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

TLDR

The isolated fixed‑point tensors and their symmetry group characterize various phases, providing a more general framework than boundary‑spin or string‑order methods. The study investigates the renormalization‑group flow of statistical and quantum Lagrangians by expressing their path integrals as tensor networks. The tensor renormalization approach enables analysis of continuous phase transitions between symmetry‑breaking and topological phases. The tensor‑entanglement‑filtering renormalization approach yields a fixed‑point structure that characterizes symmetry‑breaking and topological phases—including the Haldane phase protected by time‑reversal, parity, and translation symmetries—and allows calculation of scaling dimensions and central charges at continuous phase transitions.

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 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 $({G}_{\text{sym}},{T}_{\text{inv}})$ 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 $({G}_{\text{sym}},{T}_{\text{inv}})$ 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.