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Quantum phase transitions out of a symmetry-protected topological (SPT) phasein (1+1) dimensions into an adjacent, topologically distinct SPT phaseprotected by the same symmetry or a trivial gapped phase, are typicallydescribed by a conformal field theory (CFT). At the same time, the low-lyingentanglement spectrum of a gapped phase close to such a quantum critical pointis known(Cho et al., arXiv:1603.04016), very generally, to be universal anddescribed by (gapless) boundary conformal field theory. Using this connectionwe show that symmetry properties of the boundary conditions in boundary CFT canbe used to characterize the symmetry-protected degeneracies of the entanglementspectrum, a hallmark of non-trivial symmetry-protected topological phases.Specifically, we show that the relevant boundary CFT is the orbifold of thequantum critical point with respect to the symmetry group defining the SPT, andthat the boundary states of this orbifold carry a quantum anomaly thatdetermines the topological class of the SPT. We illustrate this connectionusing various characteristic examples such as the time-reversal breaking"Kitaev chain" superconductor (symmetry class D), the Haldane phase, and the$\mathbb{Z}_8$ classification of interacting topological superconductors insymmetry class BDI in (1+1) dimensions.

Relationship between Symmetry Protected Topological Phases and Boundary Conformal Field Theories via the Entanglement Spectrum