 Alright, hello everyone. Thank you for coming and thanks to organizers for giving me this opportunity to present our work. Today I'm going to try to convince you that the physics of the magic angle is actually incredibly generic to semi-metals in the presence of a quasi-periodic potential. There's going to be a very different perspective on this problem than anybody else has put forth, and so I'm very excited to tell you about it today. Let me just first start by thanking the collaborators in this work. My graduate student, Yixing Fu, as well as a post-doc, Ilya Koning at Rutgers, and he's been one of the main drivers of this work, he's in the audience here today. He also drove me here from Milan, so he's also one of the drivers of this talk, you could say. Justin Wilson from Caltech, as well as Yang Zi from Boulder, and the only collaborator older than 35 is David Hughes from Princeton, and then the last Sarangal Prakrishnan from the City University of New York. So the first part is going to be on this PRL, and then most of the talk will be on this unpublished work that should be appearing on the archive in a couple of weeks. So where we're going today, first I'm going to just give a brief introduction to the nodal semi-metals, and I'll probably skip the discussion on these experiments on magic angle graphene, because Oscar just went through this very carefully, but I'll jump into phase diagram of quasi-periodic semi-metals, starting from the most robust semi-metal, which is wild in 3D, and by introducing a quasi-periodic potential we can drive semi-metal to metal phase transitions, and what I'm going to show you is that at this phase transition it's actually the same thing as tuning this system to a magic angle. I'll then extend this down to lower dimensions to even 1D, and in 2D we can make the connection to magic angle quite clear. I'll then show that the excitations become flat, and then from that I can construct effective Hubbard models to give rise to very large effective interactions, where U over T can be on the order of 3,000 the bare value of U over T. Okay, so today I'm just thinking about nodal semi-metals, and so this is something between a metal and an insulator, so just generically we have some energy momentum, and we have bands that touch linearly at isolated points in the brilliant zone. This has been known for a very long time as is pointed out all the way back in 1937 that you could have accidental degeneracies in band theory. And so if I'm focusing on linear touchings in dimensions greater than 1, your excitations around this Fermi energy are gapless, so it looks like a metal, but then if you look at your density of states at that Fermi energy it's negligible, and that actually mimics an insulator, hence the name semi-metal. And so this actually gives rise to unique power law signatures in the density of states, and then, for example, in the specific heat. This can also give rise to, for example, a power law insulator behavior that gives rise to connectivity in three dimensions. So probably the most well-known example of Dirac points would be in graphene, and so if you just solve tight-binding electrons on the honeycomb lattice, here's the band structure energy versus momentum. If you focus on one of these linear touching points, you can just write down an effective K dot P theory, which gives rise to a linear dispersion with the velocity of this Dirac cone, just characterized by V. This has been seen directly in ARPES, so this is the ARPES spectra, and as you can gate this material, which tunes the Fermi energy, distinct from the solid state example, this has also been seen in cold atoms, because they can engineer optical lattices that have a honeycomb lattice, and they can then probe the band structure by band mapping techniques, for example, putting a cloud of atoms and kicking it, transferring atoms from the lower band to the upper band due to the touching point in the band structure. More recently, there's actually dynamical measurements of the berry curvature itself, which shows large berry curvature near your K prime point, and so this can be, this is rather recent, and so you can directly image the berry curvature in this honeycomb optical lattice. So more recently, we now have these 3D examples, such as the discovery of sodium-3-bismuth or cation arsenide, which are Dirac sodium metals, and so I'm just showing you the ARPES data for sodium-3-bismuth, and these are dispersions in three different momentum directions, displaying a 3D nodal band structure. So near this Dirac point, you can write down an effective alpha.k theory, so alpha is the 4x4 Dirac matrices we know from high energy physics, and then even the year after that, then while sodium metals were discovered, which also showed in your touching point, which is non-degenerate away from the touching point. So the main question I want to ask in this talk is how do I take a weakly correlated semi-metal and make it strongly correlated? I want to do this in a very universal fashion that is applicable in a wide array of systems, and we now know, actually as we just learned from Oscar's talk, and we've known for a while, the twisted bilayer graphene is one way to do this, whereas you twist these two bilayers at particular magic angles, the effective velocity of these Dirac cones goes to zero at the various magic angles. So there's not just one magic angle, there's a whole sequence of them, and it vanishes linearly near these magic angles. Okay? And so this has been known now for seven, eight years, but it was really this experiment... Oh, excuse me. So the reason why this is expected to produce very large interactions is because as the velocity goes to zero, the velocity is effectively the hopping, and whatever bear-use scale you had in the problem, the ratio of u of t should then go to infinity. So intuitively, this is why we think they should give strong correlations, promote correlations relative to a weakly correlated starting point, which is what we know graphene is, right? Okay. So this really got, I'd say, revolutionized by the announcement in March of these apparent mott-insulating phases at half-filling of these moray bands, and as Oscar really explained this quite nicely, upon doping or gating, this thing gives rise to superconductors, which may or may not be unconventional, out of these mott-insulating phases. Okay. So one point I'd like to really raise is that most angles in this twisted biolographic system are actually not commensurate whatsoever, and as a result, that's going to produce an incommensurate potential for the Dirac nodes, right? And so the question we're going to ask is what is the minimal model to capture this phenomenon at the single particle level? I want to write down simple Hamiltonians that I can solve concretely in physics in a more general setting than just twisted biolographic. And what I'm going to convince you of is that all it takes is a quasi-periodic potential and nodal semimetal band structures. Okay. So with that, let me now move on to the phase diagram of quasi-periodic semimetals. So let's start with the most robust semimetal, which is going to be a 3D-wile semimetal, and the reason why it's the most robust is because we've exhausted the three polymatrices. So any additional polymatics we add about this band structure. Let me just write down a simple lattice model. So this is hopping on a simple cubic lattice in 3D. The hopping comes with a phase and a spin-o-bit coupling. And then I'm just going to add a potential that's quasi-periodic, and I'll define this in a second. But in the absence of this potential, this band structure represents a model for an inversion-broken-wile semimetal. So if you just diagonalize this band structure by hand, you see that you get eight-wile cones at the time of versatile invariant momenta and linearizing about these cones, you see that the velocity, the bare level, is set by the hopping times your lattice spacing, which I've set to one. Let me now define this quasi-periodic potential. So this is a 3D potential, and I'm going to now introduce, it has a strength w, and it has a three random phases, which is just shifting this potential relative to my overall lattice. These are random phases, which we can average over, but they're correlated, and this is a free lattice site. This model would be random if these phases were random at each site, but they're not, they're incredibly correlated. You could say infinitely correlated. Then we introduce a quasi-periodic wave vector or incommensurate wave vector, and for numerical calculations, it's very advantageous to take something called a rational approximate, where you take this to be 2 pi times a ratio of Fibonacci numbers such that as n goes to infinity, this becomes an irrational, and by choosing our system sizes to be given by Fibonacci number, you tie the finite size rounding effects due to system size to the finite rational rounding because it's not truly incommensurate. So in the thermodynamic limit, this actually becomes incommensurate, and so this is a very nice tool for numerical calculations, as well as some analytic calculations. So in the context of twisted bilayer graphene, you can loosely think w is playing the role of some interlayer coupling, and q is playing the role of some twist angle, which you could either vary one or the other, and so for some calculations I'll fix q and vary w and vice versa. And so what are we going to look at? We're going to start by looking at the structure of eigenvalues themselves, and so we can do first the density of states, and we can use something known as the chronal polynomial method, which allows us to evaluate this on very large system sizes such as 144 cubed in a matter of maybe a couple of minutes, but going to larger system sizes, you've run into memory issues, so you have to worry about that. So L of 144 is eighth of a notch. The next one is too big to do in 3D. And so in the semi-metal phase, the semi-metal is going to be characterized by an e squared loanergy density of states, and you can show that the coefficient here is one over the velocity cubed. There's another way we can do that, which I'll explain later on, and we're going to average over twisted boundary conditions that are going to reduce finite size effects in these calculations. This last piece is not necessary, it's just a nice trick. In addition, we're going to use ed to look at level statistics, so this is something known as the adjacent gap ratio, which avoids an unfolding issue which ties level stats to the density of states. This is like a local unfolding of your level spacing. This is because we have to use ed. And then in addition, if we expect the semi-metal phase to be stable to quasi-periodic potential, what that tells you is that the structure of the wave functions also have to be stable. So if I'm just thinking I'm in a Dirac semi-metal phase, then all my wave functions are just plain wave states. That's just a delta function in k space. So there are other ways of localizing k space. So I defined the inverse participation ratio in momentum space, and so this will then be an indicator if my wave functions will delocalize in k space. So it's kind of like Anderson localization or delocalization but in momentum space. And so the reason why this is useful is because if I'm in a ballistic plain wave phase, this is going to be constant independent of my system size, whereas if I delocalize in k space, this is going to like 1 over my volume. Okay? So I now compute the density of states, and this is the density of states at zero energy function of this w, and I fixed q to be given by the ratio of Fibonacci's and as you can see the density of states remains semi-metallic at weak w, then it becomes metallic back to a semi-metal, becomes metallic again. So we find two transitions to the function of w, and Anderson localization occurs at much, much larger w. Okay? If that's a worry in the back of your mind, please don't worry about that because we're confident that happens at a much larger scale. So if I now look at the IPR momentum space uh, excuse me, so the two transitions, now we move to the IPR momentum space as you can see where the density of states is zero, your wave functions are localizing k space through L-independent. As the density of states gets generated, the IPR goes like the volume, and then it goes back to a semi-metal phase localized and then delocalizes again. So if we now look at this in energy, so this is w versus energy, and the color is my level stats. Since I've averaged over twisted battery conditions, I've broken time-reversal symmetry, and so my level stats satisfy GUE level statistics if it's diffusive, and indeed I see a very clear diffusive region, and this dashed line is actually a diffusive ballistic mobility edge where this ballistic phase is my plane wave eigenstates, and since I've averaged over the twist, these are actually localized in k space so they satisfy Poisson level statistics in good agreement with our expectation. It's very nice to then look at this with the density of states, so this now is the density of states on the left, and what you realize is that we actually have a hard gap here, and so as I tune w, so this is rho v as a function of energy for various w. So as you tune w, you open gaps, you find that energy, and you form a semi-metal miniband, just like he was talking about in the previous slide, yeah? Yes, that's right, so this is 2pi the fn minus 2 over fn, or fn is the Fibonacci, so this is fixed q. L is equal to fn, so q is 2pi fn minus 2 over L. Oh, so every L is, so this L is 144, so there's like, let's see, 55 periods, because that's the fn minus 2. Yeah, but it's a long wavelength at this stage, and so this is going to, our direct nodes are separated by pi, and this is going to be internode scattering because it's large q. I'll come to that a bit later, but, okay. So these gaps open, and you get a semi-metal miniband, so you see this e squared survives, and this gap separates them with a hard gap, a large enough w, and what you see is that this gap tries to close, it shoves all these states down to a metal, and then it opens it back up. Okay, and so if you want to understand what's going on microscopically, you can think that these band gaps are passing through each other, and maybe the states of positive energy actually invert and become the same states of negative energy on the other side of the transition. So you can test this quantitatively by forming a projector onto the positive side of the miniband, and alternatively the negative side of the miniband, and you can then look at this object, which is going to basically take the difference of these projectors, so what does this look like? So green projects onto the positive miniband, so these states at positive energy are clearly projector 1, whereas states at the negative manage band are going to be projector 2. As you go through this transition, they all mix up, and as you see what comes out is an inverted semimetal. In the language of buried curvature, this means the buried curvature of each wild mode changes sign as you move through this transition. We call this an inverted semimetal phase. It's just like passing through the magic angle and going back to a semimetal phase. Okay, so since we have these two transitions, we've checked and they have the same universal properties, so let me just focus on one of them and go through this critical properties with you in detail. So first, let me just look at row of zero as a function of w, so this KPM method expands the density of states to some finite order, so this is the order, and this order acts like a finite size effect. So larger and larger expansion orders is approaching the thermodynamic limit, and as you can see the density of states continues to sharpen, which suggests the density of states is actually jumping from zero to a non-zero value. If you now look at this energy dependence, so here's our hard gap, here's our semimetal miniband and this goes like e squared, there's corresponding fits over here which allows you to extract the velocity. As you now move through the transition, these squish together and form a metallic phase at low energy. You don't see any critical scaling regime in energy, which suggests the dynamic exponent actually not playing a role in this transition. That's an interesting question that we are still sorting out. So now if you want to understand the analytic properties of the density of states, you can ask does the density of states become non-analytic at this transition? You can assume it does not. So you could tailor expand your density of states and this coefficient, which goes like one over v cubed, would have to remain finite if the density of states remains analytic. But indeed we see as we go to large expansion orders this singularity is very sharp. On the order of 10 to the 7, which strongly suggests in our numerical ability that this actually diverges and the velocity goes to 0 at this transition. So the wild cone's velocity has gone to 0 at this single particle quantum phase transition. How does it go to 0? Goes to 0 in a power law fashion which then tells you we can extract this exponent beta which tells you the velocity vanishes in a non-trivial fashion. And 3 is given by the dimension. So this is beta over d. So what have we learned so far? We've learned quasi periodicity drives a semi-metal to metal phase transition. The density of states becomes non-analytic and jumps. And there's no critical scaling regime in energy and kind of surprisingly this transition coincides with the delocalization of momentum space wave functions. So you can then ask yourself can this transition survive down to lower dimensions? And so if you're thinking about disorder which becomes more strong as you go to lower and lower dimensionality which is this RG equation I've crossed out here this doesn't apply to quasi-periodic systems as it's well known. For example, the Auberion dream model can have a delocalized ballistic phase in 1D. So we can just rewrite our model in 2D and redo this analysis and now the connection to twist to biolography is much more clear because I'm in 2D so again this is like my interlayer coupling and again this is like a twist angle. And indeed you find another type of transition but a different universality class because you're in a different dimension. So this is W versus energy and the color is the density of states. Here's the semimetal phase and the hard gaps that squish down your density of states and here rho v scales like some coefficient times mod e. This coefficient goes like 1 over v squared and so as a result we find that rho prime diverges quadratically which gives a velocity that vanishes linearly at the critical point which is exactly the same power law as magic angle graphene which suggests these are actually the same universality class at the single particle level. And what's actually rather interesting which is going well beyond any continuum theory is that this is actually an intermediate metallic phase and not a single point. So if you linearize your direct cones and then you turn on this coupling between the two sheets of graphene you're going to miss this effect which is actually rather sharp because there's empirical evidence that this is actually a phase and not a point. So there's actually two transitions between the semimetal to a metal back to a semimetal. And so you can actually once you start realizing this is the same phenomena as magic angle graphene you can borrow the analytic methods used by McDonald for example so we have done that by integrating at momentum states outside my mini-zone we've gone to fourth order in perturbation theory and you look at this Gori expression but you can then basically at fixed q vanishes linearly in agreement with the numerical data. If you now put this all together so this is W versus q and the color is the density states at zero energy and the grad dash green line is this fourth order perturbative theory they agree remarkably well. So this is the semimetal here's a metallic phase here's a metallic phase so if you fix q you can have sequences of transitions the three, four transitions just like you have three, four, five magic angles. So these dashed lines are just showing two different cuts so here I'm showing in red the IPR momentum space in blue is the density of states so whenever the density of states becomes non-zero the IPR goes to zero. So indeed this is a delocalization of your momentum space wave functions this is a function of q I have like four or five transitions as a function of w I have two. And I'm in 2D so I can just directly look at my wave functions so what do these look like? They look like plain wave eigenstates in my metallic phase they develop this highly non-trivial structure as they've delocalized but they haven't delocalized across the entire brilliant zone which implies actually a rather interesting non-trivial structure and then you go back to the seminal phase which then relocalizes in case space with some satellite peaks so you can make this quantitative by borrowing methods from Anderson localization where you can define some sort of multifracticality of my momentum space wave functions and so this tau m of q plays a role of this multifractal exponent and the q dependence of this exponent tells you the nature of the wave functions so it turns out that these plain wave functions can be characterized as something called a frozen wave function in momentum space because this tau m vanishes at some finite q that's not zero if it was Anderson localized tau m would go to zero and q is equal to zero and b zero the whole way. In this metallic phase this develops a weak non-linearity in q which suggests these are weakly multifractal eigenstates in momentum space in 2D the quasi-periodicity is not strong enough to give you diffusion so in 3D we saw this was diffusive random matrix three level stats here in 2D quasi-periodicity is some sense that 2D is like the marginal dimension so it's not strong enough to give you bona fide diffusion. When you go back to the seminal these refreeze so you can think of this transition as some novel unfreezing transition in momentum space which goes right to this non-trivial structure of the wave functions at e equal to zero. Can we go down to 1D? To do that you have to think for a second because at d equal to one you no longer have a steady metal because you will find a density of states of the Fermi energy. In 1D what we can do is take an arbitrary power law and provided that this is less than one-half we do find that the density of states transition in 1D coincides with the delocalizational momentum space so here's my density of states this is the IPR momentum space there's nothing else to happen so 1D this is the real space IPR, 1D just localizes so it goes from the semi-metal to an Anderson insulator with the finite density of states and rather surprisingly we have a very generic result that the velocity vanishes like beta over d but for an arbitrary power law you have to generalize your effective dimensionality based on the scaling dimension which you can do rather concretely what we find is that beta is effectively 2 for a wide class of models that have Dirac points as well as this 3D-wild problem as well as this 1D model and there's a remarkable universality associated with these transitions where d just dictates the ratio of this exponent so what are the implications of this non-analytic density of states I've suggested that the velocity goes to zero but let's see this explicitly does this really give you flat bands and so to test for this since we don't have translational symmetry we're going to take our boundary conditions and we're going to twist them in the x-direction via twist and we're going to look at how our low energy eigenstates disperse as a function of the twist so now I'm sitting in the semi-metal phase so I have a little semi-metal miniband goes like mod e and indeed this is energy eigenvalues as a function of the twist I have Dirac nodes at the time reversal in Variable Menta which is well described by my twist dispersion I'm now going to my metallic phase and indeed the bands are incredibly flat so just look at the bandwidth changes by an order of magnitude and if I just draw a Fermi energy to equal to zero my bands are incredibly flat by going back to the semi-metal phase they go back to being linear with Dirac cones at the time reversal in Variable Menta okay so with that I can summarize that this transition survives in 1d, 2d and 3d and it generates flat bands in the metallic phase let me just do a direct comparison to theory on twisted bilayer graphene so this has been computed by Casciunato's group and here's the density of states the red lines are the untwisted density of states which is linear you see it opens up gaps with a semi-metal miniband at the magic angle you produce a large density of states with a Fermi energy here's our simple model we have linear density of states we open up gaps that are hard we have a semi-metal miniband at the metallic phase we generate a large density of states we can just put everything side by side so here's dispersions here's the corresponding density of states with the vanishing velocity here's our twist dispersions you can see these things look basically the same so this leads us to argue that this is the same phenomenon and we are basically arguing then that models for twisted bilayer graphene at the magic angle are actually sitting at a single particle quantum phase transition which is a universal phenomenon second of all, this opens up this phenomenon for a wide class of experimental systems because Dirac points in quasi-peric potentials can be emulated in cold atoms and trapped ions quite naturally so trapped ions is this power law hopping model in 1D I discussed and in 2D I already mentioned that you can do honeycomb optical lattices you can also do pi flux models with cold atoms in 2D so by realizing how generic this phenomenon is allows us to generalize it to other systems okay so this brings me to the last part of the talk where I want to convince you that I can construct effective Hubbard models and this actually gives rise to very strongly correlated systems I haven't actually shown that, I've just shown the velocity goes to zero because you also changes as you do this the 1A functions become spread out so you also could decrease so you have to be cautious so what do we want to do as I said we want to determine the renormalized interactions due to passing through this semi-metal transition by varying the quasi-peric potential so I'm going to add an on-site Hubbard U to my model the goal now is to take translationally in varying Hubbard models by focusing on commensurate models where all the quasi-peric systems I just told you about are like a supercell so I imagine you're tiling my whole space with supercells that have quasi-peric potentials inside them just make that a little bit more clear so I take a Q that has this ratio and I can just multiply it by some arbitrary integer and now I take my full system size to be this but the supercell is just this ratio so my supercell is going to be what I was just doing but I now have a full system and I have a continuum of momentum in this reduced brilliant zone let me just give you some intuition behind this physics so let me just start in the little lattice so I'm focusing on one supercell of size 144 squared so why are these hard gaps opening we can understand that quite clearly back to Alini's question so we have a large Q scattering Q is close to Pi and the direct nodes are separated by this is like 0,0 and this is like Pi,0 so these have opposite chirality so as a result this side of the cone and this side of the cone have the same spin they can mix and they can gap out and the states that lie within this gap have to be part of a mini brilliant zone of size Pi minus Q by Pi minus Q just because that's the size of what's left over after this this inter-node scattering process how do I check that? I can just do counting I have two spins, I have four cones this is the size of my new brilliant zone relative to the unit in momentum space and I get a analytic result which agrees remarkably well with just integrating the numerical data so indeed this is basically just dictated by inter-node scattering that's why I'm opening these gaps perturbatively so now let me zoom out to a full system where I'm now tiling my system with these supercells and I'm going to now do basically a 1A construction again following Marzari and Vanderbilt so the idea now is we're working with exact eigenstates so we can just form our projector onto this miniband and by projecting the position operator onto this miniband and simultaneously diagonalizing x and y which is a non-trivial thing to do you can then find your 1A states which are the eigenfunctions eigenfunctions these matrices will have very small off diagonal elements that you're trying to minimize to be zero numerically and the eigenvalues are my 1A centers and now I'm considering a full system with some big supercell actually I'm going to start with a small supercell to explain what's going on and so let me now think about this in the full system so I've taken now a rather large value of m this is energy versus w so this is like the spectra it's another way, we're not looking at the density states now we're looking at the direct spectra and these are hard gaps the fact that I have hard gaps allows me to find this projector in a well-defined manner and then I can then compute the 1A functions and so you see the 1A centers land precisely where the L-DOS which I've integrated with my miniband is bright so the brightest points of the L-DOS is exactly where my 1A centers land if I look at 1 1A function it's exponentially localized to these new lattice sites so I've indeed found exponentially localized 1A states which is provided because I have these gaps so following the Masari construction I can definitely find exponentially localized 1A states and then using these 1A states we can now compute the effective model, the Hubbard model properties but before doing that there's actually a finite size effect due to taking a small supercell which is this additional band here and we find is that in this gap if I form a projector oh sorry before I move on to that this is the dispersion from this projector and I have self-similarity so the model I get back from my 1A construction is the model I started from but with a very normalized T and U and so that self-similarity is rather remarkable and you can see that these three bands are completely self-similar this is the same model we had just shifted by finite chemical potential and that's an incredibly flat band at finite energy so these bands up here have already been distorted so they're no longer self-similar that's an evidence of some mobility edge coming in at finite energy which is a little bit harder to see because we don't have diffusion in 2D sorry now we move on to this, this is going to be a finite size effect but it's important to understand it so for a small supercell there's a region where there's an additional gap where you get hybridized bands and these hybridized bands actually have three 1A centers that make up my new unit cell but the really important thing is that this regime dramatically shrinks by increasing my supercell size so this is just basically a finite size effect by taking too small of a supercell let me just now present to you the last piece of evidence we have so from the 1A functions we now want to compute model parameters that are self-similar I want to focus on bands that give me back my original model with renormalized T and U and as I go to larger and larger supercells I can shrink this region so I only have to focus on this gap that can get all the way up to the transition and this is actually rather dramatic so this is U effective over T effective in units of the bare value and these are small supercells you get enhancements on the order of 10 but as we go to very large supercell you can see the enhancement becomes massive so as I pass through this metallic quantum phase transition at the single particle sector my Hubbard model becomes incredibly strongly correlated so strongly correlated it can be on the order of 3,000 times the bare value of U over T and with that I can conclude so I've shown you that quasi-periodicity stabilizes the same ultimatal quantum phase transition which can be extended down to two and even one dimension this quasi-periodicity generates flat bands through the same phenomenon that happens in twisted bilayer graphene and as a result we can construct effective Hubbard models that have dramatically enhanced interactions so thank you for your attention