 the morning session. We're going to have three talks in total and the first talk given by Debanjan Choudhuri from Cornell University and he's going to talk about what sets a super conducting transition temperature for interacting flat bands. So very much looking forward to your talk. Pages, yours. All right, thank you, Frank. I hope everyone can hear me and can see my cursor on the screen. So I would like to begin by first of all thanking the organizers, of course, for their kind invitation. Now, I was very much looking forward to actually being in person at this event. But unfortunately, the weather in Denver on Sunday, combined with, you know, United Airlines made it impossible for me to arrive in Trieste. So now I have the pleasure of giving this talk at 3am, my local time. So I'm going to tell you about this very interesting question that we have been thinking about for the last couple of years. I'm sure there are many people in the audience who have also been thinking about this question. And so I will try and give some of my own perspective on this problem. And this work has really been made possible by collaborations with a number of people, including a long term collaborator who's sorry, I can't see my cursor anymore. I don't know what happened there. That's weird. Including Erisburg, who's probably in the audience there. And I would especially like to highlight the contributions of the two young people whose pictures I'm showing. So Johannes is a postdoc at Weitzman who has been instrumental in carrying out a lot of the numerical work that I'll tell you about today. And more recently, we have been working with Dan Mao, who's a postdoc at Cornell, who has also been trying to work on some very general remarks that we can make about the problem at hand. Okay, so if I were to go back to the beginnings, lower in our field has been that if you were to start with a very simplified picture of just VCS mean theory for superconductivity, the thing that usually suppresses superconductivity is this exponential factor, the usual exponential factor with the density of states, which is set by the bandwidth, which I denote here as a w and some characteristics strength of interaction, which I denote here as u. Now the lower has been that one way to possibly crank up superconducting TC is to increase the density of states possibly by making the bandwidth smaller and smaller and smaller. And of course, this is a very worrying line of reasoning, because as you make the ratio of the bandwidth to interaction smaller, you are going further and further away from the regime where VCS theory is valid. So you're actually going to the regime of strong coupling. So in fact, you don't even know for sure whether the standard arguments that from the basis for the validity of VCS theory are valid any longer, because all hell can break loose the same interactions that drive pairing can now drive all kinds of other competing instability. So the question is, is this sort of justified? Now one can be critical of this sort of argument, but of course, there have been a number of really amazing developments in the past few years in the field of more superconductivity, where we have found examples of superconductivity. The details here are sort of unimportant of which specific material we're looking at, but the one common thing that's shared by all of these different materials is that superconductivity arises in a microscopic setting where at least at the non-interacting level we do have partially filled flat bands. So there must be something about these flat bands that make them susceptible to developing superconductivity. So the question that we're interested in this talk is, what is the physics that causes this? So coming back to this question of what it is that sets the superconducting transition temperature for flat bands in the regime of strong interactions, unfortunately, even the question is very easy to state, I can't give you a simple one line answer. And the problem is that this is a very difficult non-predurbative problem because nominally, if you were interested, let's say in the regime, you're the bandwidth is completely quenched, it's a completely interacting problem with no small parameters, and a priori mean field theory is unreliable. So the question is, how do we even go about addressing this question? So before I start talking about the kinds of microscopic settings that are relevant for the physics of these moire materials, let me start with a very simplified warm-up problem. And so the model is just going to be the paradigmatic model on the square lattice, so a Hubbard model, but importantly, not with repulsive interaction, but with attractive interactions. So again, I can't seem to find my cursor, unfortunately, but here you have attraction microscopically at every lattice site. And so U is positive. So the reason why this is an interesting model to consider is because this is something that we can solve exactly numerically using quantum Monte Carlo calculations. The model doesn't suffer from this infamous sign problem. So one can literally obtain the ground state in all the relevant correlation functions exactly. And what one finds in this model as a function of the strength of interaction U divided by T is what I've shown here on the left. So what we're plotting is the superconducting TC as a function of U over T. So when we're at weak coupling, which is on the left side of this x-axis, we find that there is indeed a superconducting instability, which asymptotically approaches the standard BCS mean field expression for the superconducting TC. However, one notices that as we crank up strength of interactions, TC eventually starts to plummet. In fact, the optimal TC is attained when the bandwidth is comparable to the strength of interactions. Now, what I'm showing here on the right is that even though you don't have superconductivity, as you crank up the strength of interactions, you still form local singlets, which is why if you look at the spin susceptibility, it starts to show a suppression at some characteristic temperature, T star, which roughly scales with U over T linear. So the question is, pair formation is not suppressed at high temperatures, but still there is no long range superconductivity. So the question is, why is that? Well, we can actually understand this problem analytically, starting from a different limit, which is you first try to solve the problem in the absence of any hopping, just solve the interaction only problem, and then do perturbation theory in the hopping. And when you do that, you may just realize that in fact, what's going on is that locally you're forming singlets, but really in order to get superconductivity, what you need is phase sclerons, meaning the cooper pairs have to move around. And in this strong coupling limit, you can essentially think of these cooper pairs as hard core bosons. And the kinetic energy of these hard core bosons can be calculated perturbatively in strength of hopping. And that gives you a scale, which is as denoted here, it's a T squared over U. And so in fact, that is exactly the asymptotic dashed line that I'm plotting here on the right, which is what the actual TC is tending towards in the regime of strong coupling. So really when the coupling is strong, what limits the long range superconductivity is the phase stiffness and not the formation of a local gap while the formation of these singlets. So this shows that the superfluid stiffness is an important quantity that we have to ensure is finite in this regime of strong interactions in order to get superconductivity. Now the physics of Moray is not really coming from a very simplified picture of the sword I just described. In fact, a cartoon for many of the Moray systems that we are dealing with is as shown here on the right. So the y-axis is energy. So what you have in the problem are a set of nearly flat bands, which I denote here as active bands in red. These are the bands that are well separated by an energy gap from the remaining bands in the problem, which is what I denote as the remote bands. And when you fill up these bands, the active bands by some partial filling, all of the physics that you see in the systems is coming from the low energy physics associated with these bands. And the regime that I'll be interested in is where the bandwidth is much smaller than the typical scale of the interactions, which is also much smaller than the gap to these other remote bands. And so in the remainder of this talk, I'm going to first start out by advertising some upcoming work where we try to form without committing ourselves to a very specific microscopic model. What are the possible constraints that you can put on the maximum TC that one can get in this particular regime of almost vanishing band with some interactions, which is well separated from the remote bands? And then I'm going to go look at a few very concrete microscopic models to address this question of, in the non-perturbative regime, what is the ground state? What is the role of the specific flat bands that you're dealing with? Specifically, if you look at the block wave functions associated with these bands, what is the role played by the geometry versus the topology associated with these flat bands? And the question that's going to come up again and again is what is it ultimately that determines the competition between the different phases that can be triggered by the same microscopic interaction that we're dealing with? And let me just say at the outset that an application of mean field theory is really unreliable to address this question. And okay, so the real question is, when are flat bands interesting? And this is something that's obviously very relevant for all of the microscopic more race type settings that we're dealing with, because it turns out that certain kinds of flat bands can be very special. So first of all, the flat bands that we're getting here, which are well isolated from these other remote bands, these arise even at the non-interacting level because of some subtle quantum interference effects. These flat bands are not coming in because you're starting with something like the Hubbard model and just taking the limit of little t or the hopping going to zero. There are more interesting quantum mechanical effects. And moreover, if you look at the one-year functions that you can construct out of the block functions associated with these active bands, they are naturally spread out, which is what I'm trying to show here in this cartoon. So basically, if you look at this cartoon, what it's trying to suggest is that, yes, so assuming that your interactions can form some local singlets, the question is, when can they naturally gain the kinetic energy? Now, in the previous example of the Hubbard model, the kinetic energy was ultimately controlled by the single electron hopping, t, which gave us the scale of t squared over u in the first place. The question is, is it possible that if the one-year functions are naturally spread out, the scale of the stiffness is also set by the strength of interactions? So ds, which is the phase stiffness scales as u instead of something like t squared over u. Now, one obvious case where we know the one-year functions are naturally spread out is if the active bands are topological, because then we know that there is a natural obstruction that one has in constructing exponentially localized one-year functions. But one of the points of what I'm going to tell you about is that topology is not essential. You can have completely trivial, topologically trivial bands, but as long as there's something else about the bands that ensures that the one-year functions can be naturally spread out, that is enough to give us this interesting regime of flatband superconductivity with the phase stiffness set by u. Let me just mention that at the mean field level, this has been addressed by a number of people in the past decade or so, including perhaps some who are already present in the audience. Okay, so before I go and start talking about very specific microscopic models, let me try to ask if you can make very generic model independent statements. And so one interesting line of work has been to address if there is a fundamental upper bound on the phase stiffness. Okay, so the question is given some generic Hamiltonian, so this could be a multi orbital model, an electronic model with spin, et cetera, et cetera, of the type that I've written here as a kinetic piece plus some interaction. And just for the sake of argument, let's take interactions to be just density-density interactions. You know, so this is a UV Hamiltonian with only density-density interactions. The reason for choosing this is that then the interaction Hamiltonian does not couple to an external electro potential. Okay, but clearly the kinetic part of the Hamiltonian will couple to an external potential via standard minimal company. The question is what is the maximum PC that one can get for such a model? And it's been pointed out that TC is going to be limited by the total integrated optical spectral weight. Okay, so if you look at the full sigma of omega as a function of t and integrate it out to infinity as a function of frequency, that is what will be an upper bound on the superconducting TC. Now, most electronic solids that we are dealing with, this will typically be to a bound on TC of the order of a few electron-volts to pin. So this is a very regular statement, but of course we know this is always hugely satisfied by any superconductor that we pick. Okay, in the context of let's say if you were to apply this bound to twisted bilayer graphene, this would mean that you're not interested in the contribution to the optical spectral weight just from the flat bands, but you have to sum this over all the, you know, all the 10,000 bands that you would get by, you know, folding your original direct dispersion in the moray in itself. Okay, all right. So this is a rigorous bound, but this is clearly not a very tight bound. So one question that we were interested in is if you can put a much tighter bound on what the superconducting facetiveness can be, you know, in a low-energy sense, meaning described by properties just of the isolated flat bands. So that's what I'm showing you here in the figure on the left. So imagine, so the figure on the left is what the system looks like at the non-interacting level where you have your active bands. Here I'm showing them to be completely empty. So this is the band insulator. Then you populate them by a finite density of electrons, which is what I'm showing you here on the right. And the question is, can we put some, at least even a conservative upper bound on the superconducting facetiveness, which only depends on the properties of these isolated flat bands. Okay, now theoretically the way to address what sets the facetiveness is to basically look at the diamagnetic response, which is the first term here on the right, which I denote as capital K, and the current current susceptibility, which is shown here as chi. Okay, and of course, we have to take a very particular limit. The order of limits is important here. So now this actually turns out to be a very complicated problem for the following reason. Now the usual way in which we actually calculate these quantities is to basically do a coil substitution, introduce a probe gauge field in the problem, and ask how the diamagnetic and the paramagnetic current susceptibilities respond if we do that. Now, when we're interested in the low energy physics associated with these flat bands, it's not as simple. And the reason is the following. So to really describe the low energy physics, what one has to do is integrate out the contributions from these remote bands, project the microscopic interactions that you're starting out with, which could be a simple density-density interaction onto these active flat bands, and then introduce a probe gauge field or carry out a gauge transformation that only restricts to the slow energy Hilbert space associated with the flat bands. So in fact, the experts will probably notice that the problem here is very reminiscent of physics that we have dealt with in the past in the context of Quantum Hall and let's say lowest land level physics. Okay, where now of course, as we carry out any of these operations, one has to worry about the effects of the analogs of a land level mixing. Okay, because these remote bands are kind of the analogs of our land allowance. Okay, so recently we have done this and I'm not going to go into the details of this, but I'm just going to give you a teaser. There are a number of moving parts that makes this problem complicated. Okay, so first of all, of course, when you project the interactions to these flat bands, that automatically generates a finite bandwidth. The bands are not really flat anymore because scale of interactions is now what sets the actual physical bandwidth in the problem. Now, if you integrate out the remote bands while something like the Schriefer-Wald transformation, you immediately see that the actual microscopic current operator associated with these flat bands is not just the projected current operator. The effect there are analogs of the effects of land level mixing, which make the current operator much more complicated. And similarly, when you try to introduce a probe gauge field in the problem to address what the diamagnetic susceptibility is going to be, you have to worry about whether the gauge transformation itself is going to, again, mix between different bands. And if you're really going to stay in the low energy manifold associated with the flat bands. And ultimately, even after you do all of this, it remains a many-body problem because the superconducting stiffness is still determined by a many-body correlation is in this flat band. So one option is to actually take specific models and try to calculate these many-body correlations or to make somewhat drastic approximations to come up with very conservative bounds. So this is going to be out yet, so I encourage you to take a look at this when it appears on the archive if you're interested. Okay, so enough of talking about generalities. So let's now look at actual microscopic models that we can solve reliably to address the question of what is it that sets the competition between superconductivity and various competing orders. So one of the first examples that we studied was the following problem, which was the problem of really topological flat bands. Okay, so here what I'm showing you here is a model of churn bands. You can write down lattice models for churn bands, where we can furthermore tune their bandwidth to be really flat. So the picture on the left shows one particular realization of, you know, a very flat band characterized by a flatness ratio, which is the bandwidth for the flat band divided by the gap to the remote bands. Here the flatness ratio is 0.01. So it's the lower band that's really flat. And the model here is time reversal symmetric. So you can really think of it as two time reverse copies of a Landau level, where let's say spin up bands form a churn one band and spin down band forms a churn minus one band. Okay, and again, microscopically, if you were to turn on an attractive interaction, just like we did earlier for our Herbert model, the question is, can this give rise to superconductivity with a TC that scales as you? And if so, what is the physics that controls it? And the method of choice for solving this problem will again be quantum Monte Carlo, because the model as written doesn't suffer from design problems. So we did this. And, you know, amazingly, we do find that, you know, given that there's literally only one energy scale in the problem in the flat band limit, which is you, you know, if there were to be superconductivity, of course, it would be set by the scale of few, which is exactly what we find. Okay, so the plot on the right shows TC as a function of you for two different bands. So the dashed line represents the really flat band that I just told you about. If you turn on a finite bandwidth, TC goes a further because the finite kinetic energy associated with the bands can, you know, further enhance TC. Now I should point out, of course, that, you know, this the TC increasing linearly with you can certainly not go on forever. This is in the regime where you is still smaller than the gap to the remote bands. If you becomes the largest energy scale in the problem, of course, TC will eventually saturate and start coming down because of the physics that I told you about earlier. And since then, you know, this type of quantum Monte Carlo has been extended to other models, where you're dealing with non-turned bands, but the bands that have more subtle topological properties and so on. Okay, so good. This is a numerical result. But the question is, what can we say more about, you know, the parent state out of which this emerges? So one immediate thing that we can notice is that the superconducting state actually is a bit fragile. What I mean by that is if you turn on infinitesimal perturbations, such as what I'm showing you here, which is a nearest neighbor attraction, you find that the system, even with a tiny nearest neighbor attraction, can actually display phase separation. So in fact, if you look at the charge compressibility, it diverges, which is what I'm showing. So the red line here denotes the inverse charge compressibility, while the blue line shows the inverse pair susceptibility. So in fact, you find that as you cool the system down from high temperatures, the system phase separates before it can become a superconductor. In fact, this can be probed by looking at other correlation functions in the normal state. So the first question we should ask is, you know, does this interesting regime of flat band superconductivity arise out of a conventional metallic state or out of some very incoherent liquid? And indeed, it's the latter. And so the normal state, first of all, has no sharp spectral features like a Fermi surface or anything of the sort, which is what the plots on the right are supposed to show. Basically, if you look at any proxy for a single particle spectral function, you'll find that it's completely featureless in the brain wall zone. Moreover, there are strong fluctuations. If you look at both the charge susceptibility and the pair susceptibility in the normal state, it looks like they're both trying to diverge as you cool the system down from high temperatures. So the question is, how do we actually make sense of all this analytically? Is there a way or a regime where we can do this without appealing to any uncontrolled approximations? So the question that I'm going to address in the remainder of my talk is if there is a sort of a solvable limit where this complex phenomenology can be understood analytically. And the fundamental trouble here is that, as I emphasized at the outset, this is a problem with no small parameter. You're basically dealing with a completely non-prodivative regime where interaction basically sets the only scale in the problem. And so this actually motivated us to come up with a model where, in fact, we can make some analytical statements. So here is how that works. Imagine I can come up with a model of perfectly flat bands, which is what I'm showing you here on the left. So I have two bands that are perfectly flat with NRDs plus minus T. And so I have to come up with a Hamiltonian, which I denote here as the H10, which will give me such bands. And then I want to turn on interactions just like I did before. So on-site attractive interactions with the strength view and nearest neighbor interactions, either attractive or repulsive with the strength of V. So the question is, what kind of non-interacting Hamiltonian can give me this? So here is what it looks like. So I'm giving you directly a model in momentum space, as shown here, which exactly gives me these two flat bands. So I'm sure, even though it's early in the morning for you, you can immediately diagonalize the Hamiltonian that I've written for you to convince yourself that this does give you two flat bands with NRDs plus minus T. So now this is a model with two orbitals, which I denote here as L, and the tau's act on the orbital labels and spin sigma. Now notice that there is an additional parameter that I have introduced here, which I denote as zeta. So this is exactly going to be eventually my small parameter. So when zeta is 0, actually the lattice really falls apart because that represents the decoupled atomic limit, where the different lattice sites actually don't talk to each other. That's the question? Yeah. It's a reason sine alpha cos alpha and alpha is again a combination of cosines of wave factors. Is it really true? Yes. So alpha k, so you can one way to imagine, actually I think someone else is sharing their screen and I think Andrei is sharing his screen and somehow I'm not able to share my screen anymore. I don't know. It tells me that I'm viewing Andrei Chubukov's screen. So there is no mistake. It's cosine alpha and alpha is again a combination of cosines. Yes. There is no mistake. That's right. Yeah. And one way you can, so basically as I said, the parameter zeta is going to be my small parameter. You can imagine expanding this in powers of zeta. And then you will see that basically this generates further and further range hoppings if you were to take the Fourier transform of this Hamiltonian in real space. But I'm actually right now dealing with a slightly different problem, which is that I am unable to share my screen because I'm seeing someone else's screen. Is there an AV person who can help me? Yeah, the tech can you? Okay. Okay. So can you see my screen now? Yes, it works. Okay. Perfect. Thank you. Right. So coming back, so what else can we say about this Hamiltonian? Well, the other thing is noticed that in the kinetic Hamiltonian there is no term with tau z. Okay. And in fact, that implies that the very curvature in this model vanishes identically. So by studying this problem, I'm not just going to address the role that topology plays. And just to give you the punchline, it doesn't. Even the very curvature or the non-trivial distribution associated with the very curvature in momentum space plays no fundamental role. Okay. Because it just vanishes identically everywhere in the Brillouin zone in this model. Now, you might ask what role does this parameter zeta really play? And it does play an important role. So in fact, if you look at what is known as the Fubini study metric, which is probably something that you've already been introduced to in this talk in this conference, that actually is set by this parameter zeta. So physically, zeta is the parameter that controls the minimal spatial extent associated with the exponentially localized one-year functions associated with these flat bands. Okay. And this will play an important role going ahead. Okay. So here's what if the one-year functions would look like if I constructed them for you for my flat bands. So they're centered around this point with the largest blue circle. So the size of these circles basically represents the magnitude of the one-year function as it decays exponentially. Okay. Now, the nice thing about this model, which lets us proceed analytically, is we can do perturbation theory in a small zeta after projecting the interactions to the flat bands. Okay. Which is what we're going to do here. So one can ask, okay, so what is the effective theory in the low energy manifold associated with the interactions projected to the flat band. Okay. And we can derive that explicitly in the limit of small zeta. Okay. So this is what it looks like. So the first few terms are, you know, an x, xe type Hamiltonian. So the spins here are really the pseudo spins, which are related to the projected operators, the electronic operators in the lower flat band, as I've shown you here. Unfortunately, I still can't bring my cursor, but hopefully this is clear. So the first terms, the sx, sx, sysy term in terms of the original language are really nothing but the pair hopping interaction between nearest neighbor sites. Okay. And importantly, you notice that this pair hopping interaction is controlled by zeta squared times u. Okay. So what this is already telling you is that the interaction is not just going to form local cooper pairs for you, but it's also going to give you the required kinetic energy for the finite superconducting phase stiffness. And of course, physically, the zeta squared term is controlling the spatial extent of the 1A functions. Of course, no topology here. We are dealing with exponentially localized 1A functions. Now, if you look at the szsd term in this effective model, that basically translates into a density-density interaction in terms of the original variables. So there is an important point here, which is when v, the nearest neighbor interaction, is set to zero, notice that the spin part of the Hamiltonian has an exact SU2 symmetry because jper becomes equal to jz. Okay. I'm going to come back to this point on the next slide. But there is another interesting thing that you get from this effective model, which is that there is an additional term that's generated, which I denote here as a h-hop, which is actually an interaction-induced bandwidth. Okay. So this is also sometimes known as a correlated hopping, which tells us that the hopping that's generated for these electrons depends on the density at some other side. Okay. So completely an interaction-induced effect. Okay. So coming back to just this approximate SU2 symmetry that's generated, it turns out there is an exact symmetry in a very specific parameter of phase space, which is when you take the limit of bandwidth going to zero, and the limit of u, the on-site attraction going to zero, and temperature going to zero, and switch off all further neighbor interactions, it turns out that's where this SU2 symmetry emerges. Okay. And in that limit, the BCS wave function for your superconducting state is in fact the exact ground state. But because of this SU2 symmetry, you know, Mormon Wagner theorem of course tells us that there can be no finite TC. So in the limit where effectively BCS mean field theory becomes exact, TC is actually zero. Okay. So now if you ask about how to think about these fluctuations that I already told you about in the normal state, there's a very nice appealing picture for that, which is basically represented in terms of the fluctuations of this nonlinear sigma model, which I have denoted here. So n is a three-component vector. Now, of course, when there is no anisotropy in the problem, there is no TC. That's this limit with the perfect SU2 symmetry. Now, there are all kinds of perturbations in this model that break the SU2 symmetry with, you know, the finite bandwidth, you know, the scale of u squared over delta, where delta is the gap to the remote bands, the nearest neighbor interactions, they all contribute anisotropy in this problem. And ultimately, the system can have a finite TC, which is determined by, you know, the scale of this anisotropy via these log corrections. And also immediately by deriving this model, I have shown you what it is that sets the scale for the superconducting phase stiffness, which is coming purely from these interactions and the finite extent of the one-year wave functions. Okay? So this is all, of course, done in the limit where we have artificially introduced a small parameter zeta. So the beauty of this model is that we can actually solve this model using quantum Monte Carlo at arbitrary strength of zeta, which is also something we did. And unsurprisingly, we do find that there is indeed superconductivity, even for zeta not small. And moreover, if you look at now TC as a function of u, it does agree with TC being proportional to u times zeta square, which is what the plot on the right is supposed to show. Okay? So I see that I'm running out of time, so I'm going to speed up a little bit. So we can actually go ahead and again analyze all of these correlation functions in the normal state. So the first thing you notice, again, is that if you look at the spin susceptibility, you find that much before you have longer in superconductivity, you are starting to form local singlets. It's just that you're not phase scoring, which is what the plot on the left is supposed to show. And in fact, you know, you see that this physics doesn't really care about zeta. Okay? So even when zeta were to be zero, and you have completely recoupled all the lattice sites, you form spins, local Cooper pairs, and the physics really remains the same, because the onset of the suppression in the spin susceptibility doesn't really change a whole lot with increasing zeta. Again, if you look at the inverse pair susceptibility and the compressibility, they both seem like they're going to diverge, which is what the plots, the two other plots here are supposed to show, which is indicative of this approximate SU2 symmetry that I told you about. So the density and pairing fluctuations are almost degenerate. Ultimately, of course, there are reasons that I told you that tip the balance in favor of either superconductivity or phase separation in this model. Okay, I haven't told you anything about all the different competing orders. So let me end this talk with the discussion of those. So far in the QMC, I had turned off the nearest neighbor interactions. Okay, let's now turn it back on. And as you can see, I'm turning on a tiny, tiny bit of a nearest neighbor repulsion here. So V is just 8% of U. Now, the first thing you notice that when you are at the commensurate density of N equals 1, which is half filling of the lower band or a quarter filling of the total system, you'll find that you generate a charge density wave at the highest temperature. And you do not have a superconducting ground state at the specific value of zeta. When you dope the system either by electrons or holes away from N equals 1, eventually these excess holes on electrons do develop superconductivity without melting the CDW. So what you see here on the left in the phase diagram is that eventually the ground state when doped away from N equals 1 is in fact, both a superconductor and a charge density wave. That is, it is a super solid. Okay, again, this is all coming purely because of interaction induced effects in a perfectly flat band. One thing that I should have mentioned is that the model by construction always has flat bands regardless of what the value of zeta is. That's one of the beauties of this model that you can tune zeta, which controls the extent of the 1A functions without affecting the bandwidth of the non-interacting bands. Okay, good. So now the question is, can we say something more about what's causing all of this competing and intertwine orders? So one way to address that is to go back and look at the single and two-particle spectra. So when V is zero, on the left part here, the red lines denote the original flat bands. Even in the presence of interactions, you split the bands because of your finite U, but nothing dramatic happens. If you look at the two-particle response, both in the pairing and the density channel, as a function of frequency and momenta, sorry, the x-axis got chopped off here, you find that omega goes as Q near the gamma point, which is again control the physics of the Goldstone mode associated with this CO2 limit. Now, when you have a finite nearest neighbor interaction, which is where all these other competing instabilities get triggered, you find that even the single-particle spectrum changes fundamentally. So notice that the actual bands here now are denoted using these orange lines, and you can clearly see that they have generated a finite bandwidth. They have finite dispersion, and this is exactly the physics of these correlated hoppings or the interaction-induced hoppings that I told you about earlier obtained in an expansion in a small zeta, but now you can explicitly see them from your Monte Carlo. Moreover, in this limit, as you can see, if you look at the pair susceptibility, they're now gapped. The system is not a superconductor, and there's a softening associated with the density spectrum at the M-point, which is exactly the wave vector associated with the CDW order. So this is just to show you the complexity associated with just your interactions predicted in the flatband. They're not just driving all kinds of instabilities, but they're fundamentally changing the dispersion associated with your non-interacting flatbands because of purely interaction with them. And the final thing I just want to mention here is that so far everything I told you about was simulations at a fixed zeta. What you can do amazingly is keeping everything else fixed. Now, vary zeta, which controls the extent of these linear functions and go from being a charge density wave at this commensurate filling of one electron per site to a super solid just by varying the spatial extent of the linear functions, which is what is being shown here on the right. So you have a transition from a CDW to a super solid so that the CDW doesn't melt, but even at this commensurate filling, the Cooper pair is actually delocalized. Okay. All right. So I think I have almost run out of my time. Let me just end again by coming back a full circle and ask about bounds on TC. So this is probably a plot that many of you have seen in the past. This is often known as the Uemura plot, which plots superconducting TC for a variety of systems as a function of some proxy for a Fermi temperature that you derive from experiment. I won't go into the details of how this is done. Now, if you look at this plot, it seems like there is really an avoided region on this plot in the upper left corner. Something seems to limit how big of a TC you can get for any system. If you look at it closely, even Magic Angle Twisted Bioliberal Fee is on there. So really, this begs the question of whether something like a Fermi temperature really sets a bound on TC. And let me just end by saying that this is something that we have analyzed recently. There is no such universal upper bound or at least no such bound that we know how to formulate in terms of some appropriately defined Fermi energy. Note that the kinds of systems we are dealing with, the Fermi energy is not given by usual N over M. One has to define it in some experimentally relevant fashion, such as maybe the difference in the chemical potentials for a field versus an empty band as shown here on the cartoon in the right. And so these are often heuristic bounds that probably have some physical meaning attached to them. But purely, theoretically, one cannot prove that there is such a bound on TC in terms of either an appropriately defined Fermi energy or a zero temperature phase difference. So if there are questions, I can come back to this later during Q&A. So let me end here with an outlook. So hopefully, I have convinced you that phase competition in flat bands in the presence of interactions with non-trivial geometry is difficult to predict. It's not topology that's fundamentally important, or it's not a necessary ingredient, but it's more the geometry associated with these flat bands. Importantly, the interactions projected to these flat bands can trigger both instabilities to a variety of competing phases, but also induce a dispersion for your electronic excitations. And certainly some elements of this physics should be relevant for the physics of even more ray systems. And maybe the real take home message is that we need new kinds of ideas and non-trivial bit of theoretical methods to address this complicated phenomenology. Thank you very much for your attention. Okay, thank you for a very nice talk. We have time for questions. Thank you. I have a question about the one-year functions. How do you build them? I mean, for building the one-year functions, do you use the flat band? Only you include other bands? Yeah, so in the exam, I mean, this is, you know, the usual story that developed by Merzari, Vanderbilt, and others. Here, when I was talking about the WANI functions, I was really talking about the WANI functions constructed out of the block wave functions for the flat band of interest. So the lower flat band that I do for you. Only for the flat band? Yes. Because, you know, the question is, in this low energy limit, the physics should be describable in terms of just that flat band and interactions projected to that flat band. So thank you for a lovely talk. So if I understand you correctly, you cannot put an upper bound on TC in flat bands, a la mojitrandera, for example. Is that just the question? No, I'm asking you. Do you have an upper bound on TC? Okay, good. So let me just quickly go back. So mojitrandera originally in 2019 proposed a bound based on this quantity here. So this is a completely rigorous upper bound. If you were to apply this directly to twisted bilayer graphene, you would have to sum over all the bands because notice the important thing here was that the interaction does not couple to the vector potential. And basically what I told you right now is if you're just interested in a bound formulated in terms of the properties of the flat bands, you have to project your interactions to the flat bands and then they will couple to the vector potential. So this story will change fundamentally. I guess last year, Mohit wrote another paper where he did come up with some bounds for interactions projected just to the flat bands. And there he was able to relate the bound to again the metric. I guess I forgot to put that reference here. My apologies on that. And so basically that is the kind of story that you're trying to extend here in this upcoming work. It's still difficult because the bound is not as simple. You still have to make some assumptions and approximations, but at least it's possible to do it eventually. And as you can see, immediately the bound will be related to the strength of interaction, the properties of your block wave functions and so on and so on. It's not as nice and simple as this story. It's much more difficult. And that's because the problem is very much like the problem of Coulomb interactions projected to a landlord. So that's the analogy of the quantum hall problem and that we know is kind of hard to deal with in general. Devajan, let me try to ask you a question a little bit like Devils Advocate. You are, I understand that there is a lot of details related to flat bands, but at the end of the day, you start with a model with attractive pairing interaction. Your U is attractive. So players are there. In case of attraction, superconductivity always compete with phase separation and with charge density wave. So the players are the same as with general case of attraction. So my question will be if probably most of the systems that we're studying with respect to TBG have repulsive and have repulsive interaction. Yes. So what would be players in your view in case of interaction, repulsive interaction? Superconductivity, first of all, it should somehow come out of repulsive interaction and then it should compete with what? Okay, that's a great question, of course. I really like to point out that, I mean, in the Moray systems, there is still electron phonon and that can cause attraction, but I agree you still have to compete with repulsive interactions. And in the TBG problem, of course, the repulsive interactions can drive all kinds of, so the Coulomb interactions projected to the TBG flat bands at commensurate fillings can drive all kinds of these quantum ferromagnet type instabilities. Whether when you dope away from these commensurate fillings, the repulsive interactions can conspire with attraction generated purely from phonons to get superconductivity is, of course, an open question, and I don't have an answer to that yet. But I will say that, again, we can actually go ahead and solve the model for electron phonon interactions in this problem in principle, you know, using many of the techniques that I described here. How to deal with repulsion is, of course, a million dollar question. Good, thanks. Unless there is another urgent question, I would like to conclude the discussion and thank you again for the very nice talk and discussion. Could you please stop sharing the screen?