Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order
Author(s)
Gu, Zheng-Cheng; Wen, Xiao-Gang
DownloadGu-2009-Tensor-entanglement.pdf (634.5Kb)
PUBLISHER_POLICY
Publisher Policy
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Terms of use
Metadata
Show full item recordAbstract
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[subscript inv] plus the symmetry group G[subscript 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+1D quantum spin-1/2 and spin-1 models. In particular, using such a (G[subscript sym],T[subscript 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[subscript sym],T[subscript 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.
Date issued
2009-10Department
Massachusetts Institute of Technology. Department of PhysicsJournal
Physical Review B
Publisher
American Physical Society
Citation
Gu, Zheng-Cheng , and Xiao-Gang Wen. “Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order.” Physical Review B 80.15 (2009): 155131. © 2009 The American Physical Society
Version: Final published version
ISSN
1550-235X
1098-0121