11:15-12:00pm: Guillaume Bal ,
Asymmetric interface transport and validity or not of the bulk-edge correspondence
Topological insulators are systems whose phase affords a topological origin. They find a growing number of applications in e.g., electronics, photonics, and the geophysical sciences. A characteristic feature of such systems is the surprising robustness to perturbation of the asymmetric transport observed along interfaces separating topologically distinct insulating bulks. This talk reviews a classification of partial differential operators and establishes a bulk-edge correspondence (BEC), which relates the quantized interface transport to the easy-to-compute index of a Fredholm operator naturally associated to the insulating bulks. While the BEC applies to general elliptic systems, we show that it fails for a specific (non-elliptic) hydrodynamic model of equatorial waves. In fact, we demonstrate that the BEC might be arbitrarily violated when a Coriolis force parameter admits discontinuities.
12:05-12:30pm:David Gontier ,
Kekulé distorsion in graphene
In this talk, we review what is known about spontaneous deformation of graphene, also called Kekulé distorsion. A result by Frank/Lieb [PRL 2011] shows that, for general Hubbard models, at most three Carbon-Carbon lengths can appear in distorted graphene. We prove that, for the simple Peierls-SSH model, two of these lengths are, actually, equal. This validates one of Kekule prediction ("Kekule-O?). This is joint work with Éric Séré and Thaddeus Roussigné.
1pm-2pm: Lunch at Gulf Stream
2:45-3:45pm: Allan MacDonald,
Introduction to Moiré Materials
Semiconductor or semimetal two-dimensional material multilayers with moiré patterns are accurately described over a limited range of energy by periodic effective Hamiltonians and act like artificial crystals with lattice constants on the 10 nm scale. These systems have become known as moiré materials. To date, the most frequently studied moiré materials have been formed in graphene or transition metal dichalcogenide two-dimensional materials. I will briefly discuss some of the most interesting properties of known moiré materials, for example the surprising appearance of nearly flat bands in graphene multilayers and of the fractional quantum anomalous Hall effect in TMD bilayers, and attempt to identify some topics that deserve more attention. Finally, I will discuss `rules’ for the construction of moiré material Hamiltonians and the potential development of new moiré material platforms.
3:45-4:15pm: Coffee break
4:15-4:40pm: Tianyu Kong ,
Modeling of electronic dynamics in twisted bilayer graphene
We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer-MacDonald PDE model, which is periodic with respect to the bilayer?s moire pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this talk, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes-Thomas estimates. We then provide extensive numerical computations which clarify the range of validity of the Bistritzer-MacDonald model.
4:45-5:10pm: Solomon Quinn ,
Higher-order corrections to the Bistritzer-MacDonald model
The first-order continuum model proposed by Bistritzer and MacDonald in 2011 accurately describes the electronic properties of twisted bilayer graphene at small twist angles. In this talk, we discuss extensions of the Bistritzer-MacDonald (BM) model to higher-order partial differential equations in the form of a systematic multiple-scales expansion. We show that the solution of these PDEs accurately approximates the corresponding tight-binding wave function under a natural choice of parameters and given initial conditions that are spectrally localized to the monolayer Dirac points. Symmetries of the higher-order models will also be discussed. This work builds on the 2023 JMP paper by Watson, Kong, MacDonald, and Luskin, which rigorously established the validity of the (first-order) BM model.
7:30pm: Dinner at Gulf Stream,
Tuesday, July 9
9:00-9:45am: Maciej Zworski ,
Classically forbidden regions for TBG now include stacking points
We show exponential decay (as the angle of twisting goes to zero) of eigenstates (protected states + eigenstates at magic angles) in the chiral model of TBG near the hexagon spanned by the stacking
points. Near interior points of the edges it follows from general results (joint with M Hitrik) and based on the geometry of Poisson brackets (with a different proof recently provided by J Sjostrand). Near the stacking points (vertices of the hexagon) it follows from an analytic hypoellipticity argument based on the specific structure of the operator (joint work with Z Tao).
9:50-10:15am: Izak Oltman ,
Random perturbations of non-self-adjoint operators in TBG
In this talk, I will briefly survey current results about random perturbations of non-self-adjoint operators arising in modeling twisted bilayer graphene. I will then present a general result about the spectral properties of random compact perturbations of Fredholm operators of index 0. This is joint work with Simon Becker and Martin Vogel.
10:20-10:45pm: Antoine Levitt ,
Numerical integration in the Brillouin zone
I will review numerical methods for the computation of Brillouin zone integrals, focusing on a recently introduced Brillouin zone complex deformation, which allows for the direct computation, without regularization, of Green functions and resonances.
10:45-11:15am: Coffee break
11:15-12:00pm: Lingrui Ge ,
Dual Lyapunov exponents, hidden partially hyperbolic structure and sharp arithmetic spectral results for type I operators
We exploit various structures of dual cocycles and develop several new concepts and tools for the study of one-frequency quasiperiodic operators with analytic potentials. As applications we solve three long-standing arithmetic conjectures: the ten martini problem, sharp phase transition in frequency and absolute continuity of the integrated density of states for all frequencies for the entire class of type I operators.
12:05-12:30pm: Ilya Kachkovskiy ,
Quasiperiodic operators with monotone potentials
Quasiperiodic operators with monotone potentials can be considered as a natural generalization of the Maryland model. This talk addresses several recent and ongoing projects, in collaboration with S. Jitomirskaya, S. Krymskii, L. Parnovski, and R. Shterenberg, that include:
1) Sharp non-perturbative arithmetic phase transition between localization and singular continuous spectra in dimension one;
2) Localization in higher dimension using perturbative methods;
3) The geometry of the spectra of such operators.
The main focus of the talk will be the one-dimensional case, where the most complete results can be obtained.
1pm-2pm: Lunch at Gulf Stream
2:30pm-6:30pm: Free discussions
7:30pm: Dinner at Gulf Stream,
Wednesday, July 10
9:30-10:15am: Andrei Bernevig ,
Mapping of Flat Bands with concentrated berry curvature to Heavy fermion models: Examples in Twisted bilayer graphene and Lieb Model.
TBA
10:20-10:45am: Zhongkai Tao ,
Dirac cones and magic angles in the Bistritzer--MacDonald TBG Hamiltonian
We demonstrate the generic existence of Dirac cones in the full Bistritzer--MacDonald Hamiltonian for twisted bilayer graphene. Its complementary set, when Dirac cones are absent, is the set of magic angles. We show the stability of magic angles obtained in the chiral limit by demonstrating that the perfectly flat bands transform into quadratic band crossings when perturbing away from the chiral limit. Moreover, using the invariance of Euler number, we show that at magic angles there are more band crossings beyond these quadratic band crossings. This is the first result showing the existence of magic angles for the full Bistritzer--MacDonald Hamiltonian and solves Open Problem No. 2 proposed in the recent survey of Zworski. This is joint work with Simon Becker, Solomon Quinn, Alexander Watson and Mengxuan Yang.
10:45-11:15am: Coffee break
11:15-11:40pm: Simon Becker ,
Gentle Introduction to the Interacting Model
In recent years, twisted bilayer graphene (TBG) has emerged as a captivating subject in condensed matter physics, offering a platform for exploring novel electronic phenomena due to its unique band structure. This talk tries to provide a gentle introduction to the interacting model of twisted bilayer graphene, aiming to set a foundation for the subsequent detailed presentation by Kevin Stubbs.
11:45-12:30pm: Kevin Stubbs ,
Exact Many-Body Ground States for Chiral Twisted Bilayer Graphene
Abstract: One of the most remarkable theoretical findings of magic angle twisted bilayer graphene (TBG) is existence of Hartree-Fock states which are exact ground states of the corresponding interacting Hamiltonian at the chiral limit. The key properties which led to this remarkable result have been idealized to define flat-band interacting (FBI) Hamiltonians. In this talk, I will discuss our recent work which characterizes all possible Hartree-Fock ground states for frustration-free FBI Hamiltonians. I will also present some new findings which prove, in certain cases, that all many-body ground states of the FBI Hamiltonian for TBG can be written as a linear combination of Hartree-Fock ground states. This talk is based on joint work with Simon Becker and Lin Lin.
1pm-2pm: Lunch at Gulf Stream
2:30-3:15pm: Oskar Vafek ,
Interacting Chern bands of twisted MoTe2 in an external magnetic field
TBA
3:20-3:45pm: Ammon Fischer ,
Atomistic modeling of correlations in moiré(-less) multilayer graphene
The experimental observation of flavor-symmetry broken phases and superconductivity in multilayer graphene has raised great theoretical interest in the competition and collaboration of various ordered phases at low electronic densities. In this density regime, the momentum 'localization' of the Fermi pockets around the graphene valleys $K^{\nu}$ motivates a pure momentum space formalism of the low-energy physics, where the locality and length scale of emergent phenomena is hidden. To tackle this problem, we bridge between existing continuum theories and \textit{ab-initio} studies of rhombohedral multilayer graphene by formulating the low-energy physics in terms of supercell Wannier functions, i.e. flavor-polarized wave-packets with mesoscopic extent. Starting from an \textit{ab-initio} electronic structure theory comprising the atomic carbon $p_z$-orbitals, the momentum 'localization' of the Fermi surface pockets around the graphene valleys $K^{\nu}$ is circumvented by backfolding the $\pi$-bands in the concomitant mini-Brillouin zone of the supercell, reminiscent of their (twisted) moiré counterparts. By projecting an orbital-resolved, dual-gated Coulomb interaction to the effective Wannier basis, we find that the low-energy physics of rhombohedral multilayer graphene is governed by weak electron-electron interactions. This opens the door for material-realistic characterization of correlated phenomena including superconductivity from first-principles within diagrammatically unbiased weak-coupling methods like the functional renormalization group.
3:45-4:15pm: Coffee break
4:15-5:00pm: Leni Bascones ,
Probing the cascades in twisted bilayer graphene
The cascades in twisted bilayer graphene involve a reorganization of the spectral weight up to several tens meV with resets, band flattening, minima of the density of states at the Fermi level in STM experiments and sawtooh peaks in the inverse compressibility. Recently, based on DMFT calculations, we showed that the formation of local moments and heavy quasiparticles, and not a symmetry breaking process, is responsible for the cascade phenomena. After introducing, the cascades and our results for the STM and inverse compressibility, I will discuss how this description compares with other observed phenomena in TBG and propose novel ways to probe the cascades.
5:05-5:50am: Mitchell Luskin ,
Modeling mechanical relaxation in incommensurate multilayer van der Waals heterostructures
The incommensurate stacking of multilayered two-dimensional materials is a challenging problem from a theoretical perspective and an intriguing avenue for manipulating their physical properties. We will present a multiscale model to obtain the mechanical relaxation pattern of twisted trilayer van der Waals (vdW) heterostructures with two independent twist angles, a generally incommensurate system without a supercell description. We adopt the configuration space as a natural description of such incommensurate layered materials, based on the local environment of atomic positions, bypassing the need for commensurate approximations.
To obtain the relaxation pattern, we perform energy minimization with respect to the relaxation displacement vectors. We show that the relaxation patterns of twisted trilayer graphene is “moiré of moiré,” as a result of the incommensurate coupling two bilayer moiré patterns. We also show that, in contrast to the symmetry-preserving in-plane relaxation in twisted bilayers, trilayer relaxation can break the two-fold rotational symmetry about the xy plane when the two twist angles are equal.
7:30pm: Dinner at Gulf Stream,
Thursday, July 11
9:10-9:55am: Marco Polini ,
Cavity quantum electrodynamics of correlated electron systems
TBA
10:00-10:45am: Christophe Mora ,
Hofstadter spectrum and Chern bands in twisted transition metal dichalcogenides
We discuss the band structure of twisted bilayer transition metal dichalcogenides, such as WSe2, under an applied magnetic field. Employing the recently developed adiabatic model of twisted transition metal dichalcogenides, we show a particularly simple band structure at one negative flux quantum per unit cell where the fictitious magnetic field is on average canceled. By varying the twist angle, we explore the evolution of the magnetic butterfly spectrum from a regime characterized by well-localized orbitals to one of nearly plane waves. We hence clarify the origin of the topological bands identified at zero magnetic field.
Furthermore, we analyze the magnetic field spectrum of twisted bilayer MoTe2, which has recently been suggested to feature higher Landau level analogues. We show that at negative unit flux per unit cell, the bands display remarkable similarity to nearly free electrons, even in regimes where the adiabatic approximation breaks down. This sheds new light on the interpretation of the recently observed topologically ordered states in MoTe2.
10:45-11:15am: Coffee break
11:15-12:00pm: Daniel Massatt ,
Electronic Structure of Incommensurate 2D Heterostructures with Mechanical Relaxation
Incommensurate 2D materials with two similar periodicities form large moiré patterns resulting in unique electronic properties including correlated insulators and superconductors. Before studying correlated effects, a thorough understanding of the single-particle picture is critical. In this work, we discuss momentum space methodologies to build algorithms for computing observables and quasi-band structure of incommensurate heterostructures for ab-initio tight-binding models. We exploit the ergodicity of the misalignment, and carefully selected perturbative expansions in momentum space with a thorough error analysis describing the effects of the large moire scale. The effects of mechanical relaxation are also considered, resulting in long-range momentum scattering.
12:05-12:30pm: Francis Nier ,
Moiré materials, a question about differentiating the tunnel effect in families
After recalling some techniques for recovering effective equations from Floquet-Bloch theory and semiclassical techniques for modulated periodic hamiltonians. I will show how questions about differentiable versions of sharp tunnel effect asymptotics in a double well problem arise in moiré materials.
1pm-2pm: Lunch at Gulf Stream
2:30pm-6:30pm: Free discussions
7pm: Conference dinner at Gulf Stream,
Friday, July 12
9:30-9:55am: Michael Hitrik ,
Spectra for non-self-adjoint operators and integrable dynamics
Non-self-adjoint operators appear in many settings, from kinetic theory and quantum mechanics to linearizations of equations of mathematical physics. The spectral analysis of such operators, while often notoriously difficult, reveals a wealth of new phenomena, compared with their self-adjoint counterparts: spectra for non-self-adjoint operators may display fascinating features, such as lattices of eigenvalues for operators of Kramers-Fokker-Planck type, say, and eigenvalues for operators with analytic coefficients in dimension one, concentrated to unions of curves in the complex spectral plane. In this talk, we shall give a broad introduction to the spectral analysis of non-self-adjoint perturbations of self-adjoint operators in dimension two, under the assumption that the classical flow of the unperturbed part is completely integrable. It turns out that spectra of such operators often have structures of distorted two-dimensional lattices, given by quantization conditions of Bohr-Sommerfeld type. This talk is based on a series of joint works with Johannes Sjöstrand.
10:00-10:45pm: Lede Xian ,
Computational design of moiré superlattices for quantum simulations
Twisted two-dimensional materials are one of the popular research topics in the field of 2D materials and condensed matter physics. Twisted stacking allows one to turn 2D materials into strongly correlated systems, providing unpretending opportunity to investigate strongly correlated physics in a relatively simple but highly tunable setup. While early research started from twisted graphene systems, using large-scale Density Functional Theory calculations, we extend the study to various other 2D materials, such as twisted BN, MoS2, GeSe, etc. Basing on the electronic structures and atomic symmetry of 2D materials, we can design twisted moiré systems to realize strongly correlated models in various lattice types and in different dimensionalities. More recently, we further design moiré systems that can realize a square-lattice Hubbard model strong frustration due to the next nearest neighbour hopping, which are expected to host unconventional superconductivity in close analogy to those of cuprate high-temperature superconductors. We believe these highly tunable 2D moiré systems will provide novel platforms for the simulation of correlated quantum state of matters.
10:45-11:15am: Coffee break
11:15-11:40pm: Solal Perrin-Roussel ,
A mathematical analysis of IPT-DMFT
In this talk, I will present the work carried out in collaboration with Eric Cancès and Alfred Kirsch. In this work, we provide a mathematical analysis of the Dynamical Mean-Field Theory, a celebrated representative of a class of approximations in quantum mechanics known as embedding methods. I will start by a mathematical formulation of the Dynamical Mean-Field Theory equations for the finite Hubbard model. After recalling the definition and properties of one-body time-ordered Green's functions and self-energies, and the mathematical structure of the Hubbard and Anderson impurity models, I will describe a specific impurity solver, namely the Iterated Perturbation Theory solver, which can be conveniently formulated using Matsubara's Green's functions. Within this framework, I will state our main result, namely that under certain assumptions, the Dynamical Mean-Field Theory equations admit a solution for any set of physical parameters. The mathematical objects are closely related to the Nevanlinna-Riesz theory, on which I will give some insights for understanding the properties of the solutions.