Ising superconductivity induced from valley symmetry breaking in twisted trilayer graphene

We show that the e-e interaction induces a strong breakdown of valley symmetry in twisted trilayer graphene, leading to a state where the two spin projections have opposite sign of the valley symmetry breaking order parameter. This leads to a spin-valley locking which has important implications for the superconductivity, explaining its protection against magnetic fields. The effect of valley symmetry breaking is validated as it reproduces the experimental observation of the reset of the Hall density at 2-hole doping. It also implies a reduction of the symmetry of the bands from C6 to C3, with an enhancement of the anisotropy of the Fermi lines which is at the origin of a KohnLuttinger (pairing) instability. The isotropy of the bands is gradually recovered, however, when the Fermi level approaches the bottom of the second valence band, explaining why the superconductivity is lost when doping beyond 3 holes per moiré unit cell in twisted trilayer graphene.