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

Hong-Hao Song, Chen Peng, Rui-Zhen Huang, Long Zhang

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.

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

Abstract

The one-dimensional (1D) domain wall of 2D topological orders is studied theoretically. The Ising domain wall model is shown to have an emergent SU(2) 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 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 -symmetry-protected topological phase and a trivial phase, thus demonstrating the holographic construction of topological transitions.

Paper Structure

This paper contains 18 equations, 4 figures.

Figures (4)

  • Figure 1: Schematic illustration of the bond and spin operator indices in $\tilde{H}_{0}$ and $\tilde{H}_{h}$.
  • Figure 2: Perturbative RG flow of (a) the domain wall model $\mathcal{H}_{1}=\mathcal{H}_{0}+\mathcal{H}_{h}$ dictated by Eq. (\ref{['eq:rg1']}) and (b) the spin chain $\mathcal{H}_{2}=\mathcal{H}_{1}+\mathcal{H}_{g}$ given by Eq. (\ref{['eq:rg2']}) in the $K_{s}=0$ limit. In (a), the region in orange is the valence-bond solid (VBS) phase of the AF Heisenberg chain Sachdev2011, which is not realized in the domain wall model.
  • Figure 3: Numerical results of the domain wall model $H_{1}$. (a) Half-chain EE $S(L/2)$ versus the chain length $L$ in the logarithmic scale in the parameter range $-1.28\leq h\leq 0$. Solid lines are the fitting to $S(L/2)=(c/6)\ln L+a$, which give an estimate of central charges listed in the legend. (b) Squared FM and AF order parameters $m_{\mathrm{FM}}^{2}$ and $m_{\mathrm{AF}}^{2}$ versus $h$ for $L=400$. All calculations are performed with the open boundary condition.
  • Figure 4: (a) Quantum phase diagram of $H_{2}$. The purple segment on the vertical axis indicates the TLL phase of the domain wall model. Phase boundaries are extracted with the crossing of $\Delta L^{z}$ evaluated with different lattice sizes, for which an example at $h=0.3$ is shown in (b).