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

TL;DR

This paper explores the emergent symmetries and phase transitions on the domain wall of 2D $ ext{Z}_2$ topological orders, revealing a conformal symmetry, a topological quantum critical point, and symmetry-breaking phenomena.

Contribution

It uncovers the emergent SU(2)$_{1}$ conformal symmetry on the domain wall and demonstrates a holographic topological transition in a 1D topological quantum critical point.

Findings

Emergent SU(2)$_{1}$ conformal symmetry due to hidden symmetry.

Magnetic field induces a transition to ferromagnetic order on the domain wall.

Identification of a 1D topological quantum critical point.

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.