much of the physics is driven by electron correlations; this has led to a number of surprises, particularly the discovery of superconductivity. It remains to be seen, he continued, whether these discoveries will lead to the same sort of technological revolution as the harnessing of semiconductor electronics did, but the work has already had a major impact on physics itself, as the materials offer a way to merge “two of the main themes of modern condensed matter physics”—unconventional superconductivity and the quantized Hall effect. So, essentially, he said, quantum Hall physics and unconventional superconductivity—two separate areas that have both been very influential in modern condensed matter physics—are being brought together in this new field of moiré quantum matter.

A number of questions naturally arise concerning these new materials, Vishwanath said. “We know the flat bands are good for getting surprising new physics,” he said. “Do we have a design principle for them?” Is it possible, for instance, to design for flat bands with a certain property? To be able to do that, he continued, it will be necessary to understand much better why the flat bands appear in the first place. Thus, he said, part of his presentation would describe recent work by his team “that has given us some insight into the flat bands at least in twisted bilayer graphene systems.” That insight has allowed his team to go beyond to the multilayers that Jarillo-Harrero had discussed and beyond the triangle geometry and using some basic principles to engineer flat bands. In the second part of his talk, he said, he would discuss a wish list of models his group would like to realize in a controlled fashion. And finally, he said, he would talk about some new models that arise from thinking about moiré physics.

UNDERSTANDING FLAT BANDS

Vishwanath began his discussion of design flat bands with a basic explanation of why flat bands appear in the magic-angle twisted bilayer graphene. “You have electrons that live in the two graphene sheets,” he said, and “occasionally the electrons will tunnel between the sheets. That’s predominantly how they see the moiré pattern.” The tunneling between the two sheets is described by a two-by-two matrix (see Figure 2-1, upper left), which is two by two because there are two sheets of graphene, and the matrix described tunneling from a particular sublattice on one sheet to a particular sublattice on the other sheet according to the parameters ω₀ and ω₁.

Early work by Bistritzer and McDonald assumed that the two tunneling parameters, ω₀ and ω₁, were the same—which is what would occur if the two sheets were perfectly rigid—and calculated the band structure shown in the lower left part of Figure 2-1. There is an extremely flat band—the moiré band—when the twist angle is the magic angle, but there are also a number of other energy bands that approach the flat band in places.

It was later realized that there are mechanical relaxations in the graphene

sheets, in which case the ω₀ parameter is smaller than the ω₁ parameter. “This has to do with mechanical stability and relaxation of the sheets,” Vishwanath said. When calculations are carried out with a smaller value for ω₀, the result is a band structure as show in the lower middle of Figure 2-1. The flat band still remains, while the other bands have pulled away from the flat band, resulting in an isolated low-energy manifold of states. It is the the filling and emptying of these bands that is being explored in many of the experiments that Jarillo-Herrero described, he said.

The question arises, then: Just why does this flat band arise. Does it exist because of some “very clever engineering” (i.e., because the twist angle is set to the magic angle), or is something deeper going on?

Vishwanath said that his group was able to answer this question by finding a further deformation of this model which has something called “chiral symmetry.” In essence, he said, this new version of the model switches off entirely the coupling between the same sublattice by setting ω₀ = 0. This new model has all the essential features of the twisted bilayer graphene except that now at the magic angle the band is perfectly flat (third energy band structure in Figure 2-1). “It has zero dispersion,” he said. “You can actually find analytic solutions for the wave functions.” The implication, he said, is that the flat band is clearly not the result of engineering. “This is something deep going on.”

This “chiral limit” model, as Vishwanath referred to it, is a very good starting point for understanding twisted bilayer graphene. Although he did not go into it any further, he commented that the zero-energy states are really zero modes, “things

that we are familiar with in condensed matter and topological phases,” and that they have some very special properties.

DESIGNING FLAT BANDS IN MOIRÉ SYSTEMS

With that foundation, Vishwanath next spoke about how to design flat bands in moiré systems, beginning with the sort of multilayer moiré systems Jarillo-Herrero had spoken about.

A group in Vishwanath’s lab led by postdoctoral fellow Eslam Khalaf explored the properties of multiple layers of graphene and, in particular, examined trilayer graphene where the top and bottom layers were aligned and the middle layer was twisted. Remarkably, Vishwanath said, what they found was that, with a particular twist angle, the electronic structure was equivalent to a magic-angle twisted graphene bilayer plus a single graphene layer, so the electronic structure contained a flat band plus a Dirac band, as Jarillo-Herrrero had explained in his talk. The group’s calculation also predicted the magic angle at which this band structure would appear, and that was approximately 1.56° (Khalaf et al. 2019). The following year, two groups, including one led by Jarillo-Herrero, created samples of this magic-angle twisted trilayer graphene and verified Khalaf’s predictions (Hao et al. 2021; Park et al. 2021; see also Wood 2021).

“We thought, given the close similarity to the bilayer graphene, something interesting should happen” with the magic-angle twisted trilayer graphene, Vishnawath said, and indeed something did. The two experimental papers reported not only that the material was an insulator under some conditions but also that under other conditions it was superconducting—and even more robust than the original bilayer system.

The group in Vishwanath’s lab also explored systems with more than three layers of graphene and calculated the magic angles for them as well. In the case of four layers, with the twists between the layers alternating first to the right, then to the left, then to the right again, the magic angle was approximately 1.78°. Whereas the ratio between the magic angle for the twisted trilayer graphene and the twisted bilayer graphene was , the ratio between the magic angle for the twisted four-layer graphene and the original bilayer material was the golden ratio, or ½(1+). It is interesting, Vishwanath said, that the golden ratio—known best for its appearance in classical architecture—should appear here. And some interesting phenomena should appear if and when an experimental group manages to create magic-angle twisted four-layer graphene.

Switching gears, Vishwanath then turned to the issue of how to design moiré systems that do not have the same triangular symmetry that is found in the graphene lattices. His group started from the insight that the flat bands in bilayer graphene appear because of a quantum interference effect. “It is not simply that

you’re trapping the electrons in little potential wells,” he said. “There is some quantum interference that prevents their propagation and gives you the flat bands.” The question, then, is whether it is possible to choreograph this quantum interference in a way that suits particular needs. “If you can do that,” he said, “you can try to design structures or platforms with the kinds of lattices that you might like to see.”

To explore how to design such structures, Vishwanath worked with Toshikaze Kariyado of Harvard and Japan’s National Institute for Materials Science, who had done theoretical work exploring flat bands in manufactured materials (Kariyado and Vishwanath 2019). What Kariyado had noticed, Vishwanath said, was that bilayer materials—graphene bilayers and others—were described by a particular Hamiltonian (see Figure 2-2, top), with two pieces of it (see Figure 2-2, upper left, lower right) describing interactions within the layers and the other two pieces (see Figure 2-2, upper right, lower left) describing interactions between the layers (i.e., tunneling).

There is a sort of “competition”between the two potentials in the Hamiltonian—ε and V—with the ε describing the electrostatics in each layer and V describing the interlayer tunneling. “The question,” Vishwanath said, “is can we control either one of them?” If so, it could be possible to balance them in such a way that it creates the flat bands seen in magic-angle twisted bilayer graphene.

It turns out that it is indeed possible to control interlayer tunneling—in particular, to shut the tunneling off altogether by shifting the lattices relative to each

other by half a lattice unit. “So there’s an interplay between symmetry and certain crystal momentum when the tunneling is shut off,” he said.

This phenomenon can be used in designing moiré lattices, Vishwanath said. There are two inputs to work with: the symmetry of the structure and the momentum location of the bands in the untwisted structure. When two planes are twisted relative to one another, there will be certain places where the stacking of the atoms in the two opposing planes is such that tunneling is canceled out in that region. “This is what we’re going to look for,” he said. “We’re going to use this to design moiré lattices.”

As an example Vishwanath should a twisted bilayer where the lattices in the two layers had rectangular symmetry. The areas where tunneling vanishes end up being quasi one-dimensional structures—long narrow parallel lines crossing the structure—along which where the electrons find it easiest to move, as opposed to moving perpendicular to those lines. “So here’s a way to get a quasi onedimensional system in a very tunable environment, and, of course, we know the interactions have a significant effect in one [dimension], so it will be very interesting to explore these structures further.”

In fact, he continued, it is possible to come up with a recipe for the different symmetry classes of twodimensional crystals and figure out what sort of lattice you would expect to get at the end of the day. “This should give us some guidance in the search for nontriangular moiré lattices,” he said.

That theory could be combined with, for instance, work by Mounet et al. (2018) which went through many known compounds in the material database, isolated those two-dimensional ones that could be easily exfoliated—shed from parent compounds—either mechanically or chemically, and then tabulated them according to their structures—cubic, hexagonal, monoclinic, etc. This would provide many different materials with different symmetry classes in the search for new types of two-dimensional materials that could be used to create quantum moiré materials.

USING MOIRÉ SYSTEMS AS A QUANTUM SIMULATOR

Next Vishwanath moved to the second part of his talk, which was about using moiré systems as a quantum simulator. As a recent review article detailed, there are good reasons to believe that twisted van der Waals heterostructures can be used as a platform for simulating various quantum phenomena of interest (Kennes et al. 2021). And recently, Vishwanath said, there has been some nice experimental progress that indicates that such quantum simulators may be quite workable. So, he said, he would describe a few of his favorite examples for such an approach.

The first is the issue of the so-called Mott transition, the quantum phase transition of a Mott insulator to a metal, which is “a major outstanding problem in

physics,” he said. The transition can happen in different settings. It can happen on an unfrustrated lattice, like a square lattice; that problem is of great interest in the cuprates. The problem on a frustrated lattice, like a triangular lattice, is of equal interest, and a recent paper using moiré materials to realize a highly controllable Hubbard model on a triangular lattice described a simulation of the Mott transition (Li et al. 2021). It represents “significant progress” on the problem, Vishwanath said.

A somewhat related problem is trying to realize the Hubbard model on a Kagome lattice rather than a triangular lattice. That would be extremely interesting, Vishwanath said, because the lattice has a high degree of frustration. “We still don’t quite know what the ground state is for the spin model,” he said, “and, beyond that, what happens when you dope electrons.” It is a huge open problem, he said, “and it would be wonderful to realize this.”

In fact, Vishwanath said, there is a recent nice paper by Allan MacDonald and his group in which the researchers used ab initio calculations to describe the band structure of a twisted transition metal dichalcogenide material such as WS₂. In the band structure of this moiré material, the first and second valence bands realized an s-orbital model and a p-orbital model, respectively, on a honeycomb lattice, while the third band realized a single-orbital model on the Kagome lattice. So this calculated third band of this moiré material might serve as a Kagome-lattice Hubbard model.

It would also be nice to have a quantum simulator of the square lattice Hubbard model, he continued, and there has been some progress with cold atoms, “which has its own strengths and weaknesses,” he said, and “it would be great to have that in the moiré context as well.”

Generally speaking, he said, there are strong reasons to try to realize these various models with moiré materials. In particular, they hope is that they will provide greater control is studying things like doping and correlationdriven transitions.

Furthermore, Vishwanath said, there are models other than these traditional ones that have been motivated by the microscopic models that appear in the moiré context. In particular, he described some ideas from a recent paper out of his laboratory about a new phase of matter called a chiral spin liquid that appears in moiré bilayers (Zhang et al. 2021). In moiré systems, he said, it is very natural to get higher symmetry—rather than just SU(2), for instance, one also gets SU(4) symmetry. “So we took a slightly more careful look at the phase diagram of an SU(4) model,” he said. They found that there is an additional parameter, a three-site ring exchange term involved in understanding the ground state of this model which is in some way related to the strength of interaction. At a certain range in the value of K, “you end up getting a novel phase of matter called a chiral spin liquid.” It is fairly robust, he said, with a large gap, “and we believe this naturally can occur in one of these SU4 models on the triangular lattice.”