Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory

Quantum-disordering a discrete-symmetry breaking state by condensing domain-walls can lead to a trivial symmetric insulator state. In this work, we show that if we bind a 1D representation of the symmetry (such as a charge) to the intersection point of several domain walls, condensing such modified domain-walls can lead to a non-trivial symmetry-protected topological (SPT) state. This result is obtained by showing that the modified domain-wall condensed state has a non-trivial SPT invariant -- the symmetry-twist dependent partition function. We propose two different kinds of field theories that can describe the above mentioned SPT states. The first one is a Ginzburg-Landau-type non-linear sigma model theory, but with an additional multi-kink domain-wall topological term. Such theory has an anomalous $U^k(1)$ symmetry but an anomaly-free $Z_N^k$ symmetry. The second one is a gauge theory, which is beyond Abelian Chern-Simons/BF gauge theories. We argue that the two field theories are equivalent at low energies. After coupling to the symmetry twists, both theories produce the desired SPT invariant.