Before proceeding with the description of the VFCI results, we mention here that in the context of valleytronics, the valley isospin is dened by a three-dimensional vector V (in analogy with the regular spin vector S) which has three projections (V x , V y , V z ) [in analogy with the regular-spin projections (S x , S y , S z )]. The three valley projections V q , q = x, y, z, obey the same Lie algebra (commutation relations) as the three spin projections S q , q = x, y, z. Likewise, the valley Casimir operator is given by