Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum

Gil Young Cho, Ken Shiozaki, Shinsei Ryu, Andreas W.W. Ludwig

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20 Scopus citations


Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking 'Kitaev chain' superconductor (symmetry class D), the Haldane phase, and the classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.

Original languageEnglish (US)
Article number304002
JournalJournal of Physics A: Mathematical and Theoretical
Issue number30
StatePublished - Jul 3 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modeling and Simulation
  • Mathematical Physics
  • General Physics and Astronomy


  • Symmetry-protected topological phases
  • boundary conformal feld theory
  • entanglement spectrum


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