 So hello, everybody. I'm Alexander Chodnowsky. I want to thank the organizers for the possibility to give this talk here. And just a minute. So the title of the talk, as you see, is Superconducting Insulator Transition. And I'm going to present a model, which is a relatively simple microscopic Hamiltonian that basically reproduces important phases, which are shown here, which are typical, proper to the typical phase diagram of a high Tc superconductor. The model is relatively simple. And you can see this phase diagram, which I'm going to explain in the rest. Strange metal phase, superconducting phase, quantum critical insulating phase, Fermi-liquid boson metal. The work was done in close collaboration with Alex Kamenev. And you can find it on the content here. So the subtitle of this work is a phase diagram of the SYK plus URA. So this SYK plus URA, I will introduce in the rest. This is the model I will be considering and will be presenting, which leads to such a phase diagram. Well, I will be able. So first of all, the Sartiv Yakitayev model attracted an interest from the solid-state community because out of many properties, but one of them is that it can reproduce the strange metal behavior of the resistivity if you make an array of SYK, so Sartiv Yakitayev grains. The Hamiltonian of these grains was introduced in the work of balance and co-authors. You can see this. Basically, there are two pieces. This is the number of isolated grains. This part is the SYK random interaction term. R denotes the position of a grain in the array. And C denotes the electron's creation or annihilation of errators at each side of a grain. So each side of a grain can host a spin-full fermion. It's important. I'm dealing here with the fermionic version, not myarana fermions, but real charged fermions. This is what I'm considering. The second part is the random tunneling between the grains from each side of one grain to any other side of the other grain. There are two energy scales characterizing this model, which is the variance of the random interaction in the grain, G squared, or G. And the randomness of the variance of the random tunneling, T0, or T0 squared. And OK. So important properties of this model was, as I told already, it exhibits a linear temperature and the dependence of resistance at temperatures higher than the typical temperature scale, T squared over J. And it exhibits also the fermi-liquid T squared dependence of the resistivity below this scale. All of this was obtained in the results of work of balance, which is cited here. And further, this work, this idea was extended in a number of works. I'm apologize if I could not cite all of them. But to the theory of a Planckian method. Furthermore, because of the strange metal phases, a typical part of a phase diagram of high Tc superconductor, which is seen there, the natural desire appears to modify or extend this model in a possible minimal way in such a way to obtain other parts of the phase diagram. First of all, superconducting phase and possibly pseudo-gap phase or other phases. Such an extension were proposed in several works. And more or less all of those works in one or another way introduce effective attraction between electrons. And the work I'm considering here is not an exception. We also start with the effective attraction in the electrons. And we introduce it in the most simple form, probably most simple possible form. This is just the local Hubbard-U term, attractive Hubbard-U term, which acts at each side of a given grain. As you can see, this is minus U. And all those C operators reside at a single side of a given grain R, with corresponding spins corresponding to the Cooper-Huper pair. And so we arrive at the model at the Hamiltonian, which I will be considering in the rest of the talk. And if you can see, this is again an array where R denotes the position of a grain in the array. The only modification is that in each grain, there isn't local attractive Hubbard interactions. First of all, what are the consequences? The simplest way to deal with this model is to do the mean field. The first attempt to do the mean field leads to the following phase diagram. So here is for the comparison, the phase diagram of the SYK array without Hubbard's term, that has only two phases, basically the Fermi-Liquid phase here and the strange metal phase, that basically differ in the temperature dependence of the resistivity. As soon as you do mean field, in such a way that you introduce the superconducting water parameter, effective superconductive water parameter as anomalous averages of the on-site electrons, forming a Cooper on-site local Cooper pair, you do some mathematics and you obtain the following the solutions of the mean field equations leads to this phase diagram. There's a schematically shown here. So you can send that immediately as soon as, first of all, this is a t equals zero axis. And you can see that as soon as you turn on even infinity is a really small u, you immediately obtain a finite amplitude of the superconductor order parameter. Yeah? This is not geometry. No, there is no geometry. I can assume, I don't go into the, it is nearest neighboring or some geometry which I will not enter into details because I assume the tunneling to be anyway small. And basically the whole, this whole results I will be, I will be reporting can be obtained from the considering the tonally tunneling between the nearest neighbor grain in the incoherent array where the whole resistivity is determined by the tunneling between the two nearest neighbors grains. That's why Tij is random. It's random variable with zero mean and the variance of these Tij squared is characterized by T zero squared. Yes, u is the same in every grain at each side. U is absolutely non-random. Yeah, for simplicity. Well, so what you get is the superconducting phase which is basically as you could expect it. An interesting point is that the, of course there is a gap in the single particle spectrum of single particle fermionic excitations opening and this gap is this time not delta. This is due to, this is the difference between the usual a conventional superconductivity and this superconductivity in the SYK model with random interactions. This is delta squared divided by J, this delta one here, which means that at small u and of course delta depends on u here is delta very small. This gap is much smaller than the usual expected superconducting gap. But even this phase diagram would be the end of the story. It would be not very interesting, maybe even boring. The interesting point comes now and this is the most important point of the whole. This is the crucial physical point of the whole talk and the whole work is namely that those phase diagram before is not quite exact. Why? Because the idea, the phase fluctuations of the superconducting order parameter in this model playing a very important role and they were not treated properly in the simple mean field solution, where if I turn back here, this delta was assumed to be a constant parameter without let's say, let's put it real number equal everywhere. So which means automatically by the mean field assumption you assume that phases are frozen as soon as the superconducting amplitude appears which is basically conventional assumption for the superconductivity. But this time let me now consider the phase fluctuations in more details. Basically, I starting business. So and first of all, to understand the most important result let me first consider a single grain. Here I disregard tunneling. I'm focusing on a single grain. What happens at the single grain if I introduce this Hubbard U. So basically technically one deals with using the field theory and makes Hubbard-Stratonovich decoupling. Importantly is that the Hubbard-Stratonovich decoupling of the U-term is naturally site local because U-term is site local so that you basically introduce initially a local decoupling field for each side. And this time I will make the following assumption that this local field has a constant amplitude delta which now I can put real. However, it has also this phase factor i e to the power i phi i for buy. So at each site, there is some superconducting phase which can be can fluctuate can differ from one side to the other. Yes. Yeah, okay. I understand. I can only, the thing is that if you, you can do it for, yes, the same goes to the J terms but basically the result is without details. If you orbit you and without you, you do not get superconductivity mathematically. This is also J terms. Yeah, you do the same. Yeah, you do the same for the J terms also. Yes, that's true. Yeah. Okay. Absolutely. Exactly, yeah. Now, could you comment on the gauge invariance of the theory and why these phases are not swallowed due to the gauge invariance if there is one? Well, you can try, but this will not without going into details. You can not swallow all those phases with the gauge filled basically. I can, we can go into mathematical details and if further, but probably let just the answer is no, they are not, but the, which I will try to give in this short talk. So now, if I can proceed, so well, so now what you are doing technically. So formally, you again start with some mean field theory. Well, you have mean field delta. This delta is the result of the solutions of the mean field equations that I showed before. But now let us consider fluctuations around this mean field. So the phase fluctuations, what do they do? There are two effects coming immediately into the play. One was known and one is present. One is that the phase fluctuations even in the non-superconducting model result in the so-called one can tell on-site charging energy which is proper to the SYK model and independent on the Hubbard coupling. It is just the property of the SYK model itself and for fermions that we have this single part which is site charging energy. This is the reaction of the fermionic model to the infinitesimal change of the chemical potential. And while it was calculated before in this work and it is approximately half of the coupling of the SYK coupling strength J. If you, and immediately you understand that if you have only this coupling presence then it tries to destroy the phase coherence because this is basically the charging and it tries to destroy coherence between the superconducting phases and eliminate superconductivity. However, there is a second term which is an effective interaction between the phases, superconducting phases at different grains. Without going into details I will explain this interaction can be calculated out of this basic diagram and this diagram which is so tow imaginary times that it's imaginary time formalism. Important is following those points are the delt superconducting, local superconducting order parameters at site I and here at the another and some another site J. And now the lines, solid lines with single arrow are normal components of the grains functions. This bubble consists of anomalous grains functions with so double arrows. And those vertices are the scattering vertices due to the SYK interaction to this SYK J. So basically this diagram describes the assisted scattering of a cube per pair from a site I to a site J assisted by these SYK interactions. And one can calculate this diagram and the most important and obtain this result. For a single diagram. Importantly is this is delta squared divided by J and divided by N, which results immediately to the interaction energy. If you now sum up all those diagrams sum up the complex conjugated diagrams all together then you obtain immediately the interaction within the sites which is minus J divided by N and here some over all sites from one to N and cosine phi I minus phi J. Obviously this term tries to synchronize the phi so freeze the phases out, yeah? Yeah, but tunneling is not considered here. I, this is tunneling can see here. Yeah, you may get further corrections. You may get, this is the basic. Yeah, the they're parametrically smaller. You can get, you can get other terms of course but this is the basic, this is the basic one. Yeah, the similar question I mean without J this is just the stiffness like efficient of this term and the stiffness controlled by the density of electrons super fluid electrons. So when J goes to zero, this coupling has to go to another term which is controlled by T. No, no. So you can, you can have this term even in the system without the SYK interaction. What, what remains, sorry, sorry. Sorry, the stiffness. In some other systems possibly but if I stick by this Hamiltonian and remove the SYK interaction then my Hamiltonian is just zero. Every single, every single. We have the time, we have the hopping time. This is probably also the question that was there. Wait, wait again. I didn't say grain. Again, as I told from the very beginning let me focus here. I'm focusing on a single grain. I disregard possibility of tunneling to the other grain. I want to know what happens in a single isolated grain with you. What is this phase diagram? Without now, disregarding tunneling to other grains. And so an important point in this term is that despite what I was doing is the fluctuation expansion of large N, of a large N theory over the mean field. The result is not the one over N correction. This is important because this result, if you, this I, this is, I intentionally put here the powers of N so that you can follow the effect that the resulting proportionality is one over N, which means after multiplying N summation goes over N squared sides. So the overall total energy scales as N as it should be for the initial model, which means the fluctuations of phases is the effect of the equal strengths than as the mean field effects. The energies are proportional to N. This cannot be disregarded. This really changes the physics that you observe. And let me now. So, and the model that you arrive at, if I put those terms together, is called the quantum version of so-called Pura-Moto model. What is the Pura-Moto model? It is known as the classical model of phase synchronization. Basically, you can also consider this as a kind of a Josephson junction arrays with infinite range Josephson couplings, if you wish, because you see, so this is each I is the island, Phi is the phase of the on the island, but Josephson coupling is everywhere, everywhere with everybody. And let us now, and this is the model that describes, basically should describe the phase diagram below this line. So as soon as a finite superconducting amplitude appears, the model, the effective Hamiltonian for the model of a single grain is now can, should be formulated in terms of these phases. Now, the properties of the model are the following. This model allows exact, so exactly almost exact solution. And it exhibits the code and you see that there is, of course, the competition of charging energy and DG. And that's why in this competition results in a quantum phase transition and change of the ground state from the phase unsynchronized phase phase, which I would call, first of all, non synchronized. It is when the charging energy dominates and phase synchronization, which in this context means super, really superconducting phase with the superconducting coherence. Now let me go into details of this phase unsynchronized phase. Basically this is the most important point and most important physical result is the appearance of this phase unsynchronized phase. And the second point is here, the following. So what are, so if that one can get more insight in this phase phase in science unsynchronized phase, if one considers the correlation function of phase exponential, and one can see that at zero temperature, the correlation function, this is the sum of products of the phase exponentials on the site I and on some other site J, some over all sites, calculated in the Matsubara formalism and you can see the perfect pole structure of this propagator. And this pole structure means that there is a single mode if you now make analytical continuation to real frequency, frequencies you obtain such a retarded correlation functions that has a pole structure that had a pole and the energy which I will call epsilon one. And what is important for this energy is that this energy epsilon one, it can be calculated explicitly. E one again is a charging, G is the coupling of this Kuramoto coupling, which is proportion, which depends on you. It is not proportional, but it rises with you naturally. And therefore one can see that they, and what is important for this mode? First of all, it is a single mode because of the pole structure. Second, this mode proliferates of diagonal long-range order over the grain because this is a correlator of the phase exponentials at different sites. And the third part important point is that this mode exists in the spectrum even in the quantum disorder phase unsynchronized phase, which means that even in the quantum disorder phase, non-synchronized phase is unsynchronized. The spectrum looks like the following. The ground state has non-synchronized phases, but the reason spec with an excited state in the spectrum that bears with these long range phase coherence, one can call them in quote superconducting state. It is not superconducting state, but it is long range coherence state. Now the, and the energy of this state diminishes and goes to zero exactly at the quantum phase transition point. Moreover, this mode is the single and only critical mode that describes, it characterizes all these quantum critical transition between the non-synchronized and phase synchronized phases. And this basically, this is the main physical result. Now, from now on, I will just outline the consequences of this result for the system and for the phase diagram. What are the consequences? First of all, one can conclude, so what is this non-synchronized phase in more physical terms? One can really understand, one can easily understand this is an insulating phase. Why? Because in this phase, finite superconducting order parameter suppresses one particle conductivity because of the big gap in the one particle spectrum. However, this pair conductivity is also suppressed because the pairs at different sites are incoherent. So naturally this phase at zero temperature is insulating. From the, on the other side, if I rise the temperature, I understand that if the temperature is, by final temperature, this state, this epsilon one, which has this long range phase figuring can be thermally activated. So I can expect that thermal activation of this state should result in the increase of conductance of the system. Now, if I go to the array, of course, not within a single grid, but nevertheless, this excited state with long range phase coherent should favor the conductance of the system, of the system in the channel, which propagates few pairs, not the single particle, but cuper pair conductance. Now, and as usual for the quantum phase transitions above the quantum phase transition point, there is the quantum critical region opens whereas the temperature plays the major role in the determination of the all those quantities. So this is the phase diagram. Now this is a complete phase diagram for a single grain, not for the array. Now let me show what happens if I introduce the array. So now let us restore the tunneling. I assume that the tunneling scale is the smallest one of all energy scale. So I can, I assume the possibility of treating the tunneling perturbatively everywhere. In that case, this is the comparisons of the diagram. What are all those tools? This is an isolated grain, this is the array. First of all, of course, the Fermi-Liquid region opens. This is as usual as expected that it was, but what is more important here, the, at this part, what changes is this right hand part of the phase diagram. What is this right hand part? Initially, each grain was a perfect superconductor in producing the tunneling. I obtained naturally Josephson junction array. So now I, so this part describes the Josephson junction array, but this time at higher temperature because the tunneling is small. First, the temperature will exceed the effective Josephson coupling. One can calculate it. This is this Josephson energy. And then this array goes to the normal phase where again, where the conductance is still there, but this is the finite resistance due to the transport due to the cooper pairs. And finally, let me outline the calculation, the explicit calculation of the resistance. First of all, the most important regime is this insulating phase. The single part, one particle conductivity is suppressed to calculate. Now the two particle conductivity. First of all, one should take into account the tunneling, which leads to the broadening of this initially exact energy level epsilon one due to the possibility of the escape of the cooper pairs in the other grain. This broadening can be calculated for the Dyson occasion. I call this phase correlated the cooper one. It is of course not exact rotation, but nevertheless for physically, it is basically the same diagram. So one can calculate this Dyson occasion. This part, this vertex basically evaluates to the Josephson energy. The physical essence of this vertex is nothing else that Josephson energy between the grains. And then one obtains first the exact solution which shows you the semi-circular density of states, which you can can for simplicity model just for the level broadening. And then the conductance to the conductant, the conductivity, pair conductivity in this case, this is the major channel is calculated. The major contribution is given by the slommas of Larkin. You can recognize here a slommas of Larkin diagram. I just want to say that those vertices, are the tunneling completutes. This is averaging over the randomness. The lower part, all correlation functions or propagators in the lower part are for the grain R prime. The upper part is for some other grain R. So this is the propagation of what you prepare from R to R prime. The result depends on the regime where you calculate these diagrams. The most important result is here. If you have the regime where epsilon one, this is again this energy of this superconducting, let's say, mode, much larger than the broadening, you have the conductivity, which has clear, thermally activated behavior, exponential e to the power minus epsilon one over t. And moreover, the activation energy depends on the distance to the phase transition. I consider exactly this point as a possibility of experimental verification, whether this model is true or not. Experimentally, you should measure the conductivity as a function of temperature and you should find thermally active, but changing the distance from the critical point. This is another question, how experimentally to change the distance, but nevertheless, you should change the distance to the critical point and you should see that the activation energy diminishes closer to the critical point. So the mode which is activated is actually the critical mode. It is the difference because, for example, for a single particle transport, the single particle gap remains unchanged and you also can thermally activate it, but this channel will have the constant activation energy. So, and then finally, if you go closer to the quantum critical regime, then you have a power law dependence of conductivity, either t squared or t, depending on the relation between the t and the level broadening gamma. So I'm close to the end. And so finally, this is a possible typical dependence of resistivity on temperature, which you should encounter if you go along the phase diagram along this dotted line. So first in this regime, you have thermally activated behavior, dropping down exponentially, then you have part of the quantum critical regime and then it changes to the linear rise corresponding to the strange metal. And finally, to the right-hand side of the diagram, I will only very briefly, so you can evaluate also the conductivity, the resistivity in the incoherence of subtraction array using the RSCJ, so-called model. The result is that the conductivity is N, which is a number of channel and it is independent of the temperature. So to conclude, this is the whole phase diagram I wanted to present for you with a typical temperature dependence of the resistivity. Important point is the following. Below this dotted dashed line here, the one particle transport is suppressed and the whole, this lower part of the diagram, the transport property in this are determined basically by the transport of Cooper pairs bosons. And here I just want to put my conclusions and summary and thank you very much for your attention. Thank you very much. Yes, good questions. Thanks, very interesting. Can I just, can you repeat the calculation with a, when you replace Tij with some Pyrals factor with a magnetic field, is it possible to do such calculation so that you have a field tuned, yeah, you have a field tuned superconductor it's later. No, wait, can you? I mean, so first of all, magnetic field explicitly is not here for the very beginning. Next step, but can it be added? It can be, yeah, it can be, but it can be, there are some preliminary result but the work is in progress, but yeah. Let me ask two quick questions. First, without hook on the first part of your talk, there were dashed lines in the phase diagram. And then you said that, well, yeah, strange metal and insulator. Is it really metal insulator transition or everything is crossover as a dashed line? At finite temperature, I would rather say crossover because you have finite. At zero temperature, you don't have a transition. I mean that. Well, as you, yeah, this is a crossover. Yeah. Okay, so the answer is- It's really never metal insulator transition. It's always crossover. It's crossover, the only transition, real transition is- Sure, sure. But this is those are crossovers because you have, in terms of conductivity at least or resistivity, you have finite conductivity everywhere. Yeah. And second question is referring to what Lara was asking during your talk. Suppose there is hopping. And you said it's small. Okay, let's take some number. And at the same time you have U. And at very small U, anything related to superconductivity is the scale of U, which is small. Why don't you have a conventional superconductivity out of Fermi-Liquid with large stiffness compared to TC at very small U? Because still you have the charge, because at finite hopping, if U is very small, then at finite hopping, you have still this charging energy proper to the- Yeah, stiffness EF over four pi, the standard state. So it goes as hopping. Yes, but this hopping is much smaller than J, which determines you the charging energy. What is J then? J is the, you should go this one. The S-Y-K-J is always present. The hopping is between the two S-Y-K grains. Each S-Y-K grain irrespectively whether the hopping is there or not, has the charging energy. Each side involved of the S-Y-K grain has its own charging energy. So you hope from one side, which has a charging energy to the other side, which also has the charging energy. This charging energy, if you assume the hopping is really small, then this charging energy dominates, which prevents you from being firmly liquid, so to say. If I can rephrase in a way that proven this to your question. So usually what you put there is the compressibility of the electrons which is controlled by the hopping. If you don't have J, so the usual way we do BCS superconductivity. Here, yes, the charging energy is not controlled by the electronic compressibility, but by J. And then this is fixed and it's large. This was make the difference with respect to usual BCS superconductor without J. At least this is what we understood. Okay. Good question. So if I randomize you, does the survive in some way? Well, I mean, if you randomize you, you're keeping, it survives. The choice of the model. The simple question, it survives. It makes everything more complicated. But I saw my first, so first of all, of course, you keep, you randomize keeping non-zero average, right? Yeah, yeah, sure. It does look though. I mean, learning from York, it seemed that things will be different than the model was fixed, you know? Well, I don't know the exact answer, right? I understand that if you randomize, but so to say, make it slightly random, slightly in the sense that you have large average, average is larger than the widths, then definitely nothing special will happen, just the corrections. If you have large randomness, huge, so that you have really large regions or realizations with repulsive view instead of attractive, I don't know the answer. Let's say so. Maybe something will happen, but I don't, I cannot answer definitely what happens in that case. If you already have this type of happening in random view. Yeah, which probably nothing, but I'm, well, because I did not calculate it to the end, I cannot make it tell you for sure. Yeah, just very briefly. Continuing on those lines, what would happen if instead of you, you had an average J of minus U? Well, the thing is that if I introduce average J as attractive, for example, it immediately brings a superconductivity into the play. However, the whole structure with an insulator disappears. The difference is the following. J, as such, is responsible already for, you can understand it in the Cooper channel as a term which proliferates the hopping of the Cooper pairs, already formed Cooper pairs. So you make, if you wish, so U is local. So U creates Cooper pair, but does not act as a hopping term between the Cooper pair. The coping does not know about attraction or repulsion. However, if you make J, then you kind of, you can tell about it as a global attraction between the electrons already not at the same side effectively, but on the different sides, which gives you the superconductivity, but does not give you the insulating state. In that case, the phases are frozen together with the appearance of the superconducting order parameter from the very beginning. Yeah. Okay, very, very, very last. Quick one, why don't you have a zero temperature both metal phase? Because either it is insulating because five phases are disordered or it is just superconducting. It's just zero temperature. It is just any connection, any relation between the array, the Josephson array at zero temperature will be just superconducting if you, ah, okay, I understand. The point at this point I was missing. The important is that the Josephson array you arrive at has charging energy much smaller than the Josephson energy from the very beginning. Why? Because the charging energy, because this formula, I did not explain it here. This is this formula. The charging energy diminishes by n, oops, by n where n is a large number of sides within the grain. So it is very tiny. So that way, which means that this array at zero temperature is definitely in the superconducting state. Okay, I think that we have to close the session because at two o'clock we should start the game. We thank you again, Alex, for the nice talk. And let's see, be back at two o'clock, right? Okay, thank you. Thank you very much. Thank you.