Search for: All records

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. 

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. 

Abstract Magic angles in the chiral model of twisted bilayer graphene are parameters for which the chiral version of the Bistritzer–MacDonald Hamiltonian exhibits a flat band at energy zero. We compute the sums over powers of (complex) magic angles and use that to show that the set of magic angles is infinite. We also provide a new proof of the existence of the first real magic angle, showing also that the corresponding flat band has minimal multiplicity for the simplest possible choice of potentials satisfying all symmetries. These results indicate (though do not prove) a hidden integrability of the chiral model. 

Abstract We study the propagation of wavepackets along curved interfaces between topological, magnetic materials. Our Hamiltonian is a massive Dirac operator with a magnetic potential. We construct semiclassical wavepackets propagating along the curved interface as adiabatic modulations of straight edge states under constant magnetic fields. While in the magnetic‐free case, the wavepackets propagate coherently at speed one, here they experience slowdown, dispersion, and Aharonov–Bohm effects. Several numerical simulations illustrate our results. 

Abstract We study discrete magnetic random Schrödinger operators on the square and honeycomb lattice. For the non-random magnetic operator on the hexagonal lattice with any rational magnetic flux, we show that the middle two dispersion surfaces exhibit Dirac cones. We then derive an asymptotic expansion for the density of states on the honeycomb lattice for oscillations of arbitrary rational magnetic flux. This allows us, as a corollary, to rigorously study the quantum Hall effect and conclude dynamical delocalization close to the conical point under disorder. We obtain similar results for the discrete random Schrödinger operator on the $$\mathbb Z^2$$-lattice with weak magnetic fields, close to the bottom and top of its spectrum.