-graphene boron-nitride moire superlattices*

Rhombohedral N = 3 trilayer graphene on hexagonal boron nitride (TG/BN) hosts gate tunable valley contrasting topological nearly flat bands that could trigger spontaneous quantum Hall phases under appropriate conditions of the valley and spin polarization. Recent experiments have reported signatures of C_ν = 2 valley Chern bands at 1/4 hole filling in contrast to the predicted value of C_ν = 3 when calculated within non-interacting single particle models. We discuss the low energy models for rhombohedral N-layer graphene (N=1,2,3) aligned with hexagonal boron nitride (hBN) under the influence of off-diagonal pseudo magnetic vector potential terms that alter the 120° rotational symmetry and leads to valley Chern numbers |C_ν | ≤ N. Our analysis suggests that nematic order with broken rotational symmetry can lead to valley Chern bands in keeping with recent Hall conductivity observations.