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We study possible patterns for spontaneous symmetry breaking in a Diracfermion model, which is applicable to twisted bilayer graphene at chargeneutrality. We show how a chiral SU(4) symmetry emerges and construct thecorresponding low-energy model that includes a Fierz-complete set ofsymmetry-allowed four-fermion interactions. We employ an unbiasedrenormalization group treatment to identify the critical points that describetransitions into different ordered phases. The resulting phase diagram dependson the number of fermion flavours and we show that the coupling betweenordering channels prevents many of the possible mean-field orders from beingaccessible 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 thatcontains inter-valley coherent, spin Hall, and valley Hall states. We study howSU(4)-breaking perturbations affect the accessibility and can additionallystabilize symmetry-broken (semi-)metallic states.

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