If we have a model with more than one Higgs-doublet, when do the VEV of a scalar field of one of those doublets must induce a nonzero VEV on a scalar field of another of those doublets?
In the two Higgs doublets models, there are two cases:
- The two doublets are independent
- There is a cross potential term between these doublets a la: $\phi_1^2\phi_2^2$
For case 1, the two VEVs are independent.
For case 2, the two VEVs are are coupled to each other. But it is hard to tell which is inducing which.