Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases

We present systematic constructions of tensor-network wave functions for bosonic symmetry-protected topological (SPT) phases respecting both onsite and spatial symmetries. From the classification point of view, our results show that in spatial dimensions $d=1,2,3$, the cohomological bosonic SPT phases protected by a general symmetry group $\text{SG}$ involving onsite and spatial symmetries are classified by the cohomology group $H^{d+1}[\text{SG},\text{U}\left(1\right)]$, in which both the time-reversal symmetry and mirror-reflection symmetries should be treated as antiunitary operations. In addition, for every SPT phase protected by a discrete symmetry group and some SPT phases protected by continuous symmetry groups, generic tensor-network wave functions can be constructed which would be useful for the purpose of variational numerical simulations. As a by-product, our results demonstrate a generic connection between rather conventional symmetry-enriched topological phases and SPT phases via an anyon condensation mechanism.

Figure 1

The $Z_2$ symmetric wave function on the honeycomb lattice. Each site contains three qubits. The six qubits around each plaquette are all in the same spin state. The $Z_2$ symmetry flips spins, which acts as ${\sigma }_x$.

Figure 2

The tensor state representing the nontrivial $Z_2$ SPT wave function defined in Eq. (20). An internal leg supports two-dimensional Hilbert space. Physical states are labeled by numbers in the circle, while virtual states are labeled by numbers at the end of internal legs.

Figure 3

Symmetry conditions for the $Z_2$ symmetric state. Here, $X$ is short for ${\sigma }_x$, and $W_g$ ($W_g^{-1}$) denotes the associated gauge transformation. For the wave function defined in Eq. (20), $W_g=|11\rangle \langle 00|+i|10\rangle \langle 01|+i|01\rangle \langle 10|+|00\rangle \langle 11|$.

Figure 4

(a) Gauge transformations which leave local tensors invariant. Here, $Z$ is short for ${\sigma }_z$. (b) The condensation of visons carrying $Z_2$ symmetry charge.

Figure 5

Tensors representing the double-semion fixed-point wave function.

Figure 6

IGG for double-semion topological order.

Figure 7

(a) An example of the plaquette IGG element ${\lambda }_p$. (b) The plaquette IGG element formed by complex number $\chi$ and ${\chi }^{-1}$. This kind of plaquette IGG exists for any PEPS state. (c) A site tensor lives on the subspace which is invariant under action of IGGs. Here, $p_1,p_2,p_3,p_4$ are four neighboring plaquettes around the tensor and $\alpha ,\beta ,\gamma ,\delta$ denote legs of the tensor. The last equation indicates that a global IGG element is obtained from multiplication of plaquette IGG elements.

Figure 8

Measurement of the quantum number ${\chi }_{m_0}\left(g\right)$ carried by an $m$ particle for a local unitary symmetry $g$. According to Eqs. (40) and (52), $J$ commute with $W_g$, so we conclude that the quantum number is obtained by ${\chi }_{m_0}\left(g\right)J_p={}^{W_{g}g}{J}_p$.

Figure 9

(a) The procedure to create $|{\mathrm{\Psi }}_{ex}\rangle$ which is $\mathcal{T}·\mathcal{P}$ invariant. One first creates a pair of $m_0$ and $m_0^{†}$ from ground state, and then move away from each other. The global phase of $|{\mathrm{\Psi }}_{ex}\rangle$ by requiring the wave-function overlap between adjacent states to be real and positive. (b) $|{\mathrm{\Psi }}_{ex}^{\prime }\rangle$ is obtained by gluing between the original left half of the tensor network with the $\mathcal{T}·\mathcal{P}$ transformed left-half tensor network. The $\mathcal{T}·\mathcal{P}$ transformed left-half tensor network is obtained by transforming the physical legs of the left half via $\mathcal{T}·\mathcal{P}$, together with applying $W_{\mathcal{T}}\mathcal{T}·W_{\mathcal{P}}·{\mathcal{T}}^{-1}$ on all the virtual legs cut by the mirror line. ${\chi }_{m_0}(\mathcal{T}·\mathcal{P})$ is defined as phase difference between $|{\mathrm{\Psi }}_{ex}\rangle$ and $|{\mathrm{\Psi }}_{ex}^{\prime }\rangle$.

Figure 10

(a) The honeycomb lattice and generators of the lattice symmetry group. $u,v$ label sites while $a,b,c$ label bonds in one unit cell. (b) The IGG element formed by phases. We require ${\chi }_a·{\chi }_b·{\chi }_c=1$.

Figure 11

Symmetries and IGG in matrix product states.

Figure 12

(a) One configuration of “global” IGG element $\lambda (T_2{C}_6,T_1)$, where the dashed blue line means ${\chi }_{C_6}$ action. It is invariant under translation symmetries. (b) The decomposition into multiplication of plaquette IGG element ${\lambda }_p(T_2{C}_6,T_1)$. The plaquette IGG element is no longer translationally invariant if ${\chi }_{C_6}=-1$.