Spontaneous symmetry breaking in graphene subjected to an in-plane magnetic field

Application of the magnetic field parallel to the plane of the graphene sheet leads to the formation of electron- and holelike Fermi surfaces. Such situation is shown to be unstable with respect to the formation of an excitonic condensate even for an arbitrary weak magnetic field and interaction strength. At temperatures lower than the mean-field temperature, the order parameter amplitude is formed. The order parameter itself is a U(2) matrix allowing for the combined rotations in the spin and valley spaces. These rotations smoothly interpolate between site and bond centered spin-density waves and spin-flux states. The trigonal warping, short-range interactions, and the three-particle umklapp processes freeze some degrees of freedom at temperatures much smaller than the mean-field transition temperature, and make either Berezinskii-Kosterlitz-Thouless [Sov. Phys. JETP 32, 493 (1971); J. Phys. C 5, L124 (1972); 6, 1181 (1973)] (driven either by vortices or half-vortices) or Ising type transitions possible. Strong logarithmic renormalization for the coupling constants of these terms by the Coulomb interaction is calculated within one-loop renormalization group. It is found that in the presence of the Coulomb interaction, some short-range interaction terms become much greater than one might expect from the naive dimensionality counting.