Emergent Symmetry and Phase Transitions on the Domain Wall of ${\mathbb{Z}}_2$ Topological Orders

The one-dimensional (1D) domain wall of 2D ${\mathbb{Z}}_2$ topological orders is studied theoretically. The Ising domain wall model is shown to have an emergent SU(2)${}_1$ conformal symmetry because of a hidden nonsymmorphic octahedral symmetry. While a weak magnetic field is an irrelevant perturbation to the bulk topological orders, it induces a domain wall transition from the Tomonaga-Luttinger liquid to a ferromagnetic order, which spontaneously breaks the anomalous ${\mathbb{Z}}_2$ symmetry and the time-reversal symmetry on the domain wall. Moreover, the gapless domain wall state also realizes a 1D topological quantum critical point between a ${\mathbb{Z}}_2^T$-symmetry-protected topological phase and a trivial phase, thus demonstrating the holographic construction of topological transitions.