Coexistence and competition of nematic and gapped states in bilayer graphene

In bilayer graphene, the phase diagram in the plane of a strain-induced bare nematic term N ₀ and a top-bottom gates voltage imbalance U ₀ is obtained by solving the gap equation in the random-phase approximation. At nonzero N ₀ and U ₀, the phase diagram consists of two hybrid spin-valley symmetry-broken phases with both nontrivial nematic and mass-type order parameters. The corresponding phases are separated by a critical line of first- and second-order phase transitions at small and large values of N ₀, respectively. The existence of a critical end point where the line of first-order phase transitions terminates is predicted. For N ₀=0, a pure gapped state with a broken spin-valley symmetry is the ground state of the system. As N ₀ increases, the nematic order parameter increases, and the gap weakens in the hybrid state. For U ₀=0, a quantum second-order phase transition from the hybrid state into a pure gapless nematic state occurs when the strain reaches a critical value. A nonzero U ₀ suppresses the critical value of the strain. The relevance of these results to recent experiments is briefly discussed.