Coexistence and competition of nematic and gapped states in bilayer graphene

Abstract

In bilayer graphene, the phase diagram in the plane of a strain-induced bare nematic term, ${\mathcal{N}}_0$, and a top-bottom gates voltage imbalance, $U_0$, is obtained by solving the gap equation in the random-phase approximation. At nonzero ${\mathcal{N}}_0$ 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 ${\mathcal{N}}_0$, respectively. The existence of a critical end point, where the line of first-order phase transitions terminates, is predicted. For ${\mathcal{N}}_0=0$, a pure gapped state with a broken spin-valley symmetry is the ground state of the system. As ${\mathcal{N}}_0$ 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.

Comments: 8 pages, 4 figures; final published version; the text is slightly expanded to include the derivation of the gap equations and the free energy functional