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.

Weyl semimetals are crystalline solids that host emergent relativistic Weyl fermions and have characteristic surface Fermi-arcs in their electronic structure. Weyl semimetals with broken time reversal symmetry are difficult to identify unambiguously. In this work, using angle-resolved photoemission spectroscopy, we visualized the electronic structure of the ferromagnetic crystal Co3Sn2S2 and discovered its characteristic surface Fermi-arcs and linear bulk band dispersions across the Weyl points. These results establish Co3Sn2S2 as a magnetic Weyl semimetal that may serve as a platform for realizing phenomena such as chiral magnetic effects, unusually large anomalous Hall effect and quantum anomalous Hall effect.

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