Twisted bilayer graphene at charge neutrality: competing orders of SU(4) Dirac fermions

We study possible patterns for spontaneous symmetry breaking in a Dirac fermion model, which is applicable to twisted bilayer graphene at charge neutrality. We show how a chiral SU(4) symmetry emerges and construct the corresponding low-energy model that includes a Fierz-complete set of symmetry-allowed four-fermion interactions. We employ an unbiased renormalization group treatment to identify the critical points that describe transitions into different ordered phases. The resulting phase diagram depends on the number of fermion flavours and we show that the coupling between ordering channels prevents many of the possible mean-field orders from being accessible at relevant, small flavour numbers. We argue that, as a consequence, twisted bilayer graphene is governed by a quantum Hall state or an SU(4) manifold of insulating spin-valley orders with emergent Lorentz symmetry that contains inter-valley coherent, spin Hall, and valley Hall states. We study how SU(4)-breaking perturbations affect the accessibility and can additionally stabilize symmetry-broken (semi-)metallic states.