3/2 magic-angle quantization rule of flat bands in twisted bilayer graphene and relationship with the Quantum Hall effect

Abstract

Flat band electronic modes in twisted graphene bilayers are responsible for superconducting and other highly correlated electron-electron phases. Although some hints were known of a possible connection between the quantum Hall effect and zero flat band modes, it was not clear how such connection appears. Here the electronic behavior in twisted bilayer graphene is studied using the chiral model Hamiltonian. As a result, it is proved that for high-order magic angles, the zero flat band modes converge into coherent Landau states with a dispersion ${\sigma }^2=1/3\alpha$, where $\alpha$ is a coupling parameter that incorporates the twist angle and energetic scales. Then it is proved that the square of the hamiltonian, which is a $2\times 2$ matrix operator, turns out to be equivalent to a two-dimensional quantum harmonic oscillator. The interlayer currents between graphene's bipartite lattices are identified with the angular momentum term while the confinement potential is an effective quadratic potential. From there it is proved a limiting quantization rule for high-order magic angles, i.e., ${\alpha }_{m+1}-{\alpha }_m=3/2$ where $m$ is the order of the angle. All these results are in very good agreement with numerical calculations.

13 pages, 7 figures

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Cited by in corpus (9)

• Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands: Explicit Color-entangled Wavefunction and Exact Density Algebra
• Exact Many-Body Ground States from Decomposition of Ideal Higher Chern Bands: Applications to Chirally Twisted Graphene Multilayers
• Atomistic theory of moiré Hofstadter's butterfly in magic-angle graphene
• Fubini-Study metric and topological properties of flat band electronic states: the case of an atomic chain with $s-p$ orbitals
• Designing Moire Patterns by Shearing
• Flat bands without twists: periodic holey graphene
• Topological phase diagram of twisted bilayer graphene as a function of the twist angle
• Classically forbidden regions in the chiral model of twisted bilayer graphene. With an appendix by Zhongkai Tao and Maciej Zworski
• Moiré-driven equilibrium of perturbations in moiré systems