Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions

Traditional mean-field theory is a generic variational approach for analyzing symmetry breaking phases. However, this simple approach only applies to symmetry breaking states with short-range entanglement. In this paper, we describe a generic approach for studying two-dimensional (2D) quantum phases with long-range entanglement (such as topological phases). The method is based on (a) a general class of trial wave functions known as tensor-product states and (b) a 2D real-space renormalization group algorithm for efficiently calculating expectation values for these states. We demonstrate our method by studying several simple 2D quantum spin models exhibiting both symmetry breaking phase transitions and topological phase transitions. Our approach can be viewed as a unified mean-field theory for both symmetry breaking phases and topological phases.

Figure 1

Tensor network: a graphic representation of the tensor-product wave function (1), (a) on a 1D chain or (b) on a 2D square lattice. The indices on the links are summed over.

Figure 2

(a) The graphic representation of the inner product of TPS $⟨\Psi |\Psi ⟩$ in term of the tensor $T$ or the double tensor $\mathbb{T}$ [see Eq. (2)]. (b) After summing over $m$ and identifying $\left(i,i^{\prime }\right)\to \alpha$ and $\left(j,j^{\prime }\right)\to \beta$, we obtain the double tensor $\mathbb{T}$ from $T$ and $T^{\ast }$.

Figure 3

The indices of the double tensor have a range $D^2$. After combining the two legs on each side into a single leg, the four-linked double tensors in (a) can be viewed as a single double tensor ${\mathbb{T}}^{\prime }$ whose indices have a range $D^4$. (c) ${\mathbb{T}}^{\prime }$ can be approximately reduced to a “smaller” double tensor ${\mathbb{T}}^{″}$ whose indices have a range $D^2$ and satisfies $\text{t}\text{Tr}\left[{\mathbb{T}}^{\prime }\otimes {\mathbb{T}}^{\prime }\cdots \right]\approx \text{t}\text{Tr}\left[{\mathbb{T}}^{″}\otimes {\mathbb{T}}^{″}\cdots \right]$.

Figure 4

(Color online) (a) We represent the original rank-four tensor by two rank-three tensors, which is an approximate decomposition. (b) Summing over the indices around the square produces a single tensor ${\mathbb{T}}^{\prime }$. This step is exact.

Figure 5

(Color online) Under the TERG procedure, a tensor network is transformed into a coarse-grained tensor network.

Figure 6

The tensor trace of this simple tensor network gives us $⟨\Psi |{\widehat{O}}_i^1{\widehat{O}}_{j=NN\left(i\right)}^2|\Psi ⟩$. Again, the indices on connected links are summed over in the tensor trace.

Figure 7

A schematic plot of calculating a two-body correlation functions with separation $\left|{\mathbf{r}}_j-{\mathbf{r}}_i\right|=L=2^m$ assuming $L=\sqrt{N}/2$. The solid dots represent the impurity tensors and open dots represent the uniform tensors. We shrink the shaded region to a single point when doing coarse graining.

Figure 8

A schematic plot of possible final configurations when calculating correlation functions with a separation $\left|{\mathbf{r}}_j-{\mathbf{r}}_i\right|=L=2^m$.

Figure 9

The deformation and renormalization of double tensor ${\mathbb{T}}_{A\left(B\right)}$ on the honeycomb lattice.

Figure 10

(Color online) The tensor network on a honeycomb lattice with six impurity double tensors can also be coarse grained without generating more impurities.

Figure 11

A tensor network on Kagome lattice can be deformed into a tensor network on honeycomb lattice.

Figure 12

A tensor network on triangular lattice can be deformed into a tensor network on honeycomb lattice.

Figure 13

A schematic plot of realizing the decomposition in Fig. 12a by several SVD decompositions.

Figure 14

(Color online) (a) Magnetization along the $x$ direction $⟨{\sigma }^x⟩$ versus transverse field $h$. The derivative of magnetization has a singularity around $h\simeq 3.1$, indicating the second-order phase transition. (b) Magnetization along the $z$ direction $⟨{\sigma }^z⟩$ versus transverse field $h$. In the inset is the log plot of $⟨{\sigma }^z⟩$ versus $\left|h-h_c\right|$, where $h_c$ is the critical field.

Figure 15

Ground-state energies (with some constant shifts) of the transverse Ising model for different $D_{\text{cut}}$.

Figure 16

(a) The spin-1/2 model Eq. (15) on links of a square lattice. The dots represent the physical spin states which are labeled by $m=0,1$. The above graph can also be viewed as a tensor network where each dot represents a rank-three tensor $g$ and each vertex represents a rank-four tensor $T$. The two legs of a dot represent the $\alpha$ and $\beta$ indices in the rank-three tensor $g_{\alpha \beta }^m$. The four legs of a vertex represent the four internal indices in the rank-four tensor $T_{\alpha \beta \gamma \lambda }$. The indices on the connected links are summed over which define the tTr. (b) A tensor network of the double-line tensors, where each dot represents a double-line tensor $g$ and each vertex represents a double-line tensor $T$. The four legs of a dot represent the ${\alpha }_{1,2}$ and ${\beta }_{1,2}$ indices in the tensor $g_{{\alpha }_1{\alpha }_2;{\beta }_1{\beta }_2}^m$. The eight legs of a vertex represent the internal indices in the rank-four tensor $T_{{\alpha }_1{\alpha }_2;{\beta }_1{\beta }_2;{\gamma }_1{\gamma }_2;{\lambda }_1{\lambda }_2}$. The indices on the connected links are summed over which define the tTr.

Figure 17

(Color online) The black squares are (minimized) average energies of the $Z_2$ model (15) for the single-line tensor network Eqs. (18, 19)]. The red dots are (minimized) average energies for the double-line tensor network [Eqs. (22, 21)]. (Here again we choose $D_{\text{cut}}=18$ and total system size is $2^9\times 2^9$)

Figure 18

The graphic representation for the double-tensor ansatz in Eq. (20).

Figure 19

(Color online) $⟨B_p⟩$ versus $g/J$. The single-line variational wave function shows a jump in $⟨B_p⟩$, which indicates a first-order phase transition around $g/J=2.3$ (black squares). The double-line variational wave functions have no jump in $⟨B_p⟩$ but the discontinuity in the derivative indicates a second-order phase transition around $g/J=3.2$ (red dots).

Figure 20

(Color online) The double-semion model on the honeycomb lattice. The ground-state wave function (26) has a TPS representation given by the above tensor network. Note that $T$ and $g$ has a double-line structure. The vertices form a honeycomb lattice which can be divided into A and B sublattices.

Figure 21

$⟨B_p⟩$ versus $g/J$; the discontinuity of the derivative around $g/J=5.0$ indicates a second-order confinement-deconfinement phase transition. (Here again we choose $D_{\text{cut}}=16$ and total number of lattice sites is $2\times 3^{15}$.)