If I have a many-body Hamiltonian that is invariant under the representation of a certain group G i.e [H,U(g)] = 0, how many phases can exist through spontaneous symmetry breaking (SSB) of this symmetry by the ground state? Specifically, if a certain subgroup H∈G can occur in G multiple times, does SSB to each of these represent different phases?
To make my question clear with an example, consider a Hamiltonian invariant under some representation of Z2×Z2={e,a,x,ax}. There are three ways the ground state can break down to Z2:
1)ZA2:{e,a},2) ZB2:{e,x},3) ZC2:{e,ax}.
Does the residual symmetry being the above three Z2 representations represent different phases of matter? Or are they the same?