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Title: Gapless Symmetry-Protected Topological Order

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Resource Type:
Journal Article: Published Article
Journal Name:
Physical Review X
Additional Journal Information:
Journal Volume: 7; Journal Issue: 4; Related Information: CHORUS Timestamp: 2017-12-11 12:57:16; Journal ID: ISSN 2160-3308
American Physical Society
Country of Publication:
United States

Citation Formats

Scaffidi, Thomas, Parker, Daniel E., and Vasseur, Romain. Gapless Symmetry-Protected Topological Order. United States: N. p., 2017. Web. doi:10.1103/PhysRevX.7.041048.
Scaffidi, Thomas, Parker, Daniel E., & Vasseur, Romain. Gapless Symmetry-Protected Topological Order. United States. doi:10.1103/PhysRevX.7.041048.
Scaffidi, Thomas, Parker, Daniel E., and Vasseur, Romain. 2017. "Gapless Symmetry-Protected Topological Order". United States. doi:10.1103/PhysRevX.7.041048.
title = {Gapless Symmetry-Protected Topological Order},
author = {Scaffidi, Thomas and Parker, Daniel E. and Vasseur, Romain},
abstractNote = {},
doi = {10.1103/PhysRevX.7.041048},
journal = {Physical Review X},
number = 4,
volume = 7,
place = {United States},
year = 2017,
month =

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1103/PhysRevX.7.041048

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Cited by: 1work
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  • Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z{sub 2} symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related tomore » a nontrivial 3-cocycle of the Z{sub 2} group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z{sub 2} symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite Z{sub 2} symmetry-protected topological order.« less
  • 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{sub inv} plus the symmetry group G{sub 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, asmore » 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{sub sym},T{sub 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{sub sym},T{sub 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.« less
  • Cited by 7
  • The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. Here in this paper we study a specific class of such phase transitions in 1+1 dimensions – the phase transition between bosonic topological phases protected by Z n × Z n. We find in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transitionmore » and the other to the topological phase transition. We also obtained a constraint on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.« less