Mathematical Aspects of 2D Quantum Materials and Meta-materials

The recent observation of fractional Chern insulators—lattice analogs of fractional quantum Hall phases realized in the absence of an external magnetic field—in two‑dimensional moiré materials opens a new avenue for the study and control of anyons, their elementary quasiparticle excitations. These platforms offer two qualitatively new opportunities: stabilizing itinerant phases of anyons and deterministically trapping individual anyons. First, I will discuss the dynamics of anyons in the absence of a magnetic field. In conventional fractional quantum Hall systems, the kinetic energy of electrically charged anyons is quenched, and they tend to localize at defects or impurities. In fractional Chern insulators, by contrast, anyons can acquire a dispersion and form itinerant phases. I present a concrete example in which a non‑uniform Berry curvature generates a nontrivial band structure for anyons, leading to dispersive anyonic excitations. If time allows, I will briefly outline possible emergent phases of mobile anyons, including the prospect of anyon superconductivity in a concrete microscopic model. Finally, I will demonstrate how gate engineering in moiré materials enables the controlled trapping of anyons at user‑defined locations. Using a microscopic model, I show that anyons can be localized away from defects or impurities, providing a route toward the programmable manipulation of anyons in solid‑state platforms.

The Thouless theory of quantum pumps establishes quantized transport per cycle and determines the conditions for that. When the description is shifted to a moving frame, as prompted by recent experiments on ultracold gases, transported and existing charges mix. Their transformation is encoded in Galilean space and time, but underlying it is one of vector bundles that may be described in a number of ways, that may or may not rely on Bloch theory. In one of them, the transformation mixes strong and weak topological indices of a bundle on a 2-torus; in another one, a 3-torus is at center stage. As a further alternative, the covariance is placed in the context of the Landauer-Büttiker formalism. (Joint work with T. Esslinger and F. Santi)

This talk is based on joint work with David Gontier and Thaddeus Roussigné. We consider a tight-binding model for graphene, where distortion of the honeycomb lattice is allowed, but penalized by a quadratic energy. We prove that the optimal 3-periodic lattice configuration has Kekulé O-type symmetry, and that for a sufficiently small elasticity parameter, the minimizer is not translation-invariant. Conversely, we prove that for a large elasticity parameter the translation-invariant configuration is the unique energy minimizer. We also investigate a related continuum model involving the Dirac operator.

In this work, we study wave propagation in two-dimensional honeycomb structures with a non-commensurate line defect or edge. Our model is a Schrödinger operator which interpolates, across the edge, between two distinct bulk (asymptotic) Hamiltonians with a common spectral gap about the Dirac point of an unperturbed honeycomb operator. We seek edge states, eigenstates that are bounded and oscillatory parallel to the edge, and decaying in the transverse direction. For non-commensurate edges, the rigorous definition of these states is nontrivial due to the lack of translation invariance along the edge. To address this, we exploit quasiperiodicity along the edge by expressing the Hamiltonian as the restriction of a 3D (degenerate elliptic) Hamiltonian describing a 3D medium with a 2D interface within which there is periodicity. Using perturbation theory, we construct edge states in this 3D setting and obtain by restriction 2D edge states which are quasiperiodic along the irrational edge. These edge states are seeded by eigenfunctions of an effective Dirac operator, which has an infinite block-diagonal structure due to the non-commensurate geometry. A consequence is that infinitely many edge state eigenpairs arise, whose energies are dense in the perturbed bulk spectral gap. The construction of the edge states requires an omnidirectional non-resonance (no-fold) condition on the dispersion functions of the unperturbed honeycomb Hamiltonian, as well as a Diophantine condition on the edge slope.

The rise of moiré materials has led to experimental realizations of integer and fractional Chern insulators in small or vanishing magnetic fields. At the same time, a set of minimal conditions sufficient to guarantee a Abelian fractional state in a flat band were identified, namely "ideal" or "vortexable" quantum geometry. Such vortexable bands share essential features with the lowest Landau level, while excluding the need for more fine-tuned aspects such as flat Berry curvature. A natural and important generalization is to ask if such conditions can be extended to capture the quantum geometry of higher Landau levels, particularly the first (1LL), where non-Abelian states at are known to be competitive. The possibility of realizing these states at zero magnetic field , and perhaps even more exotic ones, could become a reality if we could identify the essential structure of the 1LL in Chern bands. In this talk I will introduce a precise definition of "higher vortexability", which is a vast generalization of the structure of the 1LL. I will demonstrate concrete models of higher vortexability based on generalizations of Bernal graphene. I will show that higher vortexability can lead to order-of-magnitude enhancements in the neutral gap of fractionalized states. Time-permitting, I will discuss aspects of the classification of vortexable bands.

When two monolayer materials are stacked with a relative twist, an emergent moiré translation symmetry arises, giving rise to electronic structures and correlation effects that are qualitatively distinct from those of the individual layers. Moiré materials have therefore become highly tunable platforms for exploring strongly correlated quantum matter. To date, however, most studies have focused on monolayers with triangular lattices whose low-energy electronic states reside near the Γ or K points of the Brillouin zone. In this talk, I introduce a new class of moiré systems based on monolayers with triangular lattices but low-energy states at the M points of the Brillouin zone. These so-called M-point moiré materials host three time-reversal-preserving valleys related by threefold rotational symmetry. I propose twisted bilayers of exfoliable 1T-SnSe2 and 1T-ZrS2 as experimentally viable realizations. Using extensive ab initio calculations, I identify twist angles that yield isolated flat conduction bands, construct accurate continuum models, and analyze their topology, charge density profiles, and emergent symmetries. A defining feature of M-point moiré Hamiltonians is the emergence of momentum-space non-symmorphic symmetries. In particular, a non-symmorphic in-plane mirror symmetry renders certain materials effectively quasi-one-dimensional within each valley, suggesting a natural route toward realizing Luttinger-liquid physics in a two-dimensional moiré platform. I further explore the strong-coupling limit of twisted SnSe2, where analytical solutions can be obtained at integer electron fillings and include dimerized phases with finite residual entropy, valence-bond solids, and quantum paramagnets. Finally, I show that the same non-symmorphic in-plane mirror symmetry guarantees the absence of a sign problem in Quantum Monte Carlo simulations at any filling, enabling an unbiased investigation of interaction effects. Over broad and experimentally realistic ranges of twist angle and interaction strength, these systems exhibit a rich phase diagram featuring correlated insulators at integer fillings—whose nature and strength depend sensitively on the twist angle—as well as Wigner–Mott insulating phases at selected commensurate fillings.

Photonic crystals -- transparent periodic dielectric structures -- have been called "semiconductors for light" in the sense that photons propagate through them much as electrons propagate through crystalline solids. In the first part of my talk, I will show how straining a photonic crystal can make photons behave as though they were electrons immersed in a magnetic field, leading to Landau levels for light. In the second part, I will employ photonic crystals to explore the scattering properties of highly spatially correlated, so-called "stealthy-hyperuniform," disorder.

Topological photonics and phononics are rapidly developing fields that apply the principles of topological insulators from condensed matter physics into periodic optical and acoustic media. A central challenge in these systems is to understand how in-gap interface modes emerge at the boundary between two bulk media in distinct topological phases. Such modes are topologically protected and remain robust under a broad class of perturbations. In this talk, we review recent progress in elucidating the mechanisms that give rise to in-gap interface modes across various topological photonic and phononic structures.

I will give an overview of the mathematics of quantum many-body systems. The main difference with single-particle systems is that many-body Hamiltonians are not short-range (their kernels are not band matrices in the position basis). As a consequence, functions of local Hamiltonians are no longer local. Instead, people have relied on propagation bounds and mean-field theories to obtain spectral information. I will detail a few classical results and give some new applications.

I will formally discuss the dispersion of waves arising from charge density oscillations near a fixed plane in 3D at zero temperature from a Partial-Differential-Equation (PDE) perspective. The goal is to describe the interplay of microscopic scales that include a binding length in the emergence of the surface plasmon (SP), a collective low-energy charge excitation in the vicinity of the plane. The starting model is a time-dependent Hartree-type PDE in 3D that aims to provide a mean-field description of a confined interacting-particle quantum system. Two main ingredients are the repulsive Coulomb interaction, and an idealized binding external potential in the vertical (z-) direction. The linearization of this equation around the ground state yields a homogeneous integral equation for a suitably defined scattering amplitude. The existence of nontrivial solutions implies a dispersion relation, which non-linearly connects the temporal frequency and the wave number of charge oscillations near the plane. This relation is obtained exactly in closed form by a transform technique. If time permits, I will discuss the effect of a periodic potential, and aspects of the semiclassical limit.

The last couple of decades witnessed tremendous experimental progress in the study of "Floquet media," crystalline materials whose properties are changed by applying a time-periodic parametric forcing. The theory of Floquet media has so far been restricted to discrete models, which are often heuristic and approximate. Understanding these materials from their underlying PDE models, such as the Schrödinger equation, remains an open problem.Specifically, semi-metals such as graphene are known to transform into "Floquet Insulators" under such periodic driving. While traditionally this phenomenon is modeled by a spectral gap, in PDE models no such gaps are conjectured to form. How do we reconcile these seemingly contradictory statements? We prove the existence of an “effective gap” – a novel and physically-relevant notion which generalizes a (proper) spectral gap. Adopting a broader perspective, we will then survey newer results on dispersion and spectral near-invariance in bulk Floquet materials.

Mechanical relaxation and phonons are known to play important roles in the remarkable properties of moire materials such as twisted bilayer graphene. I will present atomic-scale mathematical formulations of relaxation and phonons in terms of interatomic potentials and then prove the convergence of these formulations to commonly-used continuum PDE models. The formulation allows for rigorous error estimates and systematic calculation of corrections.

The index of a pair of projections on a Hilbert space was introduced in 1973 by Brown-Douglas-Filmore and connected to the integer quantum Hall effect by Avron-Seiler-Simon in 1994. For two orthogonal projections P,Q such that P-Q is compact, index(P,Q)=dimimP∩kerQ-dimimQ∩kerP. It is manifestly an integer, and enjoys norm and compactness stability, much like the related Fredholm index. Such indices played a pivotal role in describing the quantization and stability properties in the quantum Hall effect; ASS94 related the Hall conductance to the index of a Fermi projection P and its Laughlin-flux-inserted projection U*PU. What becomes of this story in the presence of interactions? To describe infinitely-many interacting electrons in infinite-volume, the Hilbert space is replaced by a unital C-* algebra A (a CAR algebra), but there is no obvious notion of a Fredholm index. We introduce a new notion, the index of a pair of pure states (on A), prove its quantization, invariance and stability properties, and relate it to the (possibly fractional) Hall conductance. We comment that Kitaev’s invertible states always have integer conductance. Joint with Bachmann and Tauber.

2D materials can be stacked in many ways allowing for a large range of tuning parameters to control electronic properties. One of these parameters includes a relative twist angle in plane between the various layers. While individual layers may exhibit periodicity, the global ensemble has no periodicity, and is then called incommensurate. Further, different species of 2D materials naturally have different periodicities, and become incommensurate regardless of choice of angle. Incommensurate 2D materials with two or more similar periodicities form large moiré patterns resulting in exotic quantum phases including for twisted bilayer graphene correlated insulation, unconventional superconductivity, and the fractional quantum Hall effect. To understand and predict properties of moiré materials, accurate ab initio models derived from first principle physics are necessary. In this talk, we will discuss how momentum space techniques can be used to compute electronic observables and quasi-band structure for ab initio tight-binding models with various geometry complexities including double-incommensurate trilayers and mechanically relaxed twisted bilayers. We also discuss how this formulation can derive periodic continuum models including the Bistritzer-MacDonald model.

In condensed matter, the problem of electrons hopping on a two-dimensional lattice naturally leads to the notion of a Chern band. Mathematically, a Chern band is described by a smooth map from the Brillouin zone—a torus labeling the unitary characters of the Abelian group of lattice translations—to a complex projective space. The geometry of these maps plays an important role in shaping physical responses of quantum materials and effective interactions, especially when the band dispersion is flat. In particular, the so-called ideal bands are quite remarkable from the physical point of view as they reproduce the physics of the lowest energy subspace—the lowest Landau level—for electrons in a uniform magnetic field. Ideal Kähler bands correspond to the case where the map is holomorphic. Recently, we have introduced the notion of generalized Landau levels (GLLs), which provide lattice analogs of higher Landau levels. Mathematically, GLLs correspond to Chern bands for which the associated map from the Brillouin zone to complex projective space is harmonic. In this talk we will review the theory, and discuss some recent results on the rigidity of such maps, in the same spirit of Calabi's rigidity theorem for isometric embeddings of complex manifolds in complex projective space.