Lattice relaxation, mirror symmetry and magnetic field effects on ultraflat bands in twisted trilayer graphene

Twisted graphene multilayers exhibit strongly correlated insulating states and superconductivity due to the presence of ultraflat bands near the charge neutral point. In this paper, the response of ultraflat bands to lattice relaxation and a magnetic field in twisted trilayer graphene(tTLG) with different stacking arrangements is investigated by using a full tight-binding model. We show that lattice relaxations are indispensable for understanding the electronic properties of tTLG, in particular, of tTLG in the presence of mirror symmetry. Lattice relaxations renormalize the quasiparticle spectrum near the Fermi energy and change the localization of higher energy flat bands. Furthermore, different from the twisted bilayer graphene, the Hofstadter butterfly spectrum can be realized at laboratory accessible strengths of magnetic field. Our work verifies tTLG as a more tunable platform than the twisted bilayer graphene in strongly correlated phenomena.