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

By Z. Gu, X. Wen. 2009

703 citations recorded by the source corpus.

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 fixedpoint tensors Tinv plus the symmetry group Gsym 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 Gsym,Tinv 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 Gsym,Tinv 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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/7a251b8092402f8761a15fe5e2b9cd2c7671ac7c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Ryan Thorngren
2. D. Else
3. E. Witten
4. R. Verresen
5. Juven C. Wang
6. Ashvin Vishwanath
7. C. Nayak
8. M. Hermele

In Condensed Matter Physics

1. Ryan Thorngren
2. E. Witten
3. R. Verresen
4. Yi Zhou
5. T. Ng
6. D. Else
7. M. Hermele
8. Juven C. Wang

Citations in the corpus

• Topological Phases Protected by Point Group Symmetry (2016, methodology): “Such phases are known to obey a Z2 classification [6,7,10,11], a conclusion reproduced by our analysis here.”
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions (2017, background): “[5], based on a tensor network renormalization calculation [6], where simple fixed-point tensors characterizing quantum phases can be formulated.”
• Quantum spin liquid states (2016, background): “In modern terminology, the end states of integer spin chains are a manifestation of symmetry-protected topological (SPT) order (Chen et al., 2012; Gu and Wen, 2009; Pollmann et al., 2012), which manifests itself as a boundary action that 8 is protected by rotational (SO(3)) symmetry.1 For…”
• Gapless Topological Phases and Symmetry-Enriched Quantum Criticality (2019, background): “[39] Zheng-Cheng Gu and Xiao-Gang Wen, “Tensorentanglement-filtering renormalization approach and symmetry-protected topological order,” Phys.”
• The continuum limit of loop quantum gravity - a framework for solving the theory (2014, neutral): “Tensor network coarse graining schemes [35] implement a recursive improvement of the amplitudes as in (10) and (16).”
• Pre-thermal phases of matter protected by time-translation symmetry (2016, background): “[1] Zheng-Cheng Gu and Xiao-Gang Wen, “Tensorentanglement-filtering renormalization approach and symmetry-protected topological order,” Phys.”