Topological domain-wall pump with $\mathbb{Z}_2$ spontaneous symmetry breaking

A domain-wall pump by an extended cluster model of $S=1/2$ spins is proposed with local $U(1)$ gauge invariance. Its snapshot ground state is gapped and doubly degenerated due to $\mathbb{Z}_2$ invariance, which is broken by an infinitesimal boundary magnetic field. The ground state associated with the spontaneous symmetry breaking (SSB) is still symmetry-protected with additional spatial inversion that is characterized by the $\mathbb{Z}_2$ Berry phase. We investigate the topological domain-wall pump with/without boundaries. The topological pump associated with the inversion symmetry-breaking path induces a non-trivial Chern number of bulk and a singular behavior of edge states of the domain-wall. Generalization to the multi-spin interaction is also explicitly given.