish all the stuff with green functions. So in principle, you can express current in terms of the green functions and then write Dyson equation with fairly simple self-energy, but still this equation is not easy to solve. And then the result basically, if you call it the result, here it is, it's a one, a two, a three, a eight. Still we use somewhere this condition delta over EF, but not to the full glory of it and then what? So you still have a parameter T in front and it enters in all these green functions. Okay, so in a way, you can put it on computer and to give it to another student or you try to express things so that they are cast back in terms of the conductance. Okay, so you do the thing and then you pray. And it actually turns out that the prayer was answered before the prayer itself. So here's the second part of this paragraph. And it's been popularized as not profits. So actually, I don't think that this is an insult because if you don't understand the formula, doesn't help much having it. So the way this answer to the prayers basically was done in round 92. And that was utilization to a better way, a better utilization of this Andrea condition delta over EF much less than one. So basically you can imagine that in terms of green functions basically you have to do partial summations that would cross over from the potential of the barrier to the transmission amplitudes, to scattering matrix. And that basically what this scattering formalism does in abbreviated form. And what you do is just you tunnel the particle, it scatters off a superconductor acquiring some face and changing to a whole. And it can tunnel out or scatter again off the barrier. And then there is another process and so on. And then you basically sum the subsequent Andrea reflections. And the summation is if you did it done it's just a critical series. And here is the result. And it's already a phenomenological form because everything here is transmission coefficient. So T squared and R squared is one minus T squared. That's it, right? So, and then basically using that you plug it into the DIDV which can be written in this form. It's actually a BTK form that accounts for the particle reflection and whole reflection for the coming electron. So there are two channels. You can refer to the particles if all that's it. And basically the final result does have this form and it helps to navigate along the past two to the very same result. So in terms of S waves, I would say that's in my view ideal as long as we keep Andrea condition intact. So we tried to implement past two actually. So we did this thing. We tried to implement past two and not only giving it to postdoc but also trying ourselves and it was hopeless. So not like for the lack of trying, you know. So eventually, it was a space to think and basically one can adapt with binocular thing for the STM configuration. So the difference is that here is that basically it's a funny single mode problem because this scattering matrix element S simultaneously describes scattering of 2D electrons on a tip without going into a tip, but just back scattering. Okay, so it's like, if you look at on the elliptical, there is a section on inelastic scattering which people skip, but actually that's about, it's exactly this section. So the electron that comes onto the tip, it may go into the tip and this kind of inelastic process for from the point of view of 2D or it may be just scattering. And if you have any transmission, that means also they have scattering. It's just unitarity. So this is the thing that basically there is kind of a combination of two-dimensional scattering problem and tunneling. And one can also account for the block functions by generalizing S wave to a projection on the proper block function taken at point R zero. And then basically you do the same thing. You account for the conversion of particle into whole, but now it's directional, okay? Because delta is a function of K, the scattering phase, back-scattering phase or under-scattering phase depends on wave vector. And then basically one does the same thing with summation. It is more mathematically involved, but still as long as it's a single channel or small number of channels, it is a solution of an integral equation with a separable kernel, which we know how to do. And skipping details, basically one can generalize this expression that was in the S wave to what is shown here, where S node is scattering a matrix. So it's a complex number, mod less than one. And this A are expressible in terms of the of the under-scattering amplitudes. So, and one can do also for the, not only for the particle cold channel, but also for the particle reflection and get a scheme. It maybe it looks horrible, but you should look at it as a leg. You can explain it to the kid. So basically there are three ingredients, scattering matrix, block function, and gap, right? And out of the three blocks of Lego, you can construct an answer. So, and the answer basically leads you all the way to deriving the DIDV. So, once again, the ingredients here is a block function and gap that enters into the Andrea amplitude. Now, in the last kind of two, three minutes, I guess, right? 10 seconds, okay, plus two minutes. So, now if one looks at conductance, you can analyze quite a bit just looking at various limits. So at zero bias in arbitrary transmission, one has to analyze this expression. So it's average over the Fermi surface of mod squared of block function times delta over mod delta. And assuming that there's only crummers at given K, UK is one dimensional rep. So mod UK squared is a constant other point group symmetry. Now, it's very easy to show that actually delta of K, as long as it preserves latest point group symmetry, then delta over mod delta belongs to air irreducible representation and some statistic generalization of it is in that paper. And then the result is very simple. So if it's a S wave superconductivity trivial representation, then you have conventional Andrea reflection with twice unit conductance unit at zero bias. If it's non-trivial, then you get zero in the reflection at any reflection amplitude, any S naught. So basically here is a graph. This is the case of trivial representation in those show, but if it's not trivial representation, like D wave, then what one has is zero V shape at zero bias, no matter what at any transmission coefficient. One funny thing that actually is a log divergence becomes a log suppression at a stronger tundling and it's basically funnel resonance. Now, if you have not seen the presentation, but you break TRS, then again, it's zero at zero bias, but it's zero all the way to the mod delta, to the mod delta. Now, if the point symmetry, if delta breaks the point symmetry, then things depend on whether TRS is preserved or not. So left is an example where TRS is preserved and the gap is real and in this example, not less. Now, this is an example of point group symmetry breaking and TRS broken. And again, I'm talking about tunneling into a symmetric point. So there is no problem with the block function symmetry. And that's actually these two examples is something related to this diagram, to different phases. Now back to experiment and they're all done. So it is not clear to us how to explain. We first thought that you can fine tune the parameters to mimic the data at large conductance, but what Ali mentioned to us is that the more plausible cause actually is that the point contact regime stops somewhere along this line because actually the unit conductance, twice unit conductance somewhere here. And if you drive tip closer to the sample, basically most probably you just, you make a hole in the sample and the junction becomes large. It's not a single channel, it's just macroscopic. Again, that's the current state of affairs. I don't know whether it's correct or not. That's what we are discussing. So lastly, conclusions. So I like the fact that we developed a flexible scheme that is free of model parameters. Basically you can describe the DIDV inspecting state through a minimal knowledge of the normal state properties and we see that actually a reflection provides plenty of additional information about the parameter symmetry in addition to the density of states which we know and love and ectonally. And there are extensions that we would like to pursue to account for spin orbit coupling, additional structure or what the parameter which is actually important and the possibility of magnetic tips in the effect of potential scattering, magnetic scattering. I think I'll stop here, thanks. Yeah, just a minute. Early on. So for this final anti-resonance, it goes to zero, of course, when it's a delta function. But if there is a relaxation there would be that energy relaxation or phase relaxation there, then that adds, and in practice, particularly for the tip, it probably is there. So that's why it's happening. You don't see a real going to zero anti-resonance. Absolutely. Same happens in optics. I mean, that's exactly it. I totally agree with you. And after all, I mean, when even numerically we put it, like if the singularity is very weak, so you really have to know to go very close to see it. So anything will be important if you're right. Do you assume in the calculation that the tunneling points is a high-sinture point? Yeah, okay, good. So thank you. So I'm advertising one more that we don't assume. So for example, this expression, there is a block function at a tunneling point, okay? And it may be any point. It doesn't have to be high symmetry point. So this expression actually allows one to see the effects of symmetry of the block functions and delta separately. And actually, if you move away from a high symmetry point in the latest and this guy doesn't have a symmetry respect to K anymore, then the v-shape, for example, may start not from zero, but from finite value. That's actually one of the possibilities why people see v-shape not going all the way to zero. You're saying the Andrea reflection is still absent in when the tunneling point is- No, no, no. Sorry for talking. No, for Andrea reflection is absent. If you're at high symmetry point, I didn't try it, okay, so I'm sorry. So if you're at high symmetry point and delta respects the symmetry of latest but belongs to non-trivial representation, then it's zero. Otherwise it's not and you go away from zero. So even for high Tc, if your tunnel is not exactly into copper atom, but away, there is no guarantee that you have v-shape going all the way to zero. It's still v-shape, but it starts from some pedestal. Thank you. Okay. I have a very naive question. In the even last data that you showed, there is a symmetry between positive and negative even. Yes. The peak seen in one side and essentially no peak in seen in the other side. Right. So your theory completely symmetric with respect to sign change of v. In other words, can it be that the resonance is killed on one side and not killed on the other side? Yes, thanks. Again, it's naive question. That's again, very close to my heart question. So again, I'm not telling that the theory can describe the data, but it's not symmetric in the sign of v under the following condition. You may have potential scattering. There is no need in any kind of magnetic scattering, potential scattering, but for the range of biases where you are between the minimal and maximum value of the gap, the symmetry is present and it's a weird effect of interference between two different ways to go from a tip into a quasi-particle state in the superconductor. You can do it directly in some direction where you are above the gap or you can do it while under reflection as intermediate point into some other direction where you are below the gap. Okay. You can just cook up as a model and see how it works. And this interference actually gives you this asymmetry. The amazing thing you know that the density of states per se doesn't change. If you look at these expressions that I have shown in terms of denominators, they're all perfectly particle-symmetric and there is nothing, there is no harbinger of what will happen. So denominator here is particle-symmetric, but the numerators with interference effects it's in the numerators. Let's thank, let me check. Any question online? So let's thank Lenny. So the next, so we have a coffee break. What time should we eat? So then third talk begins, starts at 11.15. Thank you. I think people are already, I think start to sign but at the end of the day.