Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order

TL;DR

The paper introduces tensor-entanglement-filtering renormalization (TEFR), a tensor-network RG framework that, by filtering local entanglement, yields isolated fixed-point tensors that together with the Lagrangian symmetry form a universal $(G_{sym},T_{inv})$ phase classification. This approach unifies description of symmetry-breaking and topological phases, and enables quantitative access to critical data such as central charges and scaling dimensions from fixed-point tensors. Through detailed analyses of Ising, loop-gas, and spin-1/2/1 models, the authors demonstrate that the Haldane phase is a symmetry-protected topological phase stabilized by time-reversal, parity, and translation symmetries, while other transitions exhibit conformal critical behavior with calculable central charges. The TEFR framework thus provides a general, scalable method to classify phases and study transitions in strongly correlated systems across dimensions.

Abstract

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 (TEFR) approach that removes local entanglement and coarse grain the 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 $T_{inv}$ plus the symmetry group $G_{sym}$ 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 2D statistical Ising model, 2D statistical loop gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym}, T_{inv}) $ 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 $(G_{sym}, T_{inv})$ 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.

Figures (32)

• 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 $i$, $j$, . 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 combine the two legs on each side into a single leg, the four linked tensors in (a) can be viewed as a single tensor $T'$ in (b) with dimension $D^2$. $T'$ can be approximately reduced to a "smaller" tensor $T^{\prime\prime}$ in (c) with dimension $D$ and satisfies $\text{tTr} [ T^{\prime}\otimes T^{\prime} \cdots] \approx \text{tTr} [ T^{\prime\prime}\otimes T^{\prime\prime} \cdots]$.
• Figure 3: (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 $T'$. This step is exact.
• Figure 4: (Color online) RG transformation of tensor-network produces a coarse grained tensor-network.
• Figure 5: