Tensor-product representations for string-net condensed states

We show that general string-net condensed states have a natural representation in terms of tensor product states (TPSs). These TPSs are built from local tensors. They can describe both states with short-range entanglement (such as the symmetry-breaking states) and states with long-range entanglement (such as string-net condensed states with topological/quantum order). The tensor product representation provides a kind of “mean-field” description for topologically ordered states and could be a powerful way to study quantum phase transitions between such states. As an attempt in this direction, we show that the constructed TPSs are fixed points under a certain wave-function renormalization-group transformation for quantum states.

Figure 1

(Color online) $Z_2$ gauge model on a square lattice. The dots represent the physical states which are labeled by $m$. The above graph can also be viewed as a tensor network, where each dot represents a rank-3 tensor $g$ and each vertex represents a rank-4 tensor $T$. The two legs of a dot represent the $\alpha$ and $\beta$ indices in the rank-3 tensor $g_{\alpha \beta }^m$. The four legs of a vertex represent the four internal indices in the rank-4 tensor $T_{\alpha \beta \gamma \lambda }$. The indices on the connected links are summed over which define the tensor trace tTr.

Figure 2

(Color online) The double-semion model on the honeycomb lattice. The ground-state wave function (5) has a TPS representation given by the above tensor network. Note that $T$ and $g$ has a double line structure. Note that the vertices form a honeycomb lattice which can divided into $A$ sublattice and $B$ sublattice.

Figure 3

(Color online) Using the fusion rules, we can represent the coherent states $|t,s,u,\cdots ⟩$ in terms of the orthogonal string-net states.

Figure 4

The graphic representation for the tensor $G_{tsu}^{\alpha \beta \gamma }$.

Figure 5

A tensor complex formed by vertices, links, and faces. The dashed curves are boundaries of the faces. The links that connect the dots carry index $\alpha ,\beta ,\dots$ and the faces carry index $u,s,\dots$. Each trivalent vertex represents a $T$ tensor. The vertices on $A$ sublattice (open dots) represent $T_{\alpha \beta \gamma ;t,s,u}$. The vertices on $B$ sublattice (shaded dots) represent $T_{\alpha \beta \gamma ;t,s,u}^{\prime }=T_{{\alpha }^{\ast }{\beta }^{\ast }{\gamma }^{\ast };t,s,u}$. The dots on the links represent the $g^m$ tensor $g_{\alpha ,\beta }^m$. In the weighted tensor trace, the $\alpha ,\beta ,\dots$ indices on the links that connect the dots are summed over, and the $u,s,\dots$ indices on the closed faces are summed over with a weighting factor $a_u{a}_s\dots$.

Figure 6

The graphic representation of (a) the $T$ tensor, $T_{\alpha \beta \gamma ;tsu}$, and (b) the $g^m$ tensor $g_{{\alpha }_A{\alpha }_B}^m$.

Figure 7

Three $T$ tensors associated with different orientations of the legs are related: $T_{\alpha \beta \gamma ;tsu}$, $T_{\alpha \beta \gamma ;tsu}^{\prime }=T_{{\alpha }^{\ast }{\beta }^{\ast }{\gamma }^{\ast };tsu}$, and $T_{\alpha \beta \gamma ;tsu}^{″}=T_{\alpha \beta {\gamma }^{\ast };tsu}$.

Figure 8

(Color online) (a) The string-net condensed states on arbitrary trivalent graph in two dimensions can be written as (b) a weighted tensor trace over a tensor complex formed by $T$ and $g^m$ tensors.

Figure 9

A tensor complex with vertices, links, and faces is formed by the double tensors $\mathbb{T}$ and ${\mathbb{T}}^{\prime }$. (Dashed curves are boundaries of the faces.) The links that connect the dots carry index $i,j,\dots$ and the faces carry double-index $u_1{u}_2,s_1{s}_2,\dots$. Each trivalent vertex represents a double-$\mathbb{T}$ tensor. The vertices on $A$ sublattice (open dots) represents ${\mathbb{T}}_{ijk;t_1{t}_2,s_1{s}_2,u_1{u}_2}$. The vertices on $B$ sublattice (shaded dots) represents ${{\mathbb{T}}^{\prime }}_{ijk;t_1{t}_2,s_1{s}_2,u_1{u}_2}={\mathbb{T}}_{i^{\ast }{j}^{\ast }{k}^{\ast };t_1{t}_2,s_1{s}_2,u_1{u}_2}$. In the weighted tensor trace, the $i,j,\dots$ indices on the links that connect the dots are summed over and the $u_1{u}_2,s_1{s}_2,\dots$ indices on the closed faces are summed over independently with a weighting factor $a_{u_1}{a}_{u_2}{a}_{s_1}{a}_{s2}\dots$.

Figure 10

Graphic representations for the three double-$\mathbb{T}$ tensors ${\mathbb{T}}_{ijk;t_1{t}_2,s_1{s}_2,u_1{u}_2}$, ${\mathbb{T}}_{ijk;t_1{t}_2,s_1{s}_2,u_1{u}_2}^{\prime }={\mathbb{T}}_{i^{\ast }{j}^{\ast }{k}^{\ast };t_1{t}_2,s_1{s}_2,u_1{u}_2}$, and ${\mathbb{T}}_{ijk;t_1{t}_2,s_1{s}_2,u_1{u}_2}^{″}={\mathbb{T}}_{ij{k}^{\ast };t_1{t}_2,s_1{s}_2,u_1{u}_2}$.

Figure 11

Graphic representation of the basic rules for the double tensor $\mathbb{T}$: (a) the deformation rule and (b) the reduction rule.

Figure 12

A useful reduction of tensor complex.

Figure 13

(Color online) The coarse-graining procedure for honeycomb lattice.

Figure 14

(Color online) A tensor network where each tensor has triple-line structure. The tensor trace sums over all the indices on the links that connect two dots.

Figure 15

(Color online) (a) The $T$ tensor, $T_{u^{\prime }\alpha u;t^{\prime }\beta t;s^{\prime }\gamma s}=T_{\alpha \beta \gamma ;tsu}^0{\delta }_{u{t}^{\prime }}{\delta }_{t{s}^{\prime }}{\delta }_{s{u}^{\prime }}$, and (b) the $g^m$ tensor $g_{u_A^{\prime }{\alpha }_A{u}_A;u_B^{\prime }{\alpha }_B{u}_B}^m=h_{{\alpha }_A{\alpha }_B;u_A{u}_B}^m{\delta }_{u_A{u}_B^{\prime }}{\delta }_{u_B{u}_A^{\prime }}$, with a triple-line structure.

Figure 16

(Color online) The $T\text{−}g$ from TPS representations for string-net states can be deformed to standard TPS representations by splitting the matrix $g^m$ into two matrices $g^{m_A}$, $g^{m_B}$ on each link and recombining three sites around a vertex (the three states inside a dashed circle) into one site.