Origin of Magic Angles in Twisted Bilayer Graphene

Abstract

Twisted Bilayer graphene (TBG) is known to feature isolated and relatively flat bands near charge neutrality, when tuned to special magic angles. However, different criteria for the magic angle such as the vanishing of Dirac speed, minimal bandwidth or maximal band gap to higher bands typically give different results. Here we study a modified continuum model for TBG which has an infinite sequence of magic angles $\theta$ at which, we simultaneously find that (i) the Dirac speed vanishes (ii) absolutely flat bands appear at neutrality and (iii) bandgaps to the excited bands are maximized. When parameterized in terms of $\alpha \sim 1/\theta ,$ they recur with the simple periodicity of $\Delta \alpha \simeq 3/2$, which, beyond the first magic angle, differs from earlier calculations. Further, in this model we prove that the vanishing of the Dirac velocity ensures the exact flatness of the band and show that the flat band wave functions are related to doubly-periodic functions composed of ratios of theta functions. Also, using perturbation theory up to ${\alpha }^8$ we capture important features of the first magic angle $\theta \approx 1.09^{\circ }$ ($\alpha \approx 0.586$), which precisely explains the numerical results. Finally, based on our model we discuss the prospects for observing the second magic angle in TBG.

7 pages, 4 figures

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• Artificial moiré engineering for an ideal BHZ model
• Moiré fractals in twisted graphene layers
• The low energy excitation spectrum of magic-angle semimetals
• Topological Degeneracy Induced by Twisting
• Emergence of Cascading Flat Bands in Breathing Superlattices
• Excitonic Metal and Non-Fermi Liquid Behaviour in Twisted Double Bilayer Graphene near Charge Neutrality
• Vacancy-induced tunable Kondo effect in twisted bilayer graphene
• Emergent Hyper-Magic Manifold in Twisted Kitaev Bilayers
• Complex eigenvalue splitting for the Dirac operator
• Understanding Symmetry Breaking in Twisted Bilayer Graphene from Cluster Constraints
• Capturing magic angles in twisted bilayer graphene from information theory markers
• Conduction properties of extended defect states in Dirac materials
• Phonon-mediated superconductivity in strongly correlated electron systems: a Luttinger-Ward functional approach
• Stoner ferromagnetism in low-angle twisted bilayer graphene at three-quarters filling
• Anomalous Electrodynamics and Quantum Geometry in the Dirac-Harper model for a Graphene Bilayer
• Colossal Layer Nernst Effect in Twisted Moiré Layers
• Topologically nontrivial and trivial flat bands via weak and strong interlayer coupling in twisted bilayer honeycomb optical lattices for ultracold atoms
• Electronic structure and transport in materials with flat bands: 2D materials and quasicrystals
• Renormalized Magic Angles in Asymmetric Twisted Graphene Multilayers
• Artificial gauge fields and topological insulators in Moire superlattices
• Moiré amplification of highly tunable shift current response in twisted trilayer graphene
• Hofstadter-Moiré Butterfly in Twisted Trilayer Graphene
• Semiclassical quantization conditions in strained moiré lattices
• Intrinsic superstructure near atomically clean armchair-type step edge of graphite
• Nonflat bands and chiral symmetry in magic-angle twisted bilayer graphene
• Critical Filaments and Superconductivity in Quasiperiodic Twisted Bilayer Graphene
• Tuning monolayer superconductivity in twisted NbSe${}_2$ graphene heterostructures
• Hierarchical dimensional crossover of an optically-trapped quantum gas with disorder
• Exceptional topology in non-Hermitian twisted bilayer graphene
• Quantum geometry induced microwave enhancement of flat band superconductivity
• Ideal topological flat bands in chiral symmetric moiré systems from non-holomorphic functions
• Emergent topology of flat bands in a twisted bilayer $\alpha$-$T_3$ lattice
• How spectrum-wide quantum criticality protects surface states of topological superconductors from Anderson localization: Quantum Hall plateau transitions (almost) all the way down
• Band structure and optical response of Kekulé-modulated $\alpha -{\mathcal{T}}_3$ model
• Floquet Hofstadter butterfly in trilayer graphene with a twisted top layer
• Tight binding approximation for bilayer graphene and nanotube structures: from commensurability to incommensurability between the layers
• Network of chiral one-dimensional channels and localized states emerging in a moiré system
• Graphene bilayer and trilayer Moiré lattice with Rashba spin-orbit coupling
• Instanton-Induced Supersymmetry Breaking in Topological Semimetals
• Stacking-tunable multiferroic states in bilayer ScI2
• Symmetry-based classification of exact flat bands in single and bilayer moiré systems
• Anisotropic moiré band flattening in twisted bilayers of M-valley MXenes
• Electronic analogue of Fourier optics with mass-less Dirac fermions scattered by quantum dot lattice
• Van der Waals Heterostructure Pt${}_2$HgSe${}_3$/CrI${}_3$ for Topological Valleytronics
• Wavefunction textures in twisted bilayer graphene from first principles
• An effective curved space-time geometric theory of generic twist angle graphene with application to a rotating bilayer configuration
• Topological phase diagram of twisted bilayer graphene as a function of the twist angle
• Review of the tight-binding method applicable to the properties of moiré superlattices
• MoireStudio: A Universal Twisted Electronic Structure Calculation Package
• Coulomb drag in metallic twisted bilayer graphene
• Twisted bilayer graphene from first-principles: structural and electronic properties
• DFT+U+V Study of Magnetic Ordering in Single-Layer Pentahexoctite: Implications for Magnetic Device Platforms
• Moiré Superradiance in Cavity Quantum Electrodynamics with Quantum Atom Gas
• Nonmonotonic plasmon dispersion in strongly interacting Coulomb Luttinger liquids
• Dirac quantum criticality in twisted double bilayer transition metal dichalcogenides
• Correlation effects in magic-angle twisted bilayer graphene: An auxiliary-field quantum Monte Carlo study
• Unconventional gapless semiconductor in an extended martini lattice in covalent honeycomb materials
• Photon absorption in twisted bilayer graphene
• Exciton fractional Chern insulators in moiré heterostructures
• Analogue of atomic collapse for adatoms on rhombohedral multilayer graphene
• Effective tight-binding models in optical moiré potentials
• Collective excitations of the Chern-insulator states in commensurate double moiré superlattices of twisted bilayer graphene on hexagonal boron nitride
• Incommensurability-induced sub-ballistic narrow-band-states in twisted bilayer graphene
• Interference effects in polarization controlled Rayleigh scattering in twisted bilayer graphene
• Mean-field Modelling of Moiré Materials: A User's Guide with Selected Applications to Twisted Bilayer Graphene
• Gap labeling theorem for multilayer thin film heterostructures
• Localizing Transitions via Interaction-Induced Flat Bands
• Low-frequency and Moiré Floquet engineering: a review
• Electrodynamics of Thin Sheets of Twisted Material
• Quantum phase diagram and non-abelian Moore-Read state in double twisted bilayer graphene
• Geometry-Driven Moiré Engineering in Twisted Bilayers of High-Pseudospin Fermions
• Merged Four Dirac Points at the Critical Interlayer Distance in Commensurately Twisted Bilayer Graphene: the Origin of the Zero Velocity
• Unconventional Fractional Phases in Multi-Band Vortexable Systems
• Non-Abelian line graph: A generalized approach to flat bands
• Moiré spintronics: Emergent phenomena, material realization and machine learning accelerating discovery
• Energy Spectrum Theory of Incommensurate Systems
• Quantum Monte Carlo study of magnetism and chiral d+id-wave superconductivity in twisted bilayer graphene
• Magic distances in twisted bilayer graphene
• Classically forbidden regions in the chiral model of twisted bilayer graphene. With an appendix by Zhongkai Tao and Maciej Zworski
• Designing a Transition Photonic Band with a Synthetic Moire Sphere
• Real space representation of topological system: twisted bilayer graphene as an example
• Entanglement smectic and stripe order
• Magic angle and plasmon mode engineering in twisted trilayer graphene with pressure
• Robust Flat Bands with Tunable Energies in Honeycomb Superlattices
• Magic-angle twisted bilayer graphene under orthogonal and in-plane magnetic fields
• Effective K valley Hamiltonian for TMD bilayers under pressure and application to twisted bilayers with pressure-induced topological phase transitions
• Electrostatic interactions in twisted bilayer graphene
• Chiral topological superconductivity in twisted bilayer and double bilayer graphene
• Tri-component-pairing chiral superconductivity on the honeycomb lattice with mixed $s$- and $d$-wave symmetries
• Efficient prediction of topological superlattice bands with spin-orbit coupling
• Buckling and flat bands in twisted bilayer graphene
• Incommensurate Twisted Bilayer Graphene: emerging quasi-periodicity and stability
• Instability of Laughlin FQH liquids into gapless power-law correlated states with continuous exponents in ideal Chern bands: rigorous results from plasma mapping
• Particle-hole asymmetric phases in doped twisted bilayer graphene
• Floquet engineering and non-equilibrium topological maps in twisted trilayer graphene
• First Principle Study of Alkaline Metal Intercalation in Twisted Bilayer Graphene
• Theory of engineering flat bands in graphene using doubly-periodic electrostatic gating
• A simple real-space scheme for periodic Dirac operators
• Squeezing Quantum States in Three-Dimensional Twisted Crystals
• Quasi-two-dimensional spin helix and magnon-induced singularity in twisted bilayer graphene
• Relaxation and Its Effects on Electronic Structure in Twisted Systems: An Analytical Perspective
• Straintronics and twistronics in bilayer graphene
• Manipulate Quantum Emission by Interface States between Multi-component Moiré Lattice and Metasurface
• General Many-Body Perturbation Framework for Moiré Systems
• Coupled-wire descriptions of unconventional quantum states in twisted nanostructures
• Moiré-driven equilibrium of perturbations in moiré systems
• Robustness of real-space topology in moiré systems
• Abelian and non-Abelian fractionalized states in twisted MoTe${}_2$: A generalized Landau-level theory
• Anyon Dispersion in Aharonov-Casher Bands and Implications for Twisted MoTe${}_2$
• Gaplessness from disorder and quantum geometry in gapped superconductors