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

TL;DR

This work establishes a principled link between symmetry-protected topological phases in (1+1)D and boundary conformal field theories by exploiting the universal structure of the entanglement spectrum near quantum critical points. By treating gapped SPTs as distinct BCFT boundary conditions of a shared critical theory, it uses orbifold BCFTs and boundary-state anomalies to diagnose symmetry-protected degeneracies and topological invariants. The framework is illustrated through concrete 1D examples: the Kitaev chain (class D), the Haldane phase, and (1+1)D topological superconductors in class BDI, demonstrating how boundary anomalies encode Z2 and Z8 classifications and how entanglement spectra reflect boundary CFT data. This approach provides a conceptual and calculational bridge between quantum criticality, boundary CFT, and 1D SPT topology, offering a lens to classify proximate phases and understand their phase diagrams in theory space.

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.

Figures (3)

• Figure 1: Deformation of the domain wall. (a) A domain wall with the size $a$, which is of the lattice scale. SPT phases will localize a zero mode at the domain wall. (b) The domain wall can smoothly be deformed to a bigger spatial region. In this manipulation of the domain wall, the topological zero mode cannot be removed. (c) When we push the domain wall to $L\gg a$, we effectively find a critical mode localized at the length scale $L$. Even in this limit, the topological zero mode will be superposed with the critical mode whose level spacing will be determined by the non-topological scale $\sim 1/L$. This picture suggests that the boundary zero mode of the SPT phases can be thought as the critical mode localized at the UV scale '$a$' as mentioned in the main text.
• Figure 2: Scattering from a (1+1)-dimensional SPT phase (shaded region). $\chi^{\mathrm{in}/\mathrm{out}}_{\mathrm{I}/\mathrm{II}}$ represent the amplitudes of the in-coming/out-going single-particle states in Region I/II (each of these amplitudes being an $N$-dimensional vector representing $N$ channels).
• Figure 3: Entanglement Hamiltonian of SPT phase (from Cho et al., arXiv:1603.04016). (i): The ground state of the (1+1) dimensional SPT phase of correlation length $\xi$, in the vicinty of a quantum critical point on the infinite space $-\infty < x < +\infty$, bipartitioned into region A (positive $x$) and region B (negative $x$). (ii): The entanglement Hamiltonian (defined on a space with coordinate $u$ which is different from $x$) is that of the CFT describing the quantum critical point, but confined to a finite interval of length $\ell= \ln (\xi/a)$. On the right hand side of the interval is an interface of the CFT with the gapped SPT, providing one boundary condition. On the other side of the interval the CFT simply ends, providing another boundary conditions ("free boundary condition"). (iii): The situation depicted in (ii) is a generalization of the "expanded Jackiw-Rebbi domain wall" depicted in FIG. \ref{['Fig1']} (c), to the case of a completely general interacting SPT.