Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations

Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite $Z_2$ symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the $Z_2$ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if $Z_2$ symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite $Z_2$ symmetry-protected topological order.

Figure 1

Fixed-point ground state of 1D SPT phase with onsite symmetry of group $G$. Each site contains two spins, which form the projective representation of class $\omega$ and $-\omega$, respectively. Connected spins form a dimer which forms a one-dimensional representation of $G$. On a finite segment of the 1D chain, the boundary spins form projective representations of $G$.

Figure 2

2D “bond” state, which is short-range entangled and symmetric under the onsite symmetry of group $G$. Each site contains four spins, each forming a projective representation of $G$. Two spins connected by a bond form a projective representations of class $\omega$ and $-\omega$, respectively. The “bond” represents an entangled state of the two spins, which forms an one-dimensional representation of $G$. On a lattice with boundary, the boundary degrees of freedom are spins with projective representation $\omega$ ($-\omega$).

Figure 3

$CZX$ model. (a) Each site (circle) contains four spins (dots) and the spins in the same plaquette (square) are entangled. (b) Onsite $Z_2$ symmetry is generated by $U_{CZX}=X_1{X}_2{X}_3{X}_{4}C{Z}_{12}C{Z}_{23}C{Z}_{34}C{Z}_{41}$ (c) A local term in the Hamiltonian, which is a tensor product of one $X_4$ term and four $P_2$ terms as defined in the main text.

Figure 4

(a) $CZX$ model on a disk with boundary. (b) Boundary effective degrees of freedom form a 1D chain which cannot have an SRE symmetric state. (c) Two boundaries together can have an SRE symmetric state which is a product of entangled pairs between effective spins connected by a dashed line.

Figure 5

Reduce combination of $T\left(g_2\right)$ and $T\left(g_1\right)$ into $T\left(g_1{g}_2\right)$.

Figure 6

Different ways to reduce combinations of $T\left(g_3\right)$, $T\left(g_2\right)$, and $T\left(g_1\right)$ into $T\left(g_1{g}_2{g}_3\right)$. Only the right projection operators are shown. Their combined actions differ by a phase factor $\phi (g_1,g_2,g_3)$.

Figure 7

Reduction of the combination of $T\left(g\right)$ and $A$ into $A$. Here $T^{i,i^{\prime }}\left(g\right)$ is an MPUO, $A^i$ is a matrix product state symmetric under $T^{i,i^{\prime }}\left(g\right)$.

Figure 8

Two ways of reducing the combination of $T\left(g_2\right)$, $T\left(g_1\right)$ and $A$ into $A$. Only the right projection operators are shown. Their combined actions differ by a phase factor $\varphi (g_1,g_2)$.

Figure 9

Different ways of reducing the combination of $T\left(g_3\right)$, $T\left(g_2\right)$, $T\left(g_1\right)$, and $A$ into $A$. Only the right projection operators are shown. Their combined actions differ by a phase factor written on the arrow.