Emergent Symmetry and Phase Transitions on the Domain Wall of $\mathbb{Z}_{2}$ Topological Orders

Abstract: 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.