The route of shear to Ising superconductivity in bilayer graphene

The application of shear to bilayer graphene leads to a sequence of AB and BA stacking domains, with the consequent appearance of low-energy flat bands in the electronic spectrum. We show that some of these bands have nonvanishing Chern number when resolved in the two graphene valleys. The effect of the Coulomb interaction forces the breakdown of parity and valley symmetry at integer fillings of the flat bands, resembling the case of twisted bilayer graphene. At filling factor n=3, we find parity (time-reversal) symmetry breaking, while at n=1 the dominant phase corresponds to the breakdown of valley symmetry (for each separate spin projection). The bands lack inversion symmetry, but it is still possible to form Cooper pairs with zero momentum, as the two electron sectors with different spin projection may have opposite valley symmetry breaking. We show that the strong anisotropy of the screened Coulomb interaction in the broken-symmetry phase leads to attractive channels in the Cooper-pair vertex, opening the possibility of Kohn-Luttinger superconductivity with critical temperatures well above 1 K in the sheared graphene bilayers.