 I believe and it was all organized by you Lou so thank you you Lou and we look forward to your talk on the attraction between anti-firm magnetic quantum vortices as the origin of superconductivity and whole-doped coop rates. Okay, okay good morning everybody it's a really great pressure for me to come back and I appreciate very much that the organizers invited me to come back especially because of the celebration of 60th birthday of my long-time collaborator in organizing ICP activities and I want to say happy birthday to you peers and I think probably different people can different things so what I would do is I will spend five minutes to remind you a little bit of the high TC related activities at ICTP. I think many of you maybe not so many that may remember the Woodcock meeting, APS meeting in 1987. That was a very important event in the physics community. In fact within three months of that Woodcock meeting we had another ICTP Woodcock meeting and it was called a Geological Research Conference on High-T Temporal Superconductivity and of course there are several features of this conference. One thing is it attracted many many leading players of the field as a time including Alex Mueller, Paul Chu, Professor Tanaka, Professor Zhao Zhongxian, Bob Schrieffer, Doc Scalabino and most importantly that in the audience there were members of the Nobel physics community. In fact they were responsible for deciding to give the Nobel Prize award in 1987 just like after the publication of the paper. The other feature is that there was a big delegation from former Soviet Union. Officially there were 15 actually there were more including Vitaly Ginsberg, Alexei Yablikosov, Lev Galkov, Egor Lushinsky and many many others. In fact some of the people in the audience are the part of this delegation. That was before the fall of the Berlin War but it was already closed so it's some kind of just opening. Starting from that point the ICTP became a place of these concentrated activities on high TC related activities and the strongly correlated system. The first workshop extended workshops was in 1988. There were three annual workshops I mean in 1989, 1990 and 1991. It was organized by Baskarang Ruckenstein, Pierce Coleman's colleague and the Electoral Society and myself. And I should say that from 92 hours Pierce Coleman became the driving force of this series of the mini workshop on the strongly correlated electronic systems. I think the Phoenix community appreciates very very much your efforts. To me I think there are two features very important. One is to really attract the most active players in the game. In a sense ICTP Adriatic became a kind of focal point of high TC related activities. In fact at that time people realized that there was a urgent need to form the scientific research community on high TC and strongly correlated systems. And that community has been formed and it is really remains active for long many years. The second feature I want to mention that Pierce really emphasized a lot the pedagogical aspects of the series of activities to attract young people especially young people from developing countries. He put a lot of personal efforts to help to cultivate scientists from developing countries. And that is at the beginning the mini workshop was every year and every two years or more recently every three years. But just you close your eyes think it of it. Which kind of series of activity can last more than 30 years and always thriving. So thank you Pierce. Okay now I come back to something I want to discuss the physics. Of course you know that the high T really is part of more than 30 years efforts from theoretician the experimentalist. There are a lot of progress but there's still no consensus yet. And of course if you really ask me how many possible proposals of high TC theory or whatever I mean it's very difficult to count. So the story I'm going to tell you today is one of those and of course to get consensus is not so easy. Of course the challenges are very clear and the first is the anomalous normal state properties and the second of question is what the origin of pairing is not so clear. And then finally this is superconducting state. For a long time people thought that the superconducting state probably is more or less standard business type. But I want to tell you today that these superconducting properties are not normal and they are some specific features which are not has not been fully understood. So I don't need in this audience to really explain this phase diagram. So the story I'm going to tell you was a kind of long-term effort together with Pierre Alberto Marchetti from Padua and my long-term collaborator Su Zhaobing. And this collaboration started in 1993. When Professor Marchetti visited China first time. There are many two points. One is to really study anomalous normal state properties and the metal insulator crossover in the sugar-garde phase and the linear temperature dependence of resistivity in strange matter phase. And the second point we want to emphasize is the non-business superconductivity driven by attraction between opposite chirality spin vertices. So this is a story I want to tell you today. So this is the plan of the talk. So of course many people I wouldn't say everybody agrees that antiform fluctuation are probably the origin of superconductivity. So antiform is a really key ingredient for cooperates at least. Of course this is a natural pairing goal would be really spin fluctuating, spin waves. But if this is true, the action should be really enhanced by details of a family survey such as nesting. But as evidence it's not so clear. So what we propose as a pairing goal is another excitation also emerging from antiferromagnetism but of purely quantum origin. We called antiferromagnetic vortices. I will explain to you what we mean by antiferromagnetic vortices. So if the in antiferromagnets the spin group SU2 is broken to U1. So the quotient group SU2 over U1 is orthomorphic to S2. So it really labels direction of spins so their fluctuations are described by spin waves. So this is known to everybody. But the unbroken U1 describes an unphysical gate fluctuation. But in two dimensions one can consider vortices of a Harunov bomb type in this U1 due to antiferromagnetism. They have opposite chirality in two nail sublattices. These are really subjects we focus on. So this pairing is a due to vortex attraction. If we lower the temperature then the gas of vortices in two dimensions undergoes a causally solace like transition with the formation of a finite density of vortex pairs of opposite chirality. If vortices are centered on charges it induces a new form of charge pairing still due to antiferromagnetism. But it's different from the spin fluctuation pairing. This is different. I want to say this is not so new. Actually in the early days in the 80s similar ideas not in this explicit formulation were noted by two group of people. One is throughout the Trogman fragment you may remember that he published a very short paper where they say if two holes are sitting next to each other in the lesson then you can gain one chair. So you are winning energy. Later on in the early 90s Shryman and Shryman and the Sijia they've really studied a single hole motion of the single hole in an antiferromagnet. In the last paper of that series then explicitly mentioned the attraction due to vortex interaction due to vortex but they did not expel out its explicit. So this is a general physical idea. Now I want to just briefly outline the formula. Of course the model we are using is a single occupancy Tj model and this is the standard I don't need to explain. Of course the strong coupling approach is really to emphasize the notable occupancy constraint and we can formulate in another way using these spin charge separation slave particles. Of course different options and the option we are using is a kind of slave fermion approach and this concern is really guaranteed by poly principle. So let me just very briefly simply just what is the origin of this antiferromagnetic vortices. Physically of course this is quantum distortion due to hole in the antiferromagnetic. Mathematically in two-dimension there are many many possibilities to materialize this the charge spin charge decomposition decomposition. So what we are going to use is to use the U1 gauge to really describe the U1 charge of the degree freedom and use the SU2 to describe the spin degrees of freedom. And this is a kind of sorry a compensation of the statistics. So one is the charge flux, another is the spin flux. So one may ask a question why U1 cross L2? You guys I mean so complicated. So the reason why we came to this one is in 93 we were working on one-dimensional Tj model. It turned out that for the one-dimensional Tj model I mean using this U1 cross SU2 theory we could really recover the critical exponents either calculated by conformal theory or by the exact numerical solutions of this finite lattice. And it turned out that the hole in a one-dimensional Tj model is decomposed into hole-long and spin-long plus attached string, attached strings. Let me just explain a little bit more this one-dimensional Tj model. So if you create a hole in a one-dimensional Tj chain then if you move this hole then there's a mismatch of spin half. So this is a string of spin flips compared with the original background and up to the hollows. So if you interchange two spin-long fields due to this spin string one gets a face factor plus minus i. Hence the spin-long is a same L just in between boson and fermion. But this is easier to understand but the strange thing is that the chart of the hollow is also attached a charged string. So from the point of offer statistics it's very important if the spin-long is a semi-all and necessary that the hollow should be also a semi-all. So it has an additional effect to modify the whole then occupation statistics of the hollows. We know that usually I mean if it's a spin-less fermion we have the occupation is one. It's one. And actually because of this specific feature for which I will not elaborate it turns out that the whole statistics is one-half also in one dimension. So that is it makes this one dimension the hollows obeying the same statistics as if it were the spin-one-half for fermion. So this same I mean fermion points. So the question is how to go from 1D to 2D? This is a no trivial part, no trivial part. So naively from one dimension to 2D this doping really disturbs anti-formal background. So some kind of string confining hollow and spin-long appeal. It turns out it's not just a really confining string. And we notice that this decomposition is invariant and they get U1 gauge transformation. So this U emergent local U1 slave particle the local symmetry gives rise to this gauge field a mu and this is very important. However we cannot just just generalize simple straightforwardly the one dimensional experience to two-dimension things because in one dimension the topological defects are kings. While in two-dimension the candidate will be really the vortices. And it is natural to really look for the vortices in two dimensional TG model. So the hollow and the spin-one in two-dimension it's we really attach using the TG model theory attach a spin flux and a charge flux. A spin flux to spin-one and the charge flux to hollow to hollow. And eventually because of this statistic compensation of course the resulting statistic should be still philharmonic. And this vortices can be introduced by a coupling to the U1 cross A2 transformant theory which do not modify dynamics. And then this is a kind of exact rewriting. There should be no problem. So this is an exact but we cannot do many things exact. So we have to make a mean field approximation. So the way to make the mean field approximation really including the main idea of these things. For this U1 sector it is more straightforward and the mean field configuration is a pi flux phase which is well known. For the ACU2 sector we eventually really neglect this spin-one fluctuations. And now how can we do in the mean field approximation? We ignore this ACU2 fluctuation but we keep this U1 fluctuation in a very special way. So this U1 vortices is complementary to the S2 of the spin direction. It's like the aroma of bomb effect. And these are the chiral vortices advocated at the beginning. Of course it does not modify the anti-firm background because it costs much less energy than strings. But these gate fields do not confine it but binding at low energies. So in this gate field approach a hole is a composite particle. It's a self-generated emerging U1 gate field couples hole-ones with spin-ones. It turns out that many important physical properties are determined by this composite structure of this hole-one. Now let me describe the interaction of these vortices. The vortices is really combined this U1 gauge fluctuation with the hole-one. It's located at the hole-one position. And it gives rise to the change of anti-firm and long-range order to the short-range anti-farmatic order. I skip the mathematics details and the interaction term is really in terms of the spin-ones. And if we average over the spin vortices and it turns out that the spin-one becomes massive. It's like a localization. In a sense it's localization. Like in the line localization, in the disorder media and eventually you would just find a gap. And it's the same. The gapless anti-firmatic spin waves in the presence of this slowly moving vortices, it becomes a short range and the spin-one gains a mass. And this one actually was confirmed very early in the 80s from the ITC experiment. There's a short-range anti-farmatic order and the manganone is a bond state of the spin-one and the anti-spin-one. Now I want to go directly to the three-step scenario of the superconducting transition. What I mean by three-step, the same J term which gives rise to the finite mass of spin excitations also gives rise to attraction between vortices, between vortices. In the previous case, the average was over the vortices. And in this case, we really average over spin-one. And then we get a kind of effective interaction for hollows. And the effective hollow interaction was really induced by this hollow attraction. And it turned out that this two-dimensional coulomb attraction between vortices centered on hollows on different neosobalates use hollow pairing. I mean, if you just recall now, this is rather similar to the idea of Trugman and also Sherman and Siegel. But this is one much. So in the three-step, step one is a really causally soulless transition to form this incoherent hollow pairs due to this KT interaction of vortices. This gap equation for hollow is a business type. And it turned out to be, I mean, of D-wave symmetry because of the anti-thermal attraction is strong in horizontal and vertical interaction. So it is D-wave symmetry. But the hollow pairing alone is not enough to have real hollow pairs of hollows. So if you look at the spin-ones, I mean, for those who have been playing with the Trugman model understand in this boson, the slip-familion representation, this attraction of spin-ones in this term is repulsive. So that cannot really help to give a spin-on to play. And this is a more indirect and induced effective attraction overcomes the origin. And this is induced by the gauge field. The scenario here is very similar to the Lawton formation in superfluid helium. I will not go into the details. And this is still not enough because you can have incoherent hollow pairs and incoherent spin-on pairs. But then, of course, you need the condensation of these pairs to really give rise to superconductivity. And it turns out that the TC is really induced by the condensation of these incoherent hollow pairs. And here you go to original anti-ferromagnetic vortices, spin vortices to rear anti-ferromagnetic vortices. And it turns out that this transition isomorphic to the three-dimensional gauge XY model. And that is really give rise to the final result. Now I want to just emphasize that in this scenario, something is universal, something is universal because at the TC the gauge field is kept because of Anderson-Hicks mechanism. So this superconnection transition is of the classical XY type. The TC scale is set by the spin-on pair condensation with gain in kinetic energy. So in a sense, TC is insensitive to the Fermi surface details. So you can have a kind of universal phase diagram. So let me say this very quickly. So there are several features which really distinguish this transition from the standard BCS. In standard BCS, this exponent for superfluid density is alpha, so one-half. But if you look at this experimental data more carefully, the exponent is two-thirds, as predicted by the three-dimensional gauge XY model. And also the Wimura relation can be derived explicitly. The second thing is we can just derive from a given specified model the main features of the phase diagram. So there are four different temperatures. One is the so-called high field gap. It's really the hollow pairing temperature. Another one is the so-called lower field gap, which actually is the boundary between the field gap and the strange metal phase. Then there's a length phase. The length phase is really the pairing temperature for spin-ons. Eventually, there's a superconnection transition. So the pairing temperature for spin-ons and the lower field gap temperature T star, they are really more universal. But the other two depend a little bit on thermosurface details, but not precisely. And using this model, we can really calculate the spectral weight in the strange metal phase. And it is agreed with the ARPIS data quite well. And we can also calculate the parallel implant resistivity and find this metal insulator crossover and find this inflection point, which is nothing but the T star. And this agrees very well with the recent experiment. And this is the pairing temperature for spin-ons. It really agrees quite well with the own experiment on this lens. So just to summarize, if we combine the composite nature of John Ryan singlet and induce anti-pharmatic spin vortices centered on them, we get a three-step scenario for superconductivity, capturing many features of the cupra's phase diagram. The attraction of vortices on opposite near-sublatives give rise to a finite density of the hormone pairs. And the hormone spin-on gauge interaction give rise to finite density of RVB spin-on pairs. And eventually, condensation of the whole pairs give rise to superconductivity. Thank you very much.