All Magic Angles in Twisted Bilayer Graphene are Topological

Abstract

We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semi-metallic and topological phases. In particular we are able to prove, using an approximate low-energy particle-hole symmetry, that the gapped set of bands that exist around all magic angles have nontrivial topology stabilized by a magnetic symmetry, provided band gaps appearing at fillings of ±4 electrons per Moiré unit cell. The topological index is given as the winding number (a Z number) of the Wilson loop in the Moiré BZ. Furthermore, we also claim that, when the gapped bands are allowed to couple with higher energy bands, the Z index collapses to a stable Z2 index. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: when strongly broken, non-topological phases can appear. Our paper underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a 4-band short range tight-binding model whose 2 lower bands have the same topology, symmetry, and flatness as those of the twisted bilayer graphene, and which can be used as an effective low-energy model. We then perform large-scale (11000 atoms per unit cell, 40 days per k-point computing time) ab-initio calculations of a series of small angles, from 3 • to 1 • , which show a more complex and somewhat different evolution of the symmetry of the low-energy bands than that of the theoretical Moiré model, but which confirms the topological nature of the system.

Derivation of Wannier orbitals and minimal-basis tight-binding Hamiltonians for twisted bilayer graphene: First-principles approach

“…Another way to resolve the obstruction is to augment the flat band manifold with additional degrees of freedom which are artificial as in Refs. 21,34 which doubled the bands. This new set of topologically non-trivial bands cancels the topology of the original flat bands.…”

Flat Chern Band from Twisted Bilayer MnBi2Te4

“…Experimentally realizing such states is, however, challenging because flat topological electronic bands are generally required for electron-electron interactions to manifest.Recently, it is shown that Moiré superlattices in twisted or lattice mismatched two-dimensional (2D) materials can give rise to flat topological bands. A prime example is twisted bilayer graphene (tBLG) [32][33][34][35], where the lowest two bands carry a fragile topology [36][37][38][39][40] and become flat near the magic twist angle θ ≈ 1.1 • . In addition, flat valley Chern bands can be realized in tBLG with aligned hBN substrate [41][42][43], twisted double bilayer graphene [44][45][46], ABC trilayer graphene on hBN [47][48][49] and twisted bilayer transition metal dichalcogenides [50,51], etc.…”

Magnon magic angles and tunable Hall conductivity in 2D twisted ferromagnetic bilayers

“…The spin-orbit coupling is negligibly weak in tBLG. Nevertheless, magic angle flat bands in tBLG are topologically nontrivial [57][58][59][60] , interpreted in terms of the pseudo magnetic fields generated by the moiré potential 60 . Quantum anomalous Hall effect was observed in magic angle tBLG on hexagonal Boron Nitride (hBN) substrate [61][62][63] .…”