 I would like to thank the organizers for the invitation to come here, for the great weather. Do I need a microphone? Okay. So, again, thank you very much. So, again, thank the organizers for the invitation. And today, it is kind of continuation of a few things we heard last week on Friday about superconductor to insulator transition, except that I will try in this talk to go beyond what we have known. And I actually added this sentence of unveiling the nature of the magnetic field tuned superconductor to insulator transition. And I think that you will see what I mean as this talk goes on. This work started with the first student in my group to start it was Ali Asdami, went through Nadia Mason, Miles Steiner, and the last one, Nick Bresne. Most of this work that I will show today is due to him and, of course, lots of other characters helped. So, I'm going to start. The starting point is that at strictly zero magnetic field, increasing disorder reduces the costal established transition to zero. This was first introduced by Bisli Mui in Orlando. But then in the presence of magnetic field with this order, superconductivity exists only at t equal to zero. Now, I said this is a starting point. Let me pause for a minute and tell you a bit. I will show you my starting point of acquaintance with Boris. I met Boris first in 89. This is a picture of the two of us in Leningrad. The hermitage is behind us. So, we go. Sorry? It's the same color. The building has the same. Okay. We have several pictures. I'm not going to ask who is who. I think that at that time, at that time, at that time, at least some people said that we look alike. So, I'm going to continue beyond this other starting point on the talk. I may come back to it later. So, we are talking about the superconductor to insulator transition. And since we are talking about the fact and especially I'm going to concentrate on a finite magnetic field that the only possible phase exists at zero temperature, then we are talking about a quantum phase transition. Now, obviously, I should not forget that the first introduction of this order tuned quantum phase transition came with the work of Alan Goldman. But as I said, I find it more especially as an experimentalist in the lab. You make one film. You have one realization of this order. And therefore, the magnetic field tuned superconductor to insulator transition is I find easier to understand. So, I'm going to concentrate on that, which means that every film that I measure in the lab or my students measure means a different realization of this order and it's going to cut this zero temperature superconductor to whatever insulator transition at some point. That's the quantum phase transition. The first discussion of this problem was by Hebert and Palanen in 1990 with their fissure of letter and I'll come back to it in a minute. So, we are talking about the magnetic field tuned superconductor to insulator transition and of course, I can destroy superconductivity, global superconductivity in two ways either by amplitude dominated scenario in which simply I break cooper pairs all together as I cross the line or I can think of maintaining the character of the cooper pairs and only break the phase. This is very similar in fact to what happens near the Koster-Listau-Listau-List transition. Of course, there is just one transition that's the Koster-Listau-Listau-Listau-List transition but you can still talk about vortex, anti-vortex pairs that bind at the KT transition if you are close enough to the transition. So, in that same spirit, we can talk about just breaking the phase and maintaining cooper pairs if you want going through this transition and in fact, if you do a very simple calculation of the energy scale for phase fluctuations in two dimensions, especially films of the type that I'm going to measure you find that the scale for phase fluctuations is so low that probably this is the dominating scale for this transition. So, we are talking about a phase dominated superconductor to insulator transition in which the cast of characters are the cooper pairs associated with the cooper pairs are the vortices and since we are talking about going from a superconductor in which cooper pairs are delocalized and being a superconductor the vortices need to be localized then the insulator is where the cooper pairs are localized and the bosons will be delocalized eventually maybe bos condens. The point separating the two phases will be therefore a metal and if it's characterized by bosons it's going to be a boson metal. Talking about the quantum phase transition, I'm approaching it from a finite temperature all our measurements are done at a finite temperature looking at the quantum phase transition and therefore, unlike a classical phase transition we're just governed by a diverging length due to thermal fluctuations here I need to supplement it with the dynamics I will have some vanishing frequency that is related to the diverging correlation length via this so-called dynamical exponent but then since I am measuring at a finite temperature this vanishing frequency is going to be cut off by KT and therefore it defines a range of attraction of the quantum critical point and I will come back to that because this is a central part of this talk today. So scaling analysis for this transition was first written down by Matthew Fisher you have a diverging length I'm going to tune the superconductor to insulator transition using a magnetic field so the correlation length will diverge as I approach the critical field the vanishing frequency is going to be given with the dynamical exponent Z however, being cut off by KT it now have your competition between these two lengths which then a scaling function emerges in which there is an amplitude and then a scaling function of this reduced parameter it's basically the ratio of these lengths you can rewrite it as H minus Hc divided by this temperature to 1 over Z nu similarly if you are not too low temperatures you can also use an electric field to tune the transition and in that case the energy, the vanishing frequency is going to be cut off by the electric field and you'll get another scaling function now with respect to the electric field across the sample in fact if you measure these two then you can get Z and nu separately we did that a long time ago and then it was repeated in the quantum hole as well now what are the hallmarks therefore of this superconductor to insulator transition well if I am at a temperature which is much below the pairing temperature I am at low enough temperatures then the only thing that matters is how am I with respect to the quantum critical point call it the superconductor to insulator transition and if we look at the original data of Hebert and Pallanin then we can separate an insulating behavior from a superconducting behavior and a separatrix at which the resistance does not change as you lower the temperature now if I use that scaling function it means that if I am at that critical field looking at isotherms now then there is going to be a crossing point because at HC the argument here is zero independent of temperature well that's the original data of Hebert and Pallanin it's not a very good crossing point the recent data of many other groups have much much better data with much better crossing point and the critical resistance should be of order of the quantum of resistance for Cooper pairs which is H over 4E squared now the critical exponents Z and nu well we'll talk about it later but for the universality class that was discussed in this quantum superconductor to insulator transition we expect Z equal 1 and nu equal 2 about 2.3 but again I'll come to it later so the original results of Hebert and Pallanin were looking at some average point as the crossing point then scaling the data according to this scaling function finding that there are two branches above and below the critical field constructing this Z times nu of 1.3 okay now further work done in my group was trying to distinguish weak and strong disorder and the idea is that if I look at this phase diagram indeed there is weak disorder which is closer to the Abricosa flatis at strictly zero disorder versus strong disorder which is closer to the disappearance of superconductivity as you increase disorder so the question is whether it's the same universality class or the same behavior in all these range of magnetic field and it turns that it's not you can distinguish weak disorder in which there is an apparent critical point but the magneto resistance magneto resistance of course will tell you how insulating you are because you are increasing the magnetic field so you are going through the superconductor to insulator transition you expect the resistance to diverge if it is indeed an insulator and for weak disorder we find that this is not a very strong insulator on the other hand similar to in fact identical to the work shown by Dani Schachar on Friday strong disorder give you gigantic magneto resistance here I stopped at 200 millikelvin you'll see why soon but you go to lower temperatures and you find those mega ohms I mean those giga ohms here we are already at one and a half mega ohms at 200 millikelvin and it gets larger and it gets to some other effects that we heard before now the interesting thing is that when you get to the strong disorder then the critical resistance is very very close to H over 4E squared as an experimentalist it's very difficult to determine it exactly because although the resistivity in two dimensions is dimensionless when you go to the lab you still need to divide by the number of squares so you need to know where the current flows so there are always about 10% variations of that so the weak disorder what we found is in fact intervened by a sense of metallic phases and by now we looked at many many samples in our lab and others in other materials I will not discuss these metallic phases today this was the original data on Molygeum Menium you see that the insulating is a very weak insulator nevertheless you can do scaling in some range of temperatures and fields again I told you similar data on tantalum nitride indium oxide all these have very weak magneto resistance and then there is this appearance of metallic phase that then gives way to a true superconducting phase at lower magnetic field I'm not going to talk about that today I don't have time I will talk only about the strong disorder regime and only on experiments done on indium oxide and I'm talking about this which was the title of this possible whole insulator phase in this regime in all these cases that I'm going to talk about the critical resistance is very very close to H over 4 E square just like this example I showed you before and if you do scaling near this critical point that you cannot even see it because it's at about 6.5 kilo ohm versus we are here already at 1.5 mega ohm then you get a beautiful scaling with very small reduced parameter with Z nu of order of 2.5 okay I'll come back to this number again similar data was shown by Danny Schacher from his group Tatiana Baturina obtained on chlorinated titanium nitride so this has been seen in fact by other groups but in very small number of materials only indium oxide and this chlorinated titanium nitride okay so in fact you can collapse all the data we found in these if you simply look at the critical resistance or conductance divided by 4 E square over H and you look at very low fields extract the slope of resistance versus magnetic field extract Hc2 from that and divide your critical field by that Hc2 then all the data this is now on tantalum nitride molygium manium indium oxide and tantalum that's those that did all these measurements of scaling etc then I find that there is a regime of very weak disorder and in that regime we find these metallic phases however in the strong disorder regime we always find that the critical resistance is very close to H over 4 E square and that if we do scaling you get this critical exponent very close to about 2.5 so I'm now going to concentrate on the magnetic field-tune transition and supplement all these longitudinal resistance measurements that have been done since 1990 with whole-effect measurements now whole-effect measurements were done first by again Hebron and Paladin in 1992 and from 1992 until the preprint we put on the archive a few months ago there are no measurements of the whole effect near the superconductor to insulator transition and the reason actually is very very simple well look in Hebron and Paladin this is intermediate disorder I would say I mean you can see that already at about 50 millikelvin the resistance increased only by a little what they find is that still the whole effect starts to be finite above the superconductor to insulator transition and above a higher field the whole effect starts to splay in fact when they do high resolution whole effect they find kind of a crossing point similar to what they find here so this is peculiar and for some reason nobody picked up on that but we know why and there are lots of people still that are reluctant to measure this whole effect and the reason is that we already saw that in those very strong disorder films then the resistance is enormous now you know how you measure whole effect in the lab filled up, filled down from a sample that has the whole bar on opposite side you subtract, divide by two, you have the whole angle you have the whole effect but you can do that only if the longitudinal resistance is small because of contamination you can never really align these especially if there are also inherent inhomogeneities in the material either because you made it or because they appear there so there is a problem well we solved it a long time ago with Ali Palevsky in the lab in Tel Aviv when was it, 1876, something like that but only partially in order to extract whole effect from large resistance samples but this is still not enough but then maybe this is not important and I think this is probably one of the next two most important slides of this talk if we believe that there is a quantum critical point at zero temperature then and we are at temperatures much smaller than the pairing temperature namely all the particles are already well defined the vortices and the cooper pairs then it doesn't matter at what temperature I am in fact zero temperature from the point of view of zero temperature 5 millikelvin, 20 millikelvin, 100 millikelvin or in some cases that were discussed in high TC, 100 kelvin it doesn't matter zero temperature from the point of view of zero temperature these are the same as long as I am within the range of attraction of the quantum critical point so the whole point is why won't I stay at high temperatures try to be within the quantum critical point and therefore avoid this inability to subtract the longitudinal effects as well as as we saw on Friday there are non-equilibrium effects that appear at lower temperatures but again these non-equilibrium effects appear at some temperature I am going to stay as you remember from the IDVs I am going to stay at higher temperatures where the non-equilibrium effects do not appear yet if I am within the range of attraction of the quantum critical point I will not know about them and this is the philosophy of this talk and I think that what basically why people were reluctant to measure it is because they said oh this is a zero temperature quantum critical point I need to go to as low temperatures as possible to measure the whole effect otherwise I cannot say anything about it but you cannot do it you cannot measure at low temperatures the whole effect I can guarantee you the longitudinal resistance is too high so I am going to stay at higher temperatures and show you data and show you what one can extract from this kind of point of view so here is some data from one sample there is the critical resistance of the superconductor to insulator transition and this is the whole effect and just like in Hebron and Parliament you see now I am staying at higher temperatures and you see that there is a separation there is a crossing point maybe I chose the wrong colors kind of a crossing point because it is a crossing point of displaying namely that the divergence above the whole crossing point is much stronger than below and then that is the original result of Hebron and Parliament this is a much stronger insulator they had that at 50 mK with the same TC we have it at 200 mK as I said here is another sample this one we did we have actually several such samples we went to the magnet lab to measure at very high temperatures we wanted to make sure that all vestiges of superconductivity and pairing are gone so I can measure the actual carrier density by the low temperatures by simply looking at the slope of the whole effect so first of all here is you take one of the samples you do scaling this is the scaling with the temperature this is scaling with the electric field providing again there is another reason for the high temperature I can avoid this heating effect that may appear if I am at too low temperatures so which is by the way something I discussed with Boris long time ago the first thing he said hey did you make sure that you are not heating well we are not and you can see that you can do the scaling these are the two exponents that you have you solve this you get that nu is about 2.3 and z is 1 and every sample in which the critical resistances of order of 6.5 kOhm shows that and that's it now I am going to show you the whole effect and I am going to show you the limit of T equal to 0 so I am taking now a cut goes all the way to T equal to 0 this is decreasing temperature this is decreasing temperature on the insulating side decreasing temperature on the superconducting side I am going to show you the limit of T goes to 0 of the whole effect this is it now there is another very important thing that we could do once we have the whole effect you can invert the resistivity matrix and obtain sigma xy as well as rho xy so I am going to have rho xx, rho xy and sigma xy and look what you see here sigma xy, I mean start, rho xy seem to be 0 just like Haban and Palanen all the way to the SAT then it starts to increase this is the limit of T goes to 0 then there is a point which is the point of the crossing point of the whole effect in which the whole resistance has a break okay this appears roughly at the peak of the magneto resistance I will interpret that as going to the fermion regime after all we measure in other samples all the way to very high fields the interesting thing is that at that point exactly the rho xy is exactly h over NEC which is I find quite interesting so you see the whole resistance on the insulating side goes from 0 all the way to h over NEC which is the classical limit and then it breaks and goes up beyond that which I will interpret as wants to be an insulating adjust that in an insulating system both rho xx and rho xy go to infinity rho xy will go slower now let me zoom near the superconductor to insulator transition now you see it even better rho xy is 0 below the transition and starts to be finite above sigma xy which we inverted the matrix is 0 above the transition and then now both were by the way scaled with rho with h over 4e squared and sigma with 4e squared over h so they are both dimensionless in the same way and what you see is that you can easily have the same straight line going through both of them through the transition you go from rho to sigma of course there is more scatter than sigma because we are inverting a matrix but otherwise I think it's quite evident and 200 millikelvin so actually I think that there is some of the data goes to 100 millikelvin and most of the data that I'm showing is down to 200 millikelvin certainly if you go below 100 millikelvin you start to see non-equilibrium effects in this talk and I gave the rationale I'm avoiding this absolutely so that's basically what I meant by taking the same straight line and going through this is a manifestation of self-duality so there is no other way to explain that so let's take a field in between between these two crossing points rho xx and rho xy say 6 Tesla for example in this case look, rho xx diverges rho xy is finite well there is a name for such a phase in which rho xx diverges and rho xy is finite it's called Hall insulator this name was first invented by Kievelson, Li and Zhang in the global phase diagram of the quantum Hall effect paper of 92 the interesting thing is that at the edge of this range the rho xy is h over NEC which is the quote-unquote canonical value for a Hall insulator and all this I mean this is not there's no any massaging of the data I'm showing you rho data except that sigma xy was inverted from the rho data without smoothing without anything this is what we have so you have an insulating behavior of rho xx finite rho xy self-duality as I showed you from here well if you have self-duality and this is a Hall insulator well we have to call this side the superconducting side we have to call vortex insulator it has the exact same behavior so if here you are dominated by rho xy of the cooper pairs then here you have to be dominated by sigma xy of the vortices well it's quite interesting that if you are imposing now self-duality so you are in duality between vortex and cooper pairs then the relation between the conductivity tensor is like that with the resistivity tensor of vortices and similarly for resistivity versus conductivity of vortices self-duality at the transition implies that sigma xy at hc0 rho xy at hc0 as I showed you we find experimentally well I already discussed the fact that sigma xy well the fact that rho xy is finite that's a Hall insulator that's just like in the quantum Hall transition but now this idea that sigma xy on the superconducting side I just showed you has to be dominated well originated from vortices you invert the matrix you get that it is rho xy vortex over rho xx square plus rho xy normal inversion but on the superconducting side of course the vortex resistivity goes to infinity roughly that which means that rho xy vortex over rho xx vortex square is finite because of sigma xy vortex being finite but this relation was obtained previously in the general case by Vino Curitale which was quite interesting but I believe that this is maybe the most starting manifestation of this relation so previously observed Hall insulator phase this was observed in the quantum Hall liquid to insulator transition it was predicted in the global phase diagram by Kivilsson et al it was discovered in quantum Hall these are just examples I mean here are some examples of quantum Hall liquid to insulator transition near nu equal a third near nu equal two in both cases you do scaling and you get 2 equal to well they measured 1 over nu fine but you invert 0.43 for 1 over nu you get 2.3 very similar to what we obtain in this storm disorder regime and indeed what they find for the Hall insulator is that the rho xy is b over nec which is this limiting case before we transition into what I will call a fermi metallic phase duality was also observed before and I'll give you one example here of the group of Dan Tsu and Emerson Sheaegan Dan Shachar was the student at that time they looked at the symmetry of IV characteristics and indeed they said that these results can be interpreted as evidence for the existence of charge flux duality symmetry in the system and we showed it through the Hall effect which I believe is the most straightforward way to show it but we do find self duality as well so we have Hall insulator and we have self duality and it looks very much like the quantum Hall liquid to insulator transition so one can then get insight into the quantum Hall liquid into the superconductor to insulator transition from what we know about to insulator transition I showed you that all the characteristics including scaling are very much the same well I don't have time to discuss that because I was told that but you can cook out models that in fact there are inherent inhomogeneities just like in the quantum Hall liquid I believe first was introduced the quantum percolation model was first introduced by Choker and Codington and you can basically understand the superconductor to insulator transition in the exact same way there is always inherent inhomogeneities whether you started with a granular system or you started with a homogeneous system disordered of course we are tuning disorder and then there is always amplification of the disorder when it comes to superconductivity and you have a granular system you have something that the percolation should be a very good description for and then it could very well be that your self-duality comes from these typical junctions and self-duality in a Josephson junction is something trivial if for every so you look at the current and then you look at the voltage associated with the voltage is traversing the junction and then if for every Cooper pair goes through this going across then you can calculate the resistance you find that it is h over 4e squared times this ratio of rates but if these rates are the same as I just demonstrated you get h over 4e squared so I believe that these inhomogeneities which are then means you are dominated by this kind of junctions are behind that story now to conclude this is the phase diagram that we see if we stay above the non-equilibrium line we see a quantum critical point at zero temperature which we will call a transition from a superconductor we call it now a whole insulator because we identified the phase the insulating phase and well let's just look at that without the intervening equilibrium effects and then there are two possibilities we said that in fact our whole effect goes from zero above the superconductor to insulator all the way to h over NEC above which if I'm in a thermodynamic limit and I'm at low enough temperature we expect both rho xx and rho xy to diverge like a normal insulator but in this range in between we expect the rho xy to be finite and we're still not ruling out the possibility that in fact rho xy that we see going all the way up well maybe it's going to get as I lower the temperature and again if I could switch off non-equilibrium effect it will then go down again but in that case it's going to be a quantized whole insulator with rho xy equal to zero either one it is still a whole insulator phase but our data definitely show that the insulating phase above the superconductor to insulator transition is a whole insulator phase and at the transition there is self duality both were never in fact seen before and demonstrated before for this system so these are the conclusions I just repeated myself so you can read them and let me go back to the starting point which is 89 in Leningrad this is now just a few days ago here in Trieste so I think that in what 26 years I'm going to show only this picture without this picture this is going to be our what birthday 86 so happy birthday Boris and thank you all