Schedule for: 23w5058 - The Mathematics and Physics of Moire Superlattices

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

BIRS Lounge, PDC 2nd floor

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk I will highlight three interesting and important challenges facing moiré materials researchers: i) How can we build a better models of moiré materials ? ii) Can we open new ground in the math and physics of quasi-periodic Hamiltonians? iii) Can we develop a predictive understanding of fractional Chern insulator states in moiré materials? For i) twisted bilayer graphene seems to be the most challenging case. Current models do not fully account for the non-local nature of the interactions between flat band electrons and the negative energy sea, and this may be necessary if we are ever to achieve a quantitative understanding of the competitions between superconducting, Fermi liquid, and magnetic states. For ii), the moiré material platform allows creation of flexibly tunable quasiperiodic two-dimensional Hamiltonians. The physical properties of these systems will be influenced by interactions as well as by single-particle physics. There is an opportunity, I believe, to pose new mathematical questions. One important goal is to learn how to calculate and how to measure the quantum numbers that characterize spectral gaps. For iii), we are in early days. The discovery of fractional Chern insulator states in moiré materials is a breakthrough event in condensed matter physics. Can we use the tunability of moiré materials to stabilize fractionalized states that support non-Abelian quasiparticles?

Moiré materials provide a highly controllable platform to explore the strong electronic correlation phenomena. Specifically, Mott insulators with a lattice of local magnetic moments have been observed in semiconductor moiré bilayers in the flat band limit. In this talk, I will discuss experiments studying magnetism in doped moiré Mott insulators. One observation involves the spin polarons--bound states of a doped hole and a spin flip--in a hole-doped triangular lattice Mott insulator. The second involves the emergence of ferromagnetism at the onset of a Kondo breakdown transition in a moiré Kondo lattice (a lattice of local moments exchange-coupled to conduction electrons).

One of the most surprising theoretical discoveries of magic angle twisted bilayer graphene is that in the chiral limit, certain single Slater determinants can be the ground state of the flat-band interacting Hamiltonian. This provides an explanation of the correlated insulating phase at integer filling. We investigate the symmetry attributes of the interacting Hamiltonian and the resulting ground states. This allows us to study systems beyond TBG, including TBG-like systems with 4 flat bands per valley, and equal twist-angle trilayer graphene systems (joint work with Simon Becker and Kevin Stubbs).

We present a numerical study of an interacting BistritzerMacDonald (IBM) model of TBG using a suite of methods in quantum chemistry, including Hartree-Fock, coupled cluster singles, doubles (CCSD), and perturbative triples (CCSD(T)), as well as a quantum chemistry formulation of the density matrix renormalization group method (DMRG). At integer filling, all numerical methods agree in terms of energy and C2z T symmetry breaking. Additionally, as part of our benchmarking, we explore the impact of different schemes for removing “double-counting” in the IBM model. Our results at integer filling suggest that cross-validation of different IBM models may be needed for future studies of the TBG system. After benchmarking our approach at integer filling, we perform a systematic study of the IBM model near integer filling. In this regime, we find that the ground state can be in a metallic and C2z T symmetry breaking phase. The ground state appears to have low entropy, and therefore can be relatively well approximated by a single Slater determinant. Furthermore, we observe many low entropy states with energies very close to the ground state energy in the near integer filling regime.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The interplay between strong interactions and the unique environment created by Moire superlattices is believed to be a key for many of the most exciting phenomena seen in these two-dimensional materials. It is then important to treat both one- and two-body effects in a balanced and accurate way. This presents outstanding theoretical, algorithmic, and computational challenges. I will describe our preliminary work on developing and applying quantum Monte Carlo methods to treat Moire systems. This includes a study of the interacting Bistritzer-MacDonald model beyond specialized fillings which are sign-problem-free, calculations in a continuum model of two-dimensional electron gas in Moire potentials, and the parametrization of an exchange-correlation functional for density-functional theory calculations in two-dimensional materials.

Twisted bilayer graphene (TBG) exhibits a plethora of electronic phases. Among the variety of correlated states, the cascades in the spectroscopic properties and in the compressibility happen in a much larger energy [1,2,3], twist angle and temperature range than other effects, pointing to a hierarchy of phenomena. In this work [4], we show that the spectral weight reorganization associated to the formation of local moments and heavy quasiparticles, and not a symmetry breaking process, is responsible for the cascade phenonema. Among the phenomena reproduced in this framework are the cascade flow of spectral weight, the oscillations of the remote band energies and the asymmetric jumps of the inverse compressibility. Due to the fragile topology of TBG, we predict a strong momentum differentiation in the incoherent spectral weight. In the talk, I will also address other possible measurements which may help distinguishing the phenomenology of the cascades discussed here from proposals involving symmetry breaking. [1] Wong et al, Nature 582, 198 (2020) [2] Zondiner et al, Nature 582, 203 (2020) [3] Polski et al, arXiv:2205.05225 [4] A. Datta, M.J. Calderón, A. Camjayi, E. Bascones, Nature Comms. 14, 5036 (2023)

Understanding the phase diagram of twisted bilayer graphene and related moir\'e systems is a central theoretical challenge. While the ground states at integer fillings have been shown in many cases to be simple flavor ferromagnets, the charge excitations above such states can be non-trivial due to band topology. Conventional approaches to understand such excitations as real space topological textures fail to account for the distinct momentum space features of Chern bands and obscures their comparison to single particle excitations. Here, we present a general fully momentum space formulation for the problem of charge excitations in Chern bands. In the limit of (normal-ordered) contact interactions in an ideal flat bands, we construct exact analytical wavefunctions for the lowest energy excitation with charge $\pm e$ and spin $n + 1/2$, a spin polaron. Away from this ideal limit, we show that these analytical wavefunctions are excellent variational states describing a bound state of an electron/hole with $n$ spin flips. We show that the ansatz can be cast in the form of an antisymmetrized electron-hole geminal power and develop a diagrammatic approach to evaluate the expectation values of operators in such states, allowing us to study relatively large number of spin flips and large system sizes. We apply our formalism to study charge excitations in twisted bilayer graphene and find that (i) in the chiral limit, multispin flip polarons are the lowest energy charge excitations at charge neutrality and at non-zero integer fillings when doping towards neutrality. In the realistic limit, we find that the multispin flip states are the lowest charged excitations at $\nu = \pm (1 - \epsilon)$ for any strong coupling state and at $\nu = \pm (2 - \epsilon)$ for the time-reversal intervalley coherent state (TIVC) but not the Krammers intervalley coherent state (KIVC). We discuss the experimental implications of these results for low strain devices.

The best-established example of fractionalization starts from the partially filling flat kinetic energy dispersion, namely the fractional quantum Hall effect. We will show that twisted bilayer graphene systems present new platforms for arriving at fractionalization: fractional correlated insulating state. Various 1/3 filled twisted bilayer graphene are expected to form Chern number=0 incompressible states driven by the extended fidget spinner-shaped Wannier orbitals. The geometric frustration due to the orbital shapes leads to charge 1/3 excitations with restricted mobility. The restricted mobility limited to the sub-dimension gives fracton-like character to the charge 1/3 excitations. I will discuss theoretical predictions on how to detect such fractional correlated insulators and preliminary experimental results.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will report on 3-4 years of mathematical analysis on the chiral limit of the BMH model and report on our understanding for bilayer and multilayer systems, and what open questions remain. My understanding of the subject has been shaped by collaborations with Mark Embree, Tristan Humbert, Ryan Kim, Izak Oltman, Martin Vogel, Jens Wittsten, Mengxuan Yang, Xiaowen Zhu, and Maciej Zworski.

Fractional Chern insulators realize the remarkable physics of the fractional quantum Hall effect (FQHE) in crystalline systems with Chern bands. The lowest Landau level (LLL) is known to host the FQHE, but not all Chern bands are suitable for realizing fractional Chern insulators (FCI). Previous approaches to stabilizing FCIs focused on mimicking the LLL through momentum space criteria. Here instead we take a real-space perspective by introducing the notion of vortexability. Vortexable Chern bands admit a fixed operator that introduces vortices into any band wavefunction while keeping the state entirely within the same band. Vortexable bands admit trial wavefunctions for FCI states, akin to Laughlin states. In the absence of dispersion and for sufficiently short-ranged interactions, these FCI states are the ground state -- independent of the distribution of Berry curvature. Vortexable Chern bands emerge naturally in chiral twisted graphene, and fractional Chern insulators were subsequently observed experimentally. Recently, zero-field fractional Chern insulators, and potentially a zero-field composite Fermi liquid, were also observed in the nearly-vortexable twisted MoTe_2. New and exciting nearly-vortexable platforms are also appearing, including periodically strained graphene and helically twisted graphene.

Quantum Hall phases are the most exotic experimentally established quantum phases of matter. Recently they have been discovered at zero external magnetic field in two dimensional moire materials. I will describe recent work on their proximate phases and associated phase transitions that is motivated by the high tunability of these moire systems. These phase transitions (and some of the proximate phases) are exotic as well, and realize novel ‘beyond Landau’ criticality that have been explored theoretically for many years. I will show that these moiré platforms provide a great experimental opportunity to study these unconventional phase transitions and related unconventional phases, thereby opening a new direction for research in quantum matter.

Perfectly flat bands in moire materials have intimate connections to landau levels of Dirac fermions. In this talk, we will investigate how the robustness of zeroth Landau level to chiral disorder relates to the stability of moire flat bands against certain types of disorder. In light of moire TMDs, we will also discuss the implications for massive Dirac fermions where the chiral symmetry is explicitly broken.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The recent discovery of fractional Chern insulators (FCIs), which can exhibit the fractional quantum anomalous Hall effects, has attracted much scientific interest. I will discuss thermodynamic studies on the FCIs and the non-topological states in twisted bilayer MoTe2. I will particularly focus on the nature of the electric-field-induced transitions between the FCIs and the non-topological states. I will also compare our thermodynamic studies with recent transport studies and discuss its implications.

Superconductivity has been observed in a number of twisted and untwisted graphene multilayers. The dependence of the superconducting properties on the geometry of the graphene stack will be discussed. The possibility of novel phenomena due to non trivial order parameters will also be highlighted.

In addition to the extensive study of twisted moiré bilayers in the past decade, the scope of investigation has extended to encompass multilayer systems including three or more layers. Particular attention has recently been directed toward twisted trilayer systems which consists of three layers arranged in a specific rotational configuration. A twisted trilayer is characterized by two twist angles between adjacent layers, offering a vast parameter space that remains largely unexplored. In the first part of my talk, we will present systematic theoretical studies on the lattice relaxation and the electronic structures in general twisted trilayer graphenes [1]. We show that the relaxed lattice structure forms a patchwork of moiré-of-moiré domains, where a moiré pattern given by layer 1 and 2 and another pattern given by layer 2 and 3 become locally commensurate. The electronic band calculation reveals a wide energy window featuring sparsely distributed highly one-dimensional electron bands. These one-dimensional states exhibit a sharp localization at the boundaries between supermoiré domains, and they are identified as a topological boundary state between distinct Chern insulators. In the latter part of our discussion, we will explore the electronic structure of hBN/graphene/hBN trilayer system with arbitrary twist angles. We find that the electronic spectrum displays fractal minigaps akin to the Hofstadter butterfly. We demonstrate that each of minigaps is uniquely labeled by six topological numbers associated with the quasicrystalline Brillouin zones, and these numbers can be expressed as second Chern numbers through a formal connection with the quantum Hall effect in four-dimensional space [2,3]. [1] N. Nakatsuji, T. Kawakami, and M. Koshino, arXiv:2305.13155; Phys. Rev. X, in press. [2] M. Koshino, H. Oka, Phys. Rev. Research 4, 013028 (2022) [3] H. Oka and M. Koshino, Phys. Rev. B 104, 035306 (2021)

We propose an externally imposed superlattice potential as a platform for engineering topological phases, which has both advantages and disadvantages compared to a moiré superlattice. We show that a superlattice potential applied to Bernal-stacked bilayer graphene can generate flat Chern bands, similar to those in twisted bilayer graphene, whose bandwidth can be as small as a few meV. Further, the flat band has a favorable band geometry for realizing a fractional Chern insulator at partial filling. The superlattice potential offers flexibility in both lattice size and geometry, making it a promising alternative to achieve designer flat bands without a moiré heterostructure.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the first part of the talk I will discuss magneto-transport experiments on twisted bilayer graphene at 1.32 degree twist angle, i.e. away from the magic value. Despite the absence of correlated states at B=0, the theoretical explanation of these experiments provides insight into the origin of the Landau level degeneracy near the charge neutrality point and the role of heterostrain[1]. Equipped with this understanding, I will present a comprehensive Hatree-Fock study of interacting electrons in finite magnetic field while varying the electron density, twist angle and heterostrain. Within a panoply of correlated Chern phases emerging at a range of twist angles, I will present a unified description for the ubiquitous sequence of states with the Chern number $t$ for $(s,t)=\pm (0,4), \pm(1,3),\pm(2,2)$ and $\pm(3,1)$. Correlated Chern insulators at unconventional sequences with $s+t\neq \pm 4$ are also found, as well as with fractional $s$. I will discuss their nature.[2] [1] Xiaoyu Wang et al. PNAS2023 Vol. 120 No. 34 e2307151120 [2] Xiaoyu Wang and O.Vafek arXiv.2310.xxxx

wisted bilayer graphene (TBG) displays two seemingly contradictory characteristics: (1) quantum-dot-like behavior in STM suggesting electron localization; (2) transport experiments indicating an itinerant nature. Both aspects can be naturally captured by a topological heavy-fermion model where topological conduction electron bands interact with local moments. We study the local-moment physics and the Kondo effect within this model. We reveal that at integer fillings (nu=-2,-1,0,1,2), the RKKY interactions favor ferromagnetic states satisfying a U(4) Hund’s rule. Conversely, at non-integer fillings, the Kondo effect becomes significant, resulting in a Kondo resonance in the spectral function. Through our heavy-fermion model, we also explore the transport properties of TBG. We identify two primary types of carriers: incoherent f electrons and coherent c electrons. The coherent c electrons dominate the transport properties and give rise to a fully negative Seebeck coefficient at positive fillings. We also show that our model can also reproduce various aspects of the physics of TBG, including a natural explanation of the IKS state and its wavevector, as well as the reason for the existence of stronger correlated states for positive integer fillings, despite the bare band structure being more dispersive on that side.

The method introduced in [1] allows one to construct an approximate Kohn-Sham Hamiltonian for (incommensurate) twisted Bilayer graphene. In the first part of the talk, I will show how an effective moiré-scale continuum model, similar though not identical to the Bistritzer-MacDonald model, can be derived from this Hamiltonian by simple variational approximation [2]. In the second part of the talk, I will show that methods from semiclassical analysis can be used to study the density-of-states of this Hamiltonian in the limit of small twist angles [3]. [1] G. Tritsaris, S. Shirodkar, E. Kaxiras, P. Cazeaux, M. Luskin, P. Plechá˘c, and E.Cancès, Perturbation theory for weakly coupled two–dimensional layers, J. Mater. Res. 31 (2016) 959–966 [2] E. Cancès, L. Garrigue and D. Gontier, Simple derivation of moiré-scale continuous models for twisted bilayer graphene, Phys. Rev. B 107 (2023) 155403. [3] E. Cancès and L. Meng, Semiclassical analysis of two-scale electronic Hamiltonians for twisted bilayer graphene, in preparation

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The surprising robustness to perturbation of the asymmetric transport observed along interfaces separating distinct insulating bulks has a topological origin. This talk reviews recent classifications of partial differential operators modeling such systems. A classification by means of domain walls provides a topological invariant whose calculation as an explicit integral is straightforward. A general bulk-interface correspondence then proves that the invariant also describes the quantized aforementioned asymmetric transport. The theory is illustrated on several examples of applications and in particular gated twisted bilayer graphene models.

In this talk, I will discuss recent progress in understanding implications of the electrical conductivity tensor, coming from the Kubo formalism, for the twisted bilayer graphene (TBG) and similar heterostructures. The use of a spatially homogeneous and isotropic tensor conductivity has led us to the analytical derivation of a dispersion relation for non-retarded edge plasmons. This relation explicitly depends on the chiral response of the system. I will describe a correspondence of the chiral optical plasmon in the TBG to the magnetoplasmon in the single-layer graphene, by introducing an effective magnetic field. If time permits, I will also discuss related extensions of the theory to the twisted trilayer and quadrilayer graphene systems. In the analysis, the long-range electrostatic interaction is retained via application of the Wiener-Hopf method of factorization to systems of integral equations for scalar fields.

We formulate a plane-wave basis representation of an incommensurate bilayer linear Schrödinger equation. We use the representation to find algorithms for a number of fundamental electronic observables such as the local density of states of spatial configurations, total density of states, and the local density of states in momentum space, which is a parallel object to electronic band structure in the absence of periodicity. We further prove the equivalence of the density of states in the plane-wave formulation to that of the density of states of the real space Schrödinger equation through a properly averaged thermodynamic limit. The methodology relies on tracking the plane-wave scattering between incommensurate potentials and using these ‘hopping’ parameters to construct a matrix describing the coupling of all interacting plane-waves, which we find to be indexed by a four-dimensional lattice. The algorithm relies on truncation of the matrix via an energy truncation and hopping distance truncation with rigorous convergence rates.

I will review progress towards realistic yet manageable models of the electronic properties of twisted bilayer graphene and other moire materials. First, I will present a general approach to modeling atomic relaxation in moire materials using interatomic potentials. Then, I will discuss a general framework for deriving corrections to effective continuum models such as the Bistritzer-MacDonald model.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we will discuss the significant role that the algebraic properties of Bloch and Fermi varieties play in the study of periodic graph operators. I will begin by highlighting recent discoveries about these properties, especially the irreducibility. Then, I will show how we can use these findings, together with techniques from complex analysis and combinatorics, to study spectral and inverse spectral problems arising from periodic graph operators.

It is known that the magnetic Schr\"odinger operator on 2d lattice with irrational flux has cantor spectrum, which illustrates the well-known picture of "Hofstadter butterfly". In this talk, I will introduce the proof of cantor spectrum for another model - a 1d moire model. In particular, the morie pattern plays a key role in the exhibition of cantor spectrum in a relatively robust way that is intrinsically different from magnetic fields. This implies further potential of understanding other moire-pattern models. The talk is based on a joint work with Simon Baker and Svetlana Jitomirskaya.

In my presentation, I will explore two illustrative examples where the interplay of two moiré patterns to the formation of large moiré domains where commensuration is restored: TBG on aligned hBN and helical trilayer graphene. The latter gives rise to a topological Chern mosaic dominated by region of ABA/BAB stacking forming a periodic triangular pattern on the moiré of moiré scale. I will provide a detailed exploration of the origins of the electronic bands in the chiral limit. Exact results will reveal the existence of a Chern 2 band with unique properties that cannot be reduced to a single lowest Landau level [1-3]. Notably, our predictions are consistent with recent experimental findings [4], underscoring the significance of these moiré patterns in uncovering novel topological phenomena. [1] D. Guerci, Y. Mao, C. Mora, arXiv:2305.03702 (2023) [2] D. Guerci, Y. Mao, C. Mora, arXiv:2308.02638 (2023) [3] Y. Mao, D. Guerci, C. Mora, PRB 107, 125423 (2023) [Editors’ Suggestion] [4] L. Xia, P. Jarillo-Herrero et al., arXiv:2310.12204 (2023)

The moiré pattern observed experimentally in twisted bilayer graphene (tBLG) clearly shows the formation of different types of domains. These domains can be explained by the atomic relaxation, both in-plane and out-of-plane, using continuum elasticity theory and the Generalized Stacking Fault Energy (GSFE) concept. Moreover, the atomic relaxation significantly affects the electronic states, leading to a pair of flat bands at the charge neutrality point which are separated by band gaps from the rest. These features appear for a small range of twist angles, that we call the “magic range”, around the twist angle of 1o. We discuss how all these aspects of the system are crucial for understanding the origin of correlated states and superconductivity in tBLG. We also present a minimal model that can capture these features with 2 flat and 2 auxiliary bands and explore the implications of the model for correlated electron behavior in the context of the Hubbard model.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in breakout rooms or lounge for informal discussion. Or hike Tunnel Mountain.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.