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

TL;DR

This paper establishes a connection between symmetry-protected topological phases and boundary conformal field theories through the entanglement spectrum, revealing how boundary conditions encode topological properties.

Contribution

It demonstrates that boundary CFT boundary states, especially orbifolds, encode the topological class of SPT phases via quantum anomalies, providing a new characterization method.

Findings

Boundary CFT boundary states reflect SPT degeneracies

Orbifold boundary states carry topological quantum anomalies

Examples include Kitaev chain, Haldane phase, and BDI classification

Abstract

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 $\mathbb{Z}_8$ classification of interacting topological superconductors in symmetry class BDI in (1+1) dimensions.