Figure 1

Illustration of Wen's plaquette model and its edge state. (a) Plaquette operator ${\prod }_{\langle ij\rangle \in P}{U}_{ij}$. A plaquette operator pairs up four Majorana fermions around the square. Here, blue circles represent four Majorana fermions per site, and red lines connecting two blue circles represent $U_{ij}$ appearing in the plaquette operator. (b) In terms of the complex fermions $f=\lambda \pm i\chi$ in Eq. (6), Wen's plaquette model can be understood as the stacked one-dimensional Kitaev chains parallel to $\widehat{x}$ and $\widehat{y}$. (c) Edge states of Wen's model where the edge is along the $\widehat{y}$ direction. The plaquette operators in Hamiltonian (5) cover all the Majorana fermions except a single Majorana fermion per site on the edge. The blue circles represent dangling Majorana fermions, while red circles are Majorana fermions covered by the plaquette operators. These blue dangling modes will hop around by the magnitude $\sim t$ when we are away from the exactly solvable point. (d) The dangling Majorana fermions can be equally well understood via the stacked Kitaev chain picture. When the Kitaev chain is cut, the edge always hosts a single Majorana fermion per chain. It is clear that the edge states of (c) and (d) are exactly the same.Reuse & Permissions

Figure 2

Stability of the gapless edge states. (a) To gap out the dangling Majorana fermions, the Majorana fermions should pair up and dimerize. However, it is clear from the figure that the dimerization doubles the unit cell along the boundary. This is similar to the physics of surface states of weak topological insulators (Refs. 52 and 53). (b) Coupling of the dangling Majorana fermions to the quantum Ising chain at criticality. As the Hilbert space per site of the Ising chain has the dimension $2={\left(\sqrt{2}\right)}^2$, the Ising chain can be thought as the chains of two Majorana fermions per site. So, when the Ising chain is coupled to the dangling Majorana fermions, there are three Majorana fermions per site which is guaranteed to be stable by counting Eq. (11). (c) This picture can be further supported by looking at momentum space. The Ising chain at criticality is equivalent to a helical Majorana fermion state at $k=\pi$ and the edge state consist of the dangling Majorana fermions of the spin liquid has one mode at $k=0$ and $\pi$. Thus, there are a total of three modes at $k=\pi$ ($k=0$ is decoupled from $k=\pi$ as the coupling between $k=0$ and $k=\pi$ would double the unit cell). (d) When the interaction between the Ising chain and the edge state of the ${\mathbb{Z}}_2$ spin liquid is turned on, we will be left with one reconstructed Majorana mode at $k=\pi$ and another mode at $k=0$.Reuse & Permissions