Lattice models that realize Z_n-1-symmetry protected topological states for even n

We study the lattice model of Z_n-1-symmetry protected topological states (1-SPT) in 3+1D for even n. We write down an exactly soluble lattice model and study its boundary transformation. On the boundary, we show the existence of anyons with non-trivial self-statistics. For the n=2 case, where the bulk classification is given by an integer m mod 4, we show that the boundary can be gapped with double semion topological order for m=1 and toric code for m=2. The bulk ground state wavefunction amplitude is given in terms of the linking numbers of loops in the dual lattice. Our construction can be generalized to arbitrary 1-SPT protected by finite unitary symmetry.