The physics of “flat” electrons

Let's go for. Super 820. Thank you very much. And thank you all for sticking around. I'm Dumitru. I'm an LP fellow here at Oxford. And today I'd like to talk to you about something deceptively simple. What happens when electrons slow down dramatically? And the title of my talk is The Physics of Flat Electrons. And as you will see, this idea sits at the crossroads of, band theory, topology and strong correlations. And it leads us to some of the most exotic phases of quantum matter known today.

Before I start, however, let me make a disclaimer. I understand that the title is a bit of a clickbait. I'm not a flat electron ist. We haven't discovered the new fundamental particle. Flat electrons aren't exotic entities beyond the Standard Model. They're just the plane of the electron. Same charge, same spin. But they behave in very unusual ways inside certain materials. Now, this xkcd comic here makes a playful jab at the scientific hierarchy.

Physicists at the top, of course, but it misses something crucial. The idea that biology is just applied chemistry and so on breaks down. Once you realize that the key player is emergent behavior. As Phil Anderson shown here, famously put it, more is different. When electrons are confined to nearly flat bands, their interactions dominate and they give rise to collective phenomena that are completely invisible at the microscopic one electron level.

So yes, they're just electrons, but in the right environment they become the seeds of superconductivity, magnetism, charge, fractional ization, things that no individual electron can do on its own. So to understand what it means for electrons to be flat, we first need to talk about what it means for them to be dispersive.

So before diving into all the exotic physics, start with a quick refresher of bang theory how electrons move in a crystal, how energy bands emerge, and what we mean by dispersion. This will set the stage for understanding what it means to suppress that motion, and how that can fundamentally reshape the body landscape. So let's start from the free electron model. If we place an electron in a perfectly uniform background, as shown here on the left, there is no potential at all.

Its dispersion is just a plain, simple parabola, right? We know that the energy's h squared k squared over two m right t squared over to him. That's what we see on the left. Now when we move to a crystalline solid we break the continuous translation symmetry to discrete translation symmetry. Right. In a crystal. The momentum is only defined up to a reciprocal lattice vector. In this case it's two pi over A where a is the lattice constant.

This essentially falls back the dispersion of the free electron inside the so-called Brillouin zone. But of course, the actual crystal does have a potential with a discrete periodicity. And after we add the periodic potential as shown on the right hand side, we see gaps opening and an end bands electronic bands forming. So what do we get at the end as a band structure? Electrons still propagate, but now they do so with a modified dispersion that reflects the periodic environment.

This is the basis of the nearly free electron model. Still delocalized still momentum eigenstates, but dressed by the periodicity of the lattice. Now let's switch perspective and instead of thinking about three particles perturbed by a lattice, we can start from tightly bound electrons, say in atomic orbitals. Here I'm showing s p and the orbital levels. They're localized in discrete right. I'm showing them here on the left hand side.

The first step is to imagine copying this atomic orbitals across the lattice. Now we have one orbital per unit cell repeated periodically. That gives us a flat band structure. No hopping, no dispersion. We simply multiplied all these energy levels and add the momentum to them. Then when we allow this orbitals to hybridize, which means we allow the electrons to hop between them, then each level broadens into a band with a bandwidth controlled by the hopping amplitude.

This is the so-called tight binding model is the polar opposite of the nearly free electron case. Instead of starting from momentum eigenstates, we now start from a real state from real space orbitals and let them communicate. In the end, both routes lead us to the same concept of bands, which are just continuous energy levels, right? That the electrons occupy, which are labeled by crystalline momentum. Now let's talk about what happens when we account for electron electron interactions.

Electrons, of course, they repel one another. They carry the same charge. So we can't just take those bands and fill them with electrons as if they were non-interacting particles. Real electrons collide and they scatter and they exchange energy and momentum. But remarkably for many metals, at low enough temperature, the system behaves as if it were non-interacting. And this is the magic of land. That was Fermi liquid theory, one of the great triumphs of condensed matter physics.

What Landauer postulated and later articles of proved, is that even when electron electron interactions are present, the system can still be described in terms of long lived quasi particles. These are not bare electrons anymore. They're dressed by the interaction with the rest of the electronic see, and they require modified parameters such as modified mass and lifetime.

When I was at an undergrad in Cambridge, when I was an undergrad in Cambridge, someone explained to me the concept of quasi particles using this exact same diagram, saying the real particle is a horse and the quasi particle is a quasi horse, which at the time seemed just as helpful as it sounds.

But to their credit, it is a vivid way to see that the thing that you care about, the thing that you see, isn't the bare object, the bare electron is the bare object plus a cloud of particle hole excitation. Microscopically, this whole theory, this whole Landau Fermi liquid theory can be understood using many body perturbation theory, but critically, that only works when the electrons are dispersive enough such that the interactions stay in check.

Once that fails, the Fermi liquid theory breaks down. And that's a very good thing, because otherwise condensed matter physics would be a very boring subject. So now let's take this logic and flip it on its head. What happens when we're trying to partially fill a band that is flat? In that case, the kinetic energy, the dispersion becomes negligible. That means that this bandwidth w is much, much smaller than the interaction scale. Sometimes it's even much smaller than the energy gap.

This is the regime where Landau's theory completely breaks down. In a dispersive band, you could treat the interactions as a small perturbation on top of a dispersive band, but in a flat band there's no free motion. There is nothing to perturb against. Your unperturbed Hamiltonian is zero, right? So in fact, the situation here is reversed, right? Interactions dominate. Then kinetic energy becomes a small perturbation if it even exists at all.

So we can't start from a non-interacting solution and perturb our way out. We're forced forced to tackle the many body problem full on. Now these flat bands may feel like a hot topic now, but I've actually played quite an important role in condensed matter for decades. You could loosely trace some of these ideas back to heavy fermion materials in the 70s.

These aren't flat band systems in the strict sense, but they're often group in the same family because they exhibit narrow bands and strong electron correlation. The first truly flat bands in the in the in the sense of the term of today, appeared when Landau levels in the quantum Hall effect, both in the integer and fractional one were discovered.

These discovered discoveries were nothing short of revolutionary, because they each earned the Nobel Prize and showed that the interaction driven physics in flat bands can lead to entirely new quantum phases. The modern flat band renaissance came with twisted bilayer graphene in 2018. So if you have two layers of graphene that many of you might remember this from, last term's a morning of theoretical Physics.

If you have two layers of graphene and you rotate them at an angle which is close to the magic angle, then you can actually show that flat bands where interactions dominate emerge and you have superconductivity, correlated insulators, and more. Today, this platform is evolving. We now have fractional Chern insulators, which basically exhibits the fractional quantum Hall effect without an applied magnetic field. But there's a question which hasn't been answered until recently.

Can we get these flat bands in conversion of crystal ordinary crystals? No twisting, no stacking, no more magic, no magnetic field. This is what I'm going to try to answer in my talk. So to really motivate why flat bands are exciting, let's take a closer look at a system that recently reignited the field. Right. This is twisted bilayer of graphene. So you take two sheets of graphene.

You rotate, rotate them relative to one another at an angle of 1.1 degrees, and the resulting moiré patterns flattens the bat, flattens the bands near, charge neutrality. What you're seeing here on the left, I actually eight nearly flat degenerate flat bands. Right. Two for each value, two for each valley, two for each spin and two for each layer. They're clustered around the zero energy. Now here's the key. These gap, these black bands are gapless in the non-interacting model.

But if you try experimentally, if you try to field this bands and you fill an integer number of them, you see that the interaction effects kick in and the system develops and gaps open. These are so-called correlated insulator states labeled here in this schematic diagram on the on the upper right. Between these feelings the system can enter a superconducting phase. These are the blue SC domes which emerge entirely from interactions.

Since there is no phonon glue here, and the richness doesn't stop here actually. This is bilayer graphene has shown signatures of something called the Pomerantz effect, which means that with increasing temperature, you actually cause the system to order. This is somewhat, analogous to the permanent formation shock effect in, in, liquid helium. We're actually heating the system, makes, makes liquid helium freeze. So you have only one material, and the only knob is the carrier density.

And we see correlated insulator superconductors, strange metals, quantum dot behavior, all stemming from this flat pass. Now, to really understand the interplay between interactions and band structure, let's start with the Hubbard model. The Hubbard model is arguably the simplest model of interacting electrons on a lattice. We have fermionic creation and annihilation operators that I'm denoting here with CNC dagger. Right. And the number operator n. The Hamiltonian has two main terms.

The first one is a nearest neighbor hopping. So this one will take one electron at site R prime and hop it onto the site R, and we have an interaction term which takes the form of on site repulsion and penalizes double occupancy of a site. There's also a chemical potential term which controls the filling of the system. This model actually lets us smoothly tune between two extremes.

When T is one, or T is much larger than u, U is zero where in the non-interacting limit, as we tune just a little bit of on site repulsion, we enter into a Fermi liquid regime where perturbation theory applies. So all is well when we take T is equal to zero, then it turns out that we're in the flat band exact flat band regime. There's no dispersion. And with and interactions fully dominate. But and this is the punchline.

Just having a flat band isn't enough to guarantee interesting physics, as we will see now, both limits, both the weak, the non-interacting, and the exact flat band limit are exactly solvable, and neither captures the richness that we saw in twisted bilayer graphing. So let's begin with a very simple exercise and look at the non-interacting end of the spectrum, right? We set the order on site repulsion to zero.

In this case, the Hubbard model reduces to a textbook tight binding model that you might remember from undergrad. Right? Electrons hop between lattice sites with a certain amplitude T, and you can diagonalize this momentum. And you can diagonalize this Hamiltonian in momentum space. The ground state shown here is just a failed Fermi PSI. Right. So you simply take the electrons and occupy the lowest available momentum states up to the Fermi level. It's completely solvable and completely benign.

Now if you add interactions, but they're still weak yet non-zero when we're in familiar theory, we can apply many these perturbation theory and describe the system in terms of Landau quasi particles with renormalized parameters. This is the Fermi liquid regime that we talked about earlier. But now as we move towards stronger interactions, this description begins to fail. This the systems are to starts to develop correlations that can't really be captured by perturbation theory.

So now let's go into the opposite extreme the flat band limit, and see what's happening there. On the opposite end of the axis we have the flat band limit. We set the nearest neighbor hopping to zero. In this case, hopping is completely switched off. The system breaks up into isolated lattice sites, and there's no communication between them. So the full many body problem really reduces to solving a single site Hamiltonian. Each site has four possible states I can put.

I can have an empty state, a spin up state, spin down state, or both. Electrons, right? Doubly occupied state. The energy levels are trivial to compute, and really there's no emergent behavior. It's just it's just a four level system. And this is really the key message. A flat band on its own isn't enough, even if you have strong interaction, if the system has no structure. And I'll explain exactly what that structure means. It's basically topology. There's no connectivity, right?

You don't get anything interesting. So if you want to explain the richness of a system like twisted bilayer graphene, we need something more. And the missing ingredient is actually non-trivial band topology. And this leads me to the first proper part of my talk. Right. We've seen that neither strong interactions note nor flat bands alone are enough to explain the richness we observe in systems like twisted bilayer graphene.

So the structure that is missing is a way to characterize how bands are built and connected beyond just their energy, right? This is exactly where topological quantum chemistry comes in. It provides a systematic way to understand band structures in terms of symmetries and orbital content, telling us exactly what a band is made of and what it's allowed to do right.

In this next section, I'll introduce some of the key ideas of thick topological quantum chemistry and show how they help us navigate the space of possible flat bands, especially those that are topological. Now, depending on your background, you might look at a material like graphene from two very different angles. If you think like a chemist, you will think about graphene. In real space you have orbitals, you have bonds, you have hybridization.

But if you're a physicist, you are drawn to this spaghetti like plots which are in reciprocal space, right? So you introduce concepts such as Brillouin zones, dispersions, bands crossing and so on. But there seems to be some kind of a conceptual gap, right? On the left we have localized atomic orbitals. Right. Which feel very intuitive. But on the right we have a band structure which is completely delocalized. It's a momentum space object.

Now if we want to understand topology, this relationship between the real space and the reciprocal space is absolutely crucial. Topology, in a strict sense, is a global property of bands in momentum space. Things like Chern numbers, which require that you integrate right across the entire brain. One zone. But this but these topological features, and this is the key idea, really must have a fingerprint in real space because electrons really live in orbitals.

So what we need is a formalism that connects the real space orbital context to momentum space topology. And this is exactly what topological quantum chemistry does. So far we've been I think we've been talking about band structures. Right. In terms of eigenvalues of certain Hamiltonians. Right. But to really get the connection between real space orbital symmetry and topology, it would be actually helpful to model out the energetics, right? In other words, let's ask the following question.

Can we talk about band structure without ever writing down a Hamiltonian? And the answer is yes, of course. And the way you do that is you lean on symmetry because crystal in materials are defined by a symmetry group, right? Something called the space group which encodes all the allowed symmetry operation. Right. You have translation rotations reflection and and such and such symmetry operation. Right. Given a space group you need to add only two more ingredients.

Right. To fully characterize the crystal the position of atoms within unit cell right. And the orbitals that each atoms contribute to the band structure. Right. With just these three ingredients space group, atomic position, and orbital character, you can ask what are the possible types of band structures that are allowed by symmetry, right? What are the positive? What are the possible band structures that I can get in a real crystal?

And this is the central question of topological quantum chemistry. And the answer comes in the form of something called elementary band representations. These are symmetry based building blocks of all electronic band structures. So let's let's build some intuitive picture. Right. You fixed the space group, right? Which are the symmetries of a of a crystal. But you are still not done because there are actually there's a lot of freedom of on how you place atoms within units. Right.

And that freedom matters. Take this example. All of these three structures here share the same gravity lattice. They're built on a triangular lattice. Right. But depending on where you put the atoms you can end up with a triangular lattice, a honeycomb lattice or a Kagami lattice, each which each with different numbers of sides or occupied sides per unit cell and different symmetry properties.

And the key is that in all of these cases, the atoms really are placed in ways that respect the space group symmetry of the lattice. But this specific arrangement, right. What we call the the the weak of position, will determine how the orbitals transform under the symmetry, and therefore how the bands connect in momentum space. So any theory of elementary band representations has to take this into account. You don't just need the lattice and the orbitals.

You need to know where these orbitals sit. Once you fixed the lattice, once you fixed the atomic position, you still have another important detail, right? What orbitals you put because different orbitals transform differently under the symmetry operation, so they generate different band structures right on. In both cases shown here, I've chosen my occupied lattice sites to be to form a honeycomb lattice. Right on the left we have s orbitals right.

And this should give you something graphene like right with Dirac cones and symmetry and force and force crossings. However, on the right hand we play place p and p orbitals instead. They have very different symmetry properties, right? They point in plane, they will hybridize differently and the resulting bands will be behave very differently. So full theory of band representation has to consider not just where the orbitals sit, but also what kind of orbitals you have right?

The symmetry of the lattice, combined with the symmetry of the orbitals, determines the representation content of the band. And this is how topological quantum chemistry builds all of the bands without Hamiltonian, so not width of Hamiltonian, but from symmetry.

Ingredients. Right. So now once I've introduced all these ingredients, it's it's time to tell you what the I, the main idea of topological quantum chemistry is we want to classify all possible band structures that could arise from electron sitting in a real, in real space orbitals, right, that obey crystalline symmetric symmetries. And we want to do that without ever invoking a Hamiltonian. So the way we do that is we think of bands as being representations of space group. Right.

And these representation are these elementary band representations. So cut a long story short, what people have done and only recently in 2017, they've taken all the possible space groups, all the possible ways of arranging orbitals in them, and they've carefully tabulated every single band that can arise in such a system. Right? Every single band that can arise from localized real space orbitals, right.

And then they said, well, we know all the all the bands that have a localized real space description, and we also know all the possible bands. Right? Because we know how bands should connected momentum space. Right. And they reach the following conclusion. All sets of bands that are not induced from symmetric and localized orbitals are topologically non-trivial by design. So of course you can read a lot about topology, right? You have you have rather complicated topics such as, Chern numbers.

Right. Invariance in momentum space. But this gives you a very simple intuitive explanation. So let let me visualize that this diagram shows the the landscape of all possible band structures that are allowed by symmetry. That is, all bands that satisfy the connection, the compatibility relations in momentum space. Within this full space we have a distinguished small subset. These elementary band representation.

And these are band structures that arise from symmetric localized orbitals in real space, in other words, atomic elements. And here's the punchline not all symmetry allowed bands come from a localized real space. Description. By definition, such bands are topologically non-trivial, right? And this diagram captures the the logic of topological quantum chemistry. Let us have a very simple example. Right. Suppose you have a group of bands, right?

And you want to ask are they atomic or are they topologically right? The idea of topological quantum chemistry is to try to express these bands of interest. Right. Which are the field bands in all these cases is linear combinations of elementary band representation. Right on the left you have a band structure. There's just a single elementary band representation that's atomic. There's no surprise there, right? It's induced from localized symmetric orbitals, and it has an atomic limit.

In the middle, you have a case where the bands decompose into a fraction, like a rational non integer combination of elementary band representation. Right. By definition, in such a case your bands are topological. They do not admit, a localized real space description. Right. And of course you can have other situation where the bands are actually an integer linear combination, but one of the coefficients is negative. I will not go into the details, but those are also topologically non-trivial.

So the main takeaway of that section was that a band is topological if it doesn't have an X financially localized real space. Description. Right. Armed with this information, we're finally in a position to talk about the central question that I posed at the start of the talk. Can we realize flat bands, ideally with non-trivial topology in conventional crystalline materials? So until now, we've seen that flat bands really emerged in either engineered platforms. Right.

Your your or with strong magnetic fields. So we want to replicate the same type of physics in ordinary honest crystals. Right. With materials with no fine tuning, no mooring, no external magnetic field, none of that. Okay, so why why do we want topological flat bands? Right. I gave you this example with a simple Hubbard model. But but why do we really why do we really need the topological aspects? This is because not all flat bands are created equal. This is where the topology comes into play.

If your flat band is trivial, as it happens here on the on the left hand side right. If you Fourier transform this flat band, you get exponentially localized real space orbitals. If you try to write down an interacting theory for that, for those for that flat band, what you end up is essentially a term, a local term that looks like the the Hubbard interaction term we saw earlier. So this is so there is nothing special about this. Right.

You can still you can decouple the sides and then you can you can solve each side separately. And there's simply nothing special about about these bands. In contrast, if you have a topological flat band then you really cannot build this localized real space description either. You can't have them, either you can't have the orbitals be symmetric or in most cases, you can't have them be exponentially localized.

So when you build an interacting theory for orbitals such as these, for a topologically flat band, you find that the interacting Hamiltonian is highly non-local, even though there's no kinetic energy, the projected interaction is non-local and the physics is far richer. And this is exactly why topology matters. As I showing this schematic here on on the in the bottom right right.

The most exotic phases in condensed matter tend to emerge precisely where this nontrivial band topology meets strong correlation. And this is really the target zone. The upper right quadrant here of this phase diagram where both interactions and topology are strong. And if you analyze twisted bilayer graphene, if you analyze the the flat bands of twisted bilayer graphene, you will find that it lives exactly in this in this top right quadrant.

Right now we're trying to replicate the same thing in conventional crystals. How do we how do we even start right. Let's now look at, at the concrete example, the Kagami lattice. Right. This is this is actually named after a traditional Japanese basket weaving pattern. Right? I'm showing here on the on the bottom, left, right. And at first glance and if you remember the very beginning of the talk, you might see you might think, well, why not just have independent patterns?

Because in the independent patterns are by definition a flat band. But then those would be by definition topologically trivial. So uninteresting. Right. So we want to build flat bands in some sort of a dynamic way from quantum interference. Right in the Kagami lattice that I'm showing here, you can write something called a compact localized state. Right. And I'm showing this here in red, right. This is a linear combination of orbitals with alternating signs around the hexagon.

Now here's the here's the key. Because of the lattice geometry, any tunneling outside of this hexagon exactly cancels, right? The destructive interference is perfect. So this wave function is an exact eigenstate that I can write down for every hexagon. Every hexagonal packet of this kagami, of this kagami lattice. Right. So if you go ahead and, and Fourier transform and do the proper computation on the right hand side, you'll find that you get a flat band right here, here at energy equal to two.

I'm working in in arbitrary units where t is equal to one. Right. But actually what's what's even more interesting is if you look at this band structure on the right, you see that the flat band actually touches the dispersive bands in one single point in the brilliant zone. This touching turns out to be protected by symmetry, and you can't gap it without either breaking the symmetries of the lattice or by making the flat band topological.

And this is what I really mean by this flat bands not admitting exponentially localized symmetric quanta orbitals. At this point, you might be asking yourself, well, okay, this is this is all interesting, but is it really useful for anything? Right. And the answer is of course the affirmative. Right. I'm highlighting here a very recent paper right from last year where they synthesized one of these Kagome metals. So even though this is a 3D material, it has many active orbitals.

It has chromium, atoms, cesium atoms, antimony p orbitals. You can still see here on the right hand side a clear kagome lattice. And if you squint at the band structure here, you can sort of see near the Fermi level, you have a flat band, right? And in the band structure as a whole seems to have signatures of the Dirac band structure. Right. So we had we had this, this Dirac cone here. We also have it here we have the flat bands.

Now what's especially interesting about this compound is what happens under pressure. Right. As shown here on the phase diagram on the right. Right. The system exhibits very robust superconductivity as I apply, as I apply pressure directly above this superconducting phase. The system is in a non Fermi liquid state. Now how do I know it's a non Fermi liquid. Well experiment is can actually measure the resistance of of a sample. Right.

And if the resistance if the system is a Fermi liquid then then the resistance should various t squared right where T is the temperature. Right. So whenever you see a different exponent. And this is really what's being plotted here, you know that is a non Fermi liquid right. So this is very interesting. Very likely superconductivity here does not come from from phonons. Right. It's actually unconventional in nature and probably akin akin to one in high temperature superconductors.

So this is a very beautiful example of how flat band physics, topology and correlations all come together in real materials, not just toy models. Okay, but the this Kagami example that I gave is beautiful, is simple to understand, simple to explain, but can actually feel a little bit mysterious because I seem to just I seem to have just pulled this cargo lattice out of my pocket. Right?

So to really understand where flat bands can come from, let me start with a very simple example and then generalize that to the lattices. So take a very simple planar molecule. Right. Three outer atoms shown in cyan surrounding a central one shown in orange. All right. Now let's say that I want to build these are the atomic orbitals of the molecule right s orbitals for each atom, right s orbitals for these triangular atoms, and one s orbital for the central atoms.

Let's say that I want to build the molecular orbitals. Right. How would I do that? Well I would use concepts from group theory. Right. I can I can basically take these three atoms, these three orbitals from the exterior, make a symmetric, a c, three symmetric or a threefold rotation symmetric combination of them, and allow those that symmetric combination to to be combined with the s orbital on the central atom. Right. So these will combine right and they will split right.

Because this is this is the avoided crossing row. Right. On the other hand these orbitals here I started with three orbitals. I must have three after I hybridize them. These hybridized orbitals here that I'm denoting with e prime, which is just notation from, from group theory. They really don't have the symmetry to hybridize with the central. And so they will just stay here at zero energy. Now if you take this very simple picture put it on a lattice.

It turns out that this gives you the most general way of building flat bands in crystalline materials. And this is something called the Li lattice construction. So in crystal, in flat band materials, flat bands often appear in systems called bipartite lattices. So here's the the idea is it's actually deceptively simple. Consider two sub lattices which I'm going to call L and l tilde. And they each contain n l and nll tilde atoms.

Right. Crucially, I will assume there's all there isn't any hopping within the same sub lattice. So all the hopping the electrons can only hope from one sub lattice to the other. Right? Then a two line mathematical proof will tell you that if you have an imbalance in the number of atoms in the two sub lattices, you are guaranteed to get zero modes, zero modes which actually become flat bands. Once you Fourier transform. And this is exactly what we saw in this very simple picture, right?

We had three atoms in one of our sub lattice, one atom in the other. So we got three minus 2A3 minus one two zero amounts. Right. It's the same game but played on a lattice. Now, what we did in in a recent paper actually was to generalize this, and, and show some, some mathematical result which are very similar in spirit to, topological quantum chemistry.

What we essentially showed that it doesn't matter how you couple these two sub lattices, the topological properties of the bands are only dependent on the atoms of the two sub lattices. Right. And actually, after our study, a different team use the same concepts on all known materials. Right. And identified 7000. That's seven times ten to the three materials, right, that have flat bands and that you can actually synthesize in a lab. Right.

And actually, you can even show that the Kagami model I showed previously is actually included within this formalism. And, and I'm showing here in this slide why that is the case. If you take the Kagami lattice and call that your big sub lattice, right. And, and put in a virtual honeycomb lattice, you can see that the same wave function for the flat band in the Kagami case is a wave function of the flat band in the bipartite crystalline lattice case, and the reasoning is very similar, right?

But now with this sort of bipartite formulation, I can get all these results without ever invoking any sort of hopping hopping whatsoever. In this example here, I'm actually building two toy models of flat bands. Right? So in both cases I take my big sub lattice to be made up of s orbitals on a kagami lattice. Right. These black dots. And then I vary the the contents of the small subplots. To cut a long story short, on the left hand side I can go ahead.

Use topological quantum chemistry once again, not do a single calculation and show that the flat band must be topologically non-trivial and on the right hand side, using the same algorithmic prescription, I can do the same and show that the flat band has a symmetry protected gapless point at the gamma point, right? So it touches the dispersive band at the gamma point. And this once again, this is not just, this is not just just a mathematical toy model.

You can actually go ahead and apply this to real crystals. And, and we we've actually done that. I'm highlighting here one of the compounds that shows a flat band with a band touch point that can be diagnosed diagnose with, with this type of prescription. Actually if you want, you can go ahead and scan this QR code and it points to an online database of flat bands. And in your spare time, you might want to synthesize one of those compounds.

My personal recommendation is once you go to the website, you should go to this best flat bands, section. Because these have the cleanest, the cleanest flat bands and actually one of the one another recent experimental paper, this time from MIT, they took one of the compounds that was highlighted in this database. The compound looks something that, you would never write in a chemistry exam is calcium nickel two. Right.

And the interesting thing about this, this compound is that it actually has flat bands close to the Fermi level. Right? So there are here on on in panel A, I'm showing the result of our pitch. That's that's angle resolved, for the emission spectroscopy. Right. And this essentially probes the band structure of an actual crystal. Right. And you can see here that they've identified this flattish bands. Right. And they they even compare them to theory.

Now, if you have these flat bands quite far from the Fermi energy, then nothing really happens. But if you take the compound, you dope it. Doping means that you add different atoms with different number of electrons. You see that exactly when the flat band touches the Fermi energy, you actually get superconductivity developing, just like we did in twisted bilayer graphene. Now, I admit that six Kelvin, which is the critical temperature of this superconductor, is still not not enough.

But considering the time scale right on which on which these experiments happen, I would say that it's it's quite impressive. And before finishing, I just want to say that, maybe discarding topologically trivial bands is not, they should not be discarded straight away.

In fact, actually, in a recent work, we've shown that there's, new, fundamentally different, universality class of moiré materials in which the moral potential acts essentially as a magnetic field and realize is a very interesting algebra of the mirror and translation operators. Essentially, after you go through all these maths, you are able to show that this flat bends in twisted thin selenite, right? They have a very interesting quasi one dimensional property.

So even though the system is is two dimensional, it's actually made up of one dimensional chains. Electrons cannot hop between chains, right? But they still interact in electrostatics between chains. So this this actually paves the road to the, for example, realizing something called a Latin jr liquid. Right. And and really studying like low dimensional physics in, in real experiments. So with that I'm going to leave you with conclusions and only point one more thing.

The fact that because we know flat band compound compounds don't obey Fermi liquid theory, right? We need to come up with, with new tools on how to solve them. And one of one of one possible such new tool is actually going to be highlighted by the next talk. So make sure you stick for Dominique Stock. Thank you. So you're asking up. Yeah. Yeah. You're asking outside the perturbative regime if there are any dualities. Dualities between. Between what? Between something like this.

Well, I know that people are looking into that. I don't know specifically how relevant those are for real materials, you know, but it's it's definitely a very interesting, interesting avenue to to pursue. Yeah. But off the top of my head, I don't know of any such duality relevant for real materials. It's a it's a constantly evolving field. Right. So what isn't around today could very well be around tomorrow.

Yeah. So so the question is really about how about the experimental difficulties of actually realizing these platforms. Well, I must say that, well, there are problems with, with growing crystalline materials that experimentalists have, creatively, managed to tackle. There's still a problem of actually doping the materials, because maybe you get a flat band material, but the flat band is far from the Fermi energy. You need to dope it, and people just have to be creative.

They have to try a lot of things, because sometimes you just add the dopant and the dopant doesn't diffuse into the system, right? It just stays in one place and it's actually an impurity rather than a dopant. Right. So it's it's it's not really it's not really. But there aren't 3D recipes for that. And this is why, compared to, to engineered hetero structures.

Well, you can literally peel sheets of graphene, stack them on top of one another, put them in a dual gated setup, and just play with the electric field to dope them. For these type of conventional crystals, there is a lot more work involved. And that's why that's why results are probably, experimental results are probably appearing at a much slower pace. But people are working and people are getting more and more creative.

And, now probably sort of catalyzed by twisted bilayer graphene, they are understand that this is something worthwhile. And so so they're dedicating more time to it. And yes. So okay, so the the question is, where does where do the energetics come into play? And for that, people have, have developed computational tools which for the vast majority of materials are remarkably good. And I'm speaking, of course, about, density functional theory. Right. So it's these I'm not the DFT, the person.

But the way I understand it, I understand it is that, the programs are so powerful, you're just feeding some atom position and telling them what type of atoms you have and the distances. Then they they produce a band structure. Oftentimes, it can happen that the band structure is not accurate, especially if you have strong correlations. So then you actually need to you actually need to to get creative right with, with post-processing this DFT calculations. Right.

Sometimes sometimes the, the way you do that is while you rely on some experimental input. Right. And you, you try to work out what kind of approximation can make your, energetics agree with, with experimental measurements. Right. But there are tools for, for, finding out what the energetics are. The problem is that, you know, oftentimes this these tools are a bit like black boxes. You can't just you can't just randomly run, right, some, some atomic arrangement and hope that it works.

And this is where the topology insight comes into play, right. Because it tells you, okay, out of the myriad of, out of the infinite possibilities you could do, here's what is really worthwhile. Yeah, yeah. So the question the question is, have people if people thought about the mixed, the mixed momentum space, real space, the present day or formulation of quantum mechanics, and try to apply that in the context of topological band theory.

So the very honest answer to that is not, to my knowledge, and quite frankly, the only interaction I had with this mixed representation was during my third year of undergrad. So there's quite a there's quite some time from, from then, but, it might be something worthwhile, especially for non-interacting band theory. It might, it might actually it might turn out to be insightful. So maybe, maybe people will look at that. Yeah. But to my knowledge, I'm not aware of anything.

Well, actually. Okay. So that's that's a very interesting observation. And it's, it's related to something that we've done recently. It turns out that some of these topological flat bands. Right. You can't really you can't really, trivialize them. But you can do a basis transformation and actually convert them into a heavy fermion model. So there is a duality between some of these flat bands and, and heavy fermions, as a matter of fact, twisted bilayer graphene.

You can do, an honest to God unitary transformation on the band structure and map that to a topological heavy fermion. So then you can actually use a lot of the techniques techniques from there. Now, in for this specific experiment. Right. Obviously there are no electrons in this, in this compound. So it's not a heavy fermion, but it could very well be that you can take some of these, you can you can take some of the momentum states, right.

Do some band inversions and, and construct f like, electron state for the flat band and then re express the same sort of physics in the language of heavy fermions. But that's definitely something possible. And people have have looked into that into great detail. And thank you. Okay. I think we have to move on to that specter, which we again.