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1. Fine Structure of Flat Bands in a Chiral Model of Magic Angles

   https://doi.org/10.1007/s00023-024-01478-3  

   Becker, Simon; Humbert, Tristan; Zworski, Maciej (October 2024, Annales Henri Poincaré)

   Abstract We analyse symmetries of Bloch eigenfunctions at magic angles for the Tarnopolsky–Kruchkov–Vishwanath chiral model of the twisted bilayer graphene (TBG) following the framework introduced by Becker–Embree–Wittsten–Zworski. We show that vanishing of the first Bloch eigenvalue away from the Dirac points implies its vanishing at all momenta, that is, the existence of a flat band. We also show how the multiplicity of the flat band is related to the nodal set of the Bloch eigenfunctions. We conclude with two numerical observations about the structure of flat bands. 

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2. Dirac points for twisted bilayer graphene with in-plane magnetic field

   https://doi.org/10.4171/JST/504  

   Becker, Simon; Zworski, Maciej (June 2024, Journal of Spectral Theory)

   We study Dirac points of the chiral model of twisted bilayer graphene (TBG) with constant in-plane magnetic field. The striking feature of the chiral model is the presence of perfectly flat bands atmagic anglesof twisting. The Dirac points for zero magnetic field and non-magic angles of twisting are fixed at high symmetry pointsKandK'in the Brillouin zone, with\Gammadenoting the remaining high symmetry point. For a fixed small constant in-plane magnetic field, we show that as the angle of twisting varies between magic angles, the Dirac points move betweenK,K'points and the\Gammapoint. In particular, near magic angles, the Dirac points are located near the\Gammapoint. For special directions of the magnetic field, we show that the Dirac points move, as the twisting angle varies, along straight lines and bifurcate orthogonally at distinguished points. At the bifurcation points, the linear dispersion relation of the merging Dirac points disappears and exhibit a quadratic band crossing point (QBCP). The results are illustrated by links to animations suggesting interesting additional structure. 

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3. Mathematical results on the chiral models of twisted bilayer graphene (with an appendix by Mengxuan Yang and Zhongkai Tao)

   https://doi.org/10.4171/JST/521  

   Zworski, Maciej (August 2024, Journal of Spectral Theory)

   The study of twisted bilayer graphene (TBG) is a hot topic in condensed matter physics with special focus onmagic anglesof twisting at which TBG acquires unusual properties. Mathematically, topologically non-trivial flat bands appear at those special angles. The chiral model of TBG pioneered by Tarnopolsky, Kruchkov, and Vishwanath (2019) has particularly nice mathematical properties and we survey, and in some cases, clarify, recent rigorous results which exploit them. 

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4. Corrugation-driven symmetry breaking in magic-angle twisted bilayer graphene

   https://doi.org/10.1038/s42005-022-01013-y  

   Rakib, Tawfiqur; Pochet, Pascal; Ertekin, Elif; Johnson, Harley T. (October 2022, Communications Physics)

   Abstract The discovery of unconventional superconductivity in magic-angle twisted bilayer graphene (tBLG) supported the twist-angle-induced flat band structure predictions made a decade earlier. Numerous physical properties have since been linked to the interlayer twist angle using the flat band prediction as a guideline. However, some key observations like the nematic phase and striped charge order behind the superconductivity are missing in this initial model. Here we show that a thermodynamically stable large out-of-plane displacement, or corrugation of the bilayer, induced by the interlayer twist, demonstrates partially filled states of the flat band structure, accompanied by a broken symmetry, in the magic-angle regime and the presence of symmetry breaking associated with the superconductivity in tBLG. The distinction between low and high corrugation can also explain the observed evolution of the vibrational spectra of tBLG as a function of twist angle. Our observation that large out-of-plane deformation modes enable partial filling of states near the Fermi energy may lead to a strategy for offsetting the effects of disorder in the local twist angle, which suppresses unconventional superconductivity and correlated insulator behavior in magic-angle tBLG. 

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5. Multiple flat bands and topological Hofstadter butterfly in twisted bilayer graphene close to the second magic angle

   https://doi.org/10.1073/pnas.2100006118  

   Lu, Xiaobo; Lian, Biao; Chaudhary, Gaurav; Piot, Benjamin A.; Romagnoli, Giulio; Watanabe, Kenji; Taniguchi, Takashi; Poggio, Martino; MacDonald, Allan H.; Bernevig, B. Andrei; et al (July 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Moiré superlattices in two-dimensional van der Waals heterostructures provide an efficient way to engineer electron band properties. The recent discovery of exotic quantum phases and their interplay in twisted bilayer graphene (tBLG) has made this moiré system one of the most renowned condensed matter platforms. So far studies of tBLG have been mostly focused on the lowest two flat moiré bands at the first magic angle θ m1 ∼ 1.1°, leaving high-order moiré bands and magic angles largely unexplored. Here we report an observation of multiple well-isolated flat moiré bands in tBLG close to the second magic angle θ m2 ∼ 0.5°, which cannot be explained without considering electron–election interactions. With high magnetic field magnetotransport measurements we further reveal an energetically unbound Hofstadter butterfly spectrum in which continuously extended quantized Landau level gaps cross all trivial band gaps. The connected Hofstadter butterfly strongly evidences the topologically nontrivial textures of the multiple moiré bands. Overall, our work provides a perspective for understanding the quantum phases in tBLG and the fractal Hofstadter spectra of multiple topological bands. 

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