Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors $T_{\text{inv}}$ plus the symmetry group $G_{\text{sym}}$ of the tensors (i.e., the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional (2D) statistical Ising model, 2D statistical loop-gas model, and $1+1\text{D}$ quantum spin-1/2 and spin-1 models. In particular, using such a $\left(G_{\text{sym}},T_{\text{inv}}\right)$ characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The $\left(G_{\text{sym}},T_{\text{inv}}\right)$ characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.

Figure 1

A graphic representation of a tensor-network on a square lattice. A vertex represent a tensor $T_{ijfe}$ and the legs of a vertex carries the indices $i$, $j$, etc. Each link carries the same index. The indices on the internal links are summed over, which defines the tensor trace.

Figure 2

The tensor $T$ in the tensor network (a) has a dimension $D$. After we combine the two legs on each side into a single leg, the four linked tensors in (a) can be viewed as a single tensor $T^{\prime }$ in (b) with dimension $D^2$. $T^{\prime }$ can be approximately reduced to a “smaller” tensor $T^{″}$ in (c) with dimension $D$ and satisfies $\text{tTr}\left[T^{\prime }\otimes T^{\prime }\cdots \right]\approx \text{tTr}\left[T^{″}\otimes T^{″}\cdots \right]$.

Figure 3

(Color online) RG transformation of tensor network produces a coarse-grained tensor network.

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 $T^{\prime }$. This step is exact.

Figure 5

(a) A CDL tensor. (b) A network of CDL tensor.

Figure 6

The fixed-point tensor and its tensor network for topological phases.

Figure 7

(Color online) The TEFR procedure. We first deform the tensor network on square lattice (a) to the tensor network on lattice (b) using the transformations in Fig. 4a. We then replace the four rank-three tensors $S_1,S_2,S_3,S_4$ in (b) by a new set of simpler rank-three tensors $S_1^{\prime },S_2^{\prime },S_3^{\prime },S_4^{\prime }$ in (c) while keeping the tensor trace of the tensor network approximately the same (see Fig. 8). The index on a solid link has a dimension $D$. The index on a dashed link has a dimension $D^{\prime }$. Thus the tensor $S_i$ has a dimension $D\times D\times D$, while the tensor $S_i^{\prime }$ has a dimension $D\times D\times D^{\prime }$. The SVD method is used to go from (a) to (b) [see Fig. 4a] and from (c) to (d) (see Fig. 9). The four $S_i^{″}$’s on the small squares in (d) are traced over to obtain $T^{\prime }$ in (e).

Figure 8

(Color online) A tensor trace of a product of four rank-three tensors $S_1,S_2,S_3,S_4$ gives rise to one rank-four tensor. Such a rank-four tensor can also be produced (approximately) by a different set of rank-three tensors $S_1^{\prime },S_2^{\prime },S_3^{\prime },S_4^{\prime }$. We may choose $S_i^{\prime }$ trying to minimize the dimension of the index on the dashed links. Or we may choose $S_i^{\prime }$ trying to optimize some other quantities.

Figure 9

(Color online) The SVD method is used to implement the above two transformations.

Figure 10

TEFR transformation of a CDL tensor. Here the index on a dashed line has a dimension 1.

Figure 11

(Color online) A tensor network of CDL tensor can be reduced to a tensor network of dimension-one tensor. $\left(\text{a}\right)\to \left(\text{b}\right)$ uses the transformation in Fig. 10a. $\left(\text{b}\right)\to \left(\text{c}\right)$ uses the transformation in Fig. 10b. $\left(\text{c}\right)\to \left(\text{d}\right)$ uses the transformation in Fig. 9. $\left(\text{d}\right)\to \left(\text{e}\right)$ uses the transformation in $\left(\text{b}\right)\to \left(\text{c}\right)$ in Fig. 4. Here the index on a dashed line has a dimension 1.

Figure 12

(Color online) The temperature dependence of free energy (per site) $F\left(T\right)$ and $F^{″}={\partial }^{2}F/\partial T^2$ for the 2D statistical Ising model. A phase transition can be seen around the temperature $T\approx 2.3$. The vertical line marks the exact $T_c=2.26919$.

Figure 13

The graphic representations of (a) ${\sum }_{ru}{T}_{ruru}$, (b) ${\sum }_{ruld}{T}_{rulu}{T}_{ldrd}$, and (c) ${\sum }_{ruld}{T}_{ruld}{T}_{ldru}$.

Figure 14

(Color online) The temperature dependence of $X_1$, $X_2$, and the central charge $c$ for the 2D statistical Ising model. The vertical line marks the exact $T_c=2.26919$.

Figure 15

(Color online) The temperature dependence of $X_1$, $X_2$, and the central charge $c$ for the 2D statistical loop-gas model. The vertical line marks the exact $T_c=2.26919$.

Figure 16

(Color online) The $V$ dependence of $\text{ln}z=\text{ln}\left(Z\right)/N$ and ${\left(\text{ln}z\right)}^{″}=d^2\text{ln}z/d{V}^2$ for the generalized 2D loop-gas model. The sharpest peak for ${\left(\text{ln}z\right)}^{″}$ and the $\text{ln}z$ curve are for a tensor network with $N=4^7$ tensors. Other peaks are for tensor networks with $N=4^6$, $N=4^5$, $N=4^4$, and $N=4^3$ tensors.

Figure 17

(Color online) The $V$ dependence of $X_1$, $X_2$ for the generalized 2D loop-gas model.

Figure 18

(Color online) The $V$ dependence of the central charge $c$ for the generalized 2D loop-gas model. The sharpest peak is for a tensor network with $N=4^7$ tensors. Other peaks are for tensor networks with $N=4^6$, $N=4^5$, $N=4^4$, and $N=4^3$ tensors. The horizontal line marks $c=1$ (the expected central charge value at the critical point).

Figure 19

(a) The graphic representation of ${\left(e^{-\delta \tau H_i}\right)}_{m_i{m}_{i+1},m_i^{\prime }{m}_{i+1}^{\prime }}$. (b) $e^{-\delta \tau H_i}$ can be viewed as a rank-four tensor.

Figure 20

(Color online) A $T_a\text{−}{T}_b$ tensor network of size $L\times M$ is reduced to a $T_a^{\prime }\text{−}{T}_b^{\prime }$ tensor network of size $L\times M/2$. From $\left(\text{a}\right)\to \left(\text{b}\right)$, the local deformation Fig. 4a is used. From $\left(\text{b}\right)\to \left(\text{c}\right)$, the local deformation Fig. 9 is used. From $\left(\text{c}\right)\to \left(\text{d}\right)$, a local deformation similar to that in Fig. 4b is used.

Figure 21

(Color online) The color-map plot of $\left(r,g,b\right)=\left(X_1/3,X_2/3,1/2{\lambda }_{\text{sum}}\right)$ shows the phase diagram of our spin-1 system [Eq. (29)] which contains four phases, a trivial $S^z=0$ phase “TRI,” two $Z_2$ symmetry breaking phases $Z_2^x$ and $Z_2^y$, and a Haldane phase “Haldane,” The transition between those phases are all continuous phase transitions.

Figure 22

(Color online) The color-map plot of the central charge as a function of $U,B$. The color is given by $\left(r,g,b\right)=\left(c_8,c_9,c_{10}\right)$, where $c_i$ is the central charge obtained at the $i\text{th}$ iteration of Fig. 2. The lines with nonzero central charge are the critical lines with gapless excitations. The model [Eq. (29)] has a U(1) spin rotation symmetry along the $U=0$ line. The $U=0$ line between the two $Z_2$ phases is a gapless state.

Figure 23

(Color online) The color-map plot of $\left(r,g,b\right)=\left(X_1,X_2,1/{\lambda }_{\text{sum}}\right)$ shows the phase diagram of the spin-1 system [Eq. (34)] which contains four phases, a trivial phase TRI, two $Z_2$ symmetry breaking phases $Z_2^y$ and $Z_2^z$, and a Haldane phase Haldane. The transition between those phases are all continuous phase transitions.

Figure 24

(Color online) The central charge as a function of $U,B$. The lines with nonzero central charge are the critical lines with gapless excitations. The model [Eq. (34)] has a U(1) spin rotation symmetry along the $U=0$ line. The $U=0$ line between the two $Z_2$ phases is a gapless state.

Figure 25

(Color online) The color-map plot of $\left(r,g,b\right)=\left(X_1/3,X_2/3,1/2{\lambda }_{\text{sum}}\right)$ shows the phase diagram of the spin-1 system [Eq. (35)] which contains four phases, a trivial phase TRI, two $Z_2$ symmetry breaking phases $Z_2^x$ and $Z_2^y$, and a Haldane phase Haldane. The transitions between those phases are all continuous phase transitions.

Figure 26

(Color online) The color-map plot of the central charge as a function of $U,B$. The color is given by $\left(r,g,b\right)=\left(c_8,c_9,c_{10}\right)$, where $c_i$ is the central charge obtained at the $i\text{th}$ iteration of Fig. 2. The lines with nonzero central charge are the critical lines with gapless excitations.

Figure 27

(Color online) The fixed-point wave function of the Haldane phase can be obtained from the imaginary-time evolution of the fixed-point tensor network of the Haldane phase $T^{\text{Haldane}}=T\left({\sigma }^y,{\sigma }^y,{\sigma }^y,{\sigma }^y\right)$, which is formed by the corner matrix ${\sigma }^y$ (i.e., the boundary of the tensor network gives raise to the fixed-point wave function). The dimension 2 of the corner matrix implies that each index of the corner matrix is associated with a pseudo-spin-1/2. Thus each coarse-grained site has two spin-1/2 pseudospins. The corner matrix ${\sigma }^y$ ties the two pseudospins on the neighboring sites into a pseudospin singlets. Thus the fixed-point wave function of the Haldane phase can be viewed as a dimer state formed by nearest-neighbor pseudospins.

Figure 28

(Color online) The paradigm of topological order.

Figure 29

A rank-four tensor expressed as tensor network.

Figure 30

The entanglement-filtering procedure.

Figure 31

For the simplest CDL tensor, the outer line singular values are not scaled after one loop iteration, however, the inner line singular values are scaled to a power. This nontrivial scaling behavior is the key why the scaling-SVD algorithm can filter out those local entanglements for a CDL tensor.

Figure 32

A square-lattice tensor network in a 2D space (or $1+1\text{D}$ space time).