Lattice models that realize $\mathbb{Z}_n$-1-symmetry protected topological states for even $n$

Higher symmetries can emerge at low energies in a topologically ordered state with no symmetry, when some topological excitations have very high energy scales while other topological excitations have low energies. The low energy properties of topological orders in this limit, with the emergent higher symmetries, may be described by higher symmetry protected topological order. This motivates us, as a simplest example, to study a lattice model of $\mathbb{Z}_n$-1-symmetry protected topological (1-SPT) states in 3+1D for even $n$. We write down an exactly solvable 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.

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