Braiding statistics approach to symmetry-protected topological phases

We construct a two-dimensional (2D) quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a “symmetry-protected topological phase.” We describe a simple physical construction that distinguishes this system from a conventional paramagnet: We couple the system to a ${\mathbb{Z}}_2$ gauge field and then show that the $\pi$-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.

Figure 1

The Hamiltonians $H_0,H_1$ [Eqs. (1) and (2)] for the two spin models. (a) The Hamiltonian $H_0$ is a sum of single spin terms ${\sigma }_p^x$. (b) The Hamiltonian $H_1$ is a sum of seven spin terms $B_p=-{\sigma }_p^x{\prod }_{\langle pq{q}^{\prime }\rangle }{i}^{\frac{1-{\sigma }_q^z{\sigma }_{q^{\prime }}^z}{2}}$, where the product runs over the six triangles $\langle pq{q}^{\prime }\rangle$ containing $p$.

Figure 2

A schematic plot of the ground states ${\Psi }_0$ and ${\Psi }_1$ for the two paramagnets $H_0,H_1$. (a) In terms of domain wall configurations, the ground state ${\Psi }_0$ is a equal weight superposition of all configurations. (b) The ground state ${\Psi }_1$ is also a superposition of all domain wall configurations, but each configuration enters with a sign ${(-1)}^{N_{\text{dw}}}$, where $N_{\text{dw}}$ is the total number of domain walls.

Figure 3

The Hamiltonians ${\tilde{H}}_0,{\tilde{H}}_1$ [Eq. (8)] for the two gauged spin models. (a) The Hamiltonian ${\tilde{H}}_0$ is a sum of two terms. The first term is the gauge flux term ${\mu }_{pq}^z{\mu }_{qr}^z{\mu }_{rp}^z$ (thick triangle), where ${\mu }_{pq}^z$ denotes the ${\mathbb{Z}}_2$ gauge field on the link $\langle pq\rangle$. The second term is the spin interaction ${\sigma }_p^x{O}_p$, where $O_p={\prod }_{\langle pqr\rangle }(1+{\mu }_{pq}^z{\mu }_{qr}^z{\mu }_{rp}^z)/2$ and the product runs over the six triangles adjacent to $p$. (b) The Hamiltonian ${\tilde{H}}_1$ includes the same gauge flux term ${\mu }_{pq}^z{\mu }_{qr}^z{\mu }_{rp}^z$ but has a more complicated seven spin interaction ${\tilde{B}}_p{O}_p$ [Eq. (9)].

Figure 4

The string operator $V_{\beta }^0$ (11) is defined for any path $\beta$ on the dual honeycomb lattice and is given by a product of ${\mu }_{pq}^x$ over all links $\langle pq\rangle$ crossing $\beta$ (thickened lines). Applying this operator to the ground state $|{\Psi }_0\rangle$ creates two $\pi$ fluxes at the endpoints of $\beta$ (shaded triangles).

Figure 5

A schematic picture of the two states $V_{\beta }^0{V}_{\gamma }^0|{\Psi }_0\rangle$, $V_{\gamma }^0{V}_{\beta }^0|{\Psi }_0\rangle$. The first state (left) is obtained from a process in which two $\pi$ fluxes are created at the endpoints of $\gamma$, and then two more fluxes are created, braided around the path $\beta$ and then annihilated. The second state (right) corresponds to executing these two steps in the opposite order. We expect these two states to differ by the Berry phase $e^{2i\theta }$ associated with braiding one $\pi$-flux excitation around another. The same is true for $V_{\beta }^1,V_{\gamma }^1$.

Figure 6

The string operator $V_{\beta }^1$ [Eq. (14)] is defined for any path $\beta$ on the dual honeycomb lattice. It acts on all triangles $\langle pq{q}^{\prime }\rangle$ along the path $\beta$ (thickened lines). The action is different depending on whether $q,q^{\prime }$ are to the left of $\beta$ (purple sites) or to the right of $\beta$ (blue sites). Applying this operator to the ground state $|{\Psi }_1\rangle$ creates two $\pi$ fluxes at the endpoints of $\beta$ (shaded triangles).

Figure 7

The toric code and doubled semion models $H_{t.c},H_{d.s}$ [Eq. (19)]. In both systems, the Hilbert space is equivalent to a spin-$1/2$ model where the spins live on the links $l$ of the honeycomb lattice. (a) The toric code Hamiltonian $H_{t.c}$ is a sum of two terms. The first term $Q_v$ [Eq. (17)] is a product of ${\tau }_l^z$ over the three links adjacent to the vertex $v$. The second term involves the interaction ${\prod }_{l\in p}{\tau }_l^x$ which acts on the six links adjacent to the plaquette $p$. (b) The doubled semion Hamiltonian $H_{d.s}$ includes the same vertex term $Q_v$, but contains a more complicated plaquette term ${\prod }_{l\in p}{\tau }_l^x{\prod }_{l\in \text{legs}\text{of}p}f\left({\tau }_l^z\right)$, where $f\left(x\right)=i^{(1-x)/2}$.

Figure 8

(a) We consider a process in which two $\pi$ fluxes are created in the bulk, moved to the boundary along a path $\beta$ and then annihilated near points $a,b$. (b) We prove that the edge is protected by considering two paths $\beta ,\gamma$, and their corresponding flux creation/annihilation processes.

Figure 9

The operator $W_{\beta }$ [Eq. (23)] is defined for any path $\beta$ on the dual honeycomb lattice that joins points $a,b$ on the edge. In the interior of the path $\beta$ (shaded region), $W_{\beta }$ acts like the symmetry transformation $S={\prod }_p{\sigma }_p^x$. The operator also acts on all triangles $\langle pq{q}^{\prime }\rangle$ along the path $\beta$ (thickened lines). The action is different depending on whether $q,q^{\prime }$ are to the left of $\beta$ (purple sites) or to the right of $\beta$ (blue sites).

Figure 10

A schematic picture of the zero energy edge states in the case $H_{\text{edge}}=0$. (a) The $2^N$ zero energy edge states can be parametrized by boundary spin configurations $\{{\alpha }_1,...,{\alpha }_N\}$, where ${\alpha }_n=\uparrow ,\downarrow$. (b) For each choice of $\left\{\alpha \right\}$, the corresponding wave function ${\Psi }_{\left\{\alpha \right\}}$ is defined by ${\Psi }_{\left\{\alpha \right\}}\left(\left\{{\alpha }_{\text{int}}\right\}\right)={(-1)}^{N_{\text{dw}}}$, where $N_{\text{dw}}$ is the total number of domain walls in the system. We use a convention where we close up all the domain walls by assuming there is a ghost spin in the exterior of the disk, pointing in the $\uparrow$ direction.

Figure 11

The operator $B_n^{\uparrow }$ is defined just like $B_p$ [Eq. (2)], except with an additional ghost spin in the exterior of the disk pointing in the $\uparrow$ direction (dotted arrow). More explicitly, $B_n^{\uparrow }$ acts on the three triangles $\langle nq{q}^{\prime }\rangle$ containing the boundary spin $n$ with an action given by (37). The operator $B_n^{\downarrow }$ is defined similarly.

Figure 12

In the Turaev-Viro model (B1), the degrees of freedom $i,j,k=0,1$ live on the links of the triangulation and $G_{kln}^{ijm}$ gives a weight to each tetrahedron in the space-time triangulation. In the dual spin model, the degrees of freedom are Ising spins living on the vertices of the triangulation while the dual weight $\tilde{G}$ is defined by mapping domain walls between spins onto the link variables $i=0,1$. For example, the above configuration corresponds to $\tilde{G}(\downarrow ,\uparrow ,\downarrow ,\uparrow )=G_{110}^{110}$. The thick red lines denote links with $i=1$.