Recent experiments on graphene multilayers under displacement field have demonstrated a wide variety of electronically ordered phases, including valley and/or spin polarized phases as well as potentially unconventional superconducting phases. In addition, quantum oscillation measurements in Bernal bilayer graphene showed the presence of electronic nematic states. Here, we investigate the emergence of nematic order in this system within a self-consistent Hartree-Fock framework, using a realistic band structure with trigonal warping and a dual-gate screened Coulomb interaction. We compute the phase diagram as carrier density and displacement field are varied, and find a sequence of isospin (spin and valley) polarized states consistent with experiment, including partially polarized phases with a large Fermi pockets for isospin-majority carriers and smaller Fermi pockets for isospin-minority carriers. Within these partially isospin polarized states, we identify regions with three $C_3$-symmetric pockets for the minority carriers and regions with only one or two such pockets, implying that the system develops a spontaneous nematic order that breaks $C_3$ symmetry. We find numerically that nematicity emerges near the boundary between fully and partially polarized phases and is controlled by the strength of the screened interaction. We analytically derive a criterion for the nematic order, which agrees with our numerical results.

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• Fermi surface
• Fermions
• Nematic order
• Phase diagrams
• Valley degrees of freedom
• Multilayer graphene