Such an understanding allows one to classify all 1D gapped quantum phases: All 1D gapped phases are classified by the following three mathematical objects: ( G H , G Ψ Ψ , H 2 [ G Ψ Ψ , U ( 1 ) ] ) , where G H is the symmetry group of the Hamiltonian, G Ψ Ψ the symmetry group of the ground states, and H 2 [ G Ψ Ψ , U ( 1 ) ] the second group cohomology class of G Ψ Ψ . (Note that H 2 [ G , U ( 1 ) ] classifies the projective representations of G .) If there is no symmetry breaking (ie G Ψ Ψ = G H ), the 1D gapped phases are classified by the projective representations of symmetry group G H .