Moiré Landau levels of a $C_4$-symmetric twisted bilayer system in the absence of a magnetic field

Abstract

It is widely known that the twisted bilayer graphene (TBG) shows flat bands at magic angles, which can be well described by the effective continuum model derived by Bistritzer and MacDonald (BM). We propose in this paper a similar twisted bilayer system but defined on the square lattice with $\pi$-flux per plaquette, and study its spectrum using the BM Hamiltonian with a mass term which is originated from the staggered potential. The basic difference between the TBG and the present model is simply rotational symmetry, $C_3$ vs $C_4$, as well as a mass term. Nevertheless, the feature of the flat bands is quite different: Those of the TBG appear at magic angles only, while the present model shows many flat bands, which are reminiscent of Landau levels, quite stably at any angles even in the absence of a magnetic field other than $\pi$-flux which keeps time reversal (TR) symmetry. Moreover, flat bands emerge in the mass gap of the Dirac spectrum, and each state composing these flat bands is well-localized at the position forming the moiré lattice. It turns out that the moiré potential serves as a periodic magnetic field, which can give energies smaller that the gap around moiré lattice positions. We derive a local Hamiltonian valid around the moiré lattice sites and show that it indeed reproduces the energies of the flat bands within the mass gap. Since these mid-gap states are localized at the moiré lattice, they form degenerate levels, which may be referred to as moiré Landau levels, although the mechanism of degeneracies are different from the conventional Landau levels. Interestingly, doubled fermions of the BH Hamiltonian associated with two layers have opposite charges when they couple with the effective moiré magnetic filed, which concern TR symmetry.

arXiv comment13.1 pages, 9 figures, v2: Fig. 5 added, a reference added, v3: final version

References in corpus (36)

• Magic-angle graphene superlattices: a new platform for unconventional superconductivity
• Correlated Insulator Behaviour at Half-Filling in Magic Angle Graphene Superlattices
• Moire bands in twisted double-layer graphene
• Graphene bilayer with a twist: electronic structure
• Flat Bands in Slightly Twisted Bilayer Graphene
• Origin of Magic Angles in Twisted Bilayer Graphene
• Dirac Semimetals in Two Dimensions
• Continuum Model of the Twisted Bilayer
• All "Magic Angles" Are "Stable" Topological
• Optical Absorption in Twisted Bilayer Graphene
• Nearly Flat Chern Bands in Moiré Superlattices
• Electrically Tunable Low Density Superconductivity in a Monolayer Topological Insulator
• Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
• Nematicity and Competing Orders in Superconducting Magic-Angle Graphene
• Energy Spectrum and Quantum Hall Effect in Twisted Bilayer Graphene
• Band Structure of Twisted Bilayer Graphene: Emergent Symmetries, Commensurate Approximants and Wannier Obstructions
• Ground State and Hidden Symmetry of Magic Angle Graphene at Even Integer Filling
• Magic Angle Hierarchy in Twisted Graphene Multilayers
• Twisted Bilayer Graphene IV. Exact Insulator Ground States and Phase Diagram
• Moire Butterflies
• Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene
• Twisted bilayer graphene III. Interacting Hamiltonian and exact symmetries
• Twisted bilayer graphene. II. Stable symmetry anomaly
• High-temperature topological superconductivity in twisted double layer copper oxides
• Twisted Bilayer Graphene V: Exact Analytic Many-Body Excitations in Twisted Bilayer Graphene Coulomb Hamiltonians: Charge Gap, Goldstone Modes and Absence of Cooper Pairing
• Band Symmetries and Singularities in Twisted Multilayer Graphene
• Multiple topological transitions in twisted bilayer graphene near the first magic angle
• Twisted bilayer graphene.VI. An Exact Diagonalization Study of Twisted Bilayer Graphene at Non-Zero Integer Fillings
• Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a $k\cdot p$ 2-Band model
• Chiral Approximation to Twisted Bilayer Graphene: Exact Intra-Valley Inversion Symmetry, Nodal Structure and Implications for Higher Magic Angles
• Landau levels in twisted bilayer graphene and semiclassical orbits
• Landau Level Degeneracy in Twisted Bilayer Graphene: Role of Symmetry Breaking
• Chiral Magic-Angle Twisted Bilayer Graphene in a Magnetic Field: Landau Level Correspondence, Exact Wavefunctions and Fractional Chern Insulators
• Flat band in twisted bilayer Bravais lattices
• Hofstadter butterfly and the quantum Hall effect in twisted double bilayer graphenes
• Interference of the Bloch phase in layered materials with stacking shifts

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