Tensor product representation of a topological ordered phase: Necessary symmetry conditions

The tensor product representation of quantum states leads to a promising variational approach to study quantum phase and quantum phase transitions, especially topological ordered phases which are impossible to handle with conventional methods due to their long-range entanglement. However, an important issue arises when we use tensor product states (TPSs) as variational states to find the ground state of a Hamiltonian: can arbitrary variations in the tensors that represent ground state of a Hamiltonian be induced by local perturbations to the Hamiltonian? Starting from a tensor product state which is the exact ground state of a Hamiltonian with ${\mathbb{Z}}_2$ topological order, we show that, surprisingly, not all variations in the tensors correspond to the variation in the ground state caused by local perturbations of the Hamiltonian. Even in the absence of any symmetry requirement of the perturbed Hamiltonian, one necessary condition for the variations in the tensors to be physical is that they respect certain ${\mathbb{Z}}_2$ symmetry. We support this claim by calculating explicitly the change in topological entanglement entropy with different variations in the tensors. This finding will provide important guidance to numerical variational study of topological phase and phase transitions. It is also a crucial step in using TPS to study universal properties of a quantum phase and its topological order.

Figure 1

Hexagonal lattice where each link is occupied by two qubits. The three gray regions are configurations used for calculating topological entanglement entropy. Qubits on the boundaries are drawn explicitly while others are omitted for clarity. Notice that regions are always separated by breaking links in half. For ${\mathbb{Z}}_2$ model with string tension, numerical calculation for $S_{tp}$ is done for the three gray regions while for ${\mathbb{Z}}_2$ model with end of strings, analytically calculation is possible for any region.

Figure 2

(Color online) Tensor product representation of ${\mathbb{Z}}_2$ ground state. One tensor is assigned to every three qubits connected to the same vertex. Tensors $T_A$ are on vertices in sublattice $A$ and $T_B$ are in sublattice $B$. The out-of-plane gray links represent the physical indices of the qubits. The tensors connect according to the underlying hexagonal lattice.

Figure 3

Topological entanglement entropy $S_{tp}$ of the ${\mathbb{Z}}_2$ model with string tension calculated for the three gray regions as shown in Fig. 1. $1/g$ is the weight of each string segment relative to vacuum. $S_{tp}$ remains stable for a finite region away from the ideal ${\mathbb{Z}}_2$ model and goes sharply to zero at $g\sim 1.75$.

Figure 4

(Color online) Symmetry of the ${\mathbb{Z}}_2$ tensor. The tensor representing the ideal ${\mathbb{Z}}_2$ state is invariant under the action of $Z\otimes Z\otimes Z$ to its inner indices. The variation in string tension [Eq. (9)] does not break this symmetry and topological order is stable. The variation with end of strings [Eq. (11)] breaks this symmetry and destroys topological order

Figure 5

(Color online) Topological entanglement Renyi entropy $\left(S_{tpr}\right)$ calculated for the three gray regions as shown in Fig. 1. Size of regions grow from region 1 to 3. The calculation is done for 200 random tensors in the neighborhood of ${\mathbb{Z}}_2$ tensor. 100 of them have ${\mathbb{Z}}_2$ symmetry (plotted on the left hand side) while the other 100 have not (plotted on the right hand side). For tensors with ${\mathbb{Z}}_2$ symmetry, $S_{tpr}$ approach $-1$ very quickly as we include more and more qubits in the reduced region while for tensors without ${\mathbb{Z}}_2$ symmetry, $S_{tpr}$ goes toward 0 as the region gets larger.

Figure 6

Tensor network for calculating Renyi entropy $\left(\alpha =2\right)$ of the big site in a one-dimensional tensor product state. The contraction of the tensor network gives $\text{Tr}\left({\rho }^2\right)=\text{exp}\left[-S_2\left(\rho \right)\right]$, where $\rho$ is the reduced density matrix. The lowest level represents the state, where horizontal links represent inner indices along the one-dimensional chain and vertical links represent physical indices. Four copies of the state are stacked together and corresponding physical indices are connected between the levels. For physical indices outside the reduced region, the connection is between levels 1 and 2 and levels 3 and 4. For those in the reduced region, the connection is between levels 1 and 4 and levels 2 and 3.