Boundary conformal field theory and symmetry protected topological phases in $2+1$ dimensions

TL;DR

<p>We introduce a BCFT-based diagnostic for detecting nontrivial 2+1D SPT phases by examining the edge CFTs that reside on their boundaries. The central idea is that if the edge CFT cannot admit a symmetry-preserving Cardy boundary state, the bulk must be a nontrivial SPT; equivalently, edgeability and gappability are controlled by the existence of symmetry-invariant Ishibashi/Cardy constructions tied to Haldane's null-vector condition. The framework is applied to (i) time-reversal symmetric topological insulators, (ii) bosonic Z_N SPTs, and (iii) Z_2×Z_2 symmetric topological superconductors, revealing a correspondence between boundary state structure and bulk classification, including a Z8 result for the latter. This BCFT perspective connects modular consistency, boundary locality, and symmetry realization to SPT diagnostics, offering a concrete, edge-based route to classify and understand symmetry-protected topological phases with potential extensions to higher dimensions and dualities.</p>

Abstract

We propose a diagnostic tool for detecting non-trivial symmetry protected topological (SPT) phases protected by a symmetry group $G$ in 2+1 dimensions. Our method is based on directly studying the 1+1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests in the in-existence of boundary states that preserves both the conformal symmetry and the global symmetry $G$. We discuss the relation between edgeability, the ability to find a consistent boundary state, and gappability, the ability to gap out a CFT, in the presence of $G$. We study several cases including time-reversal symmetric topological insulators, $\mathbb{Z}_N$ symmetric bosonic SPTs, and $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric topological superconductors.

Figures (3)

• Figure 1: Edgeability and gappability of conformal field theories are closely related -- they both diagnose if they must be realized as a boundary of (topological) systems in one higher dimensions. Hence, edgeability and gappability are both related to quantum anomalies.
• Figure 2: We claim that one cannot "cut" or "make a boundary" while preserving $G$ symmetry for certain CFTs and certain symmetry implementations. These symmetry implementations correspond precisely to $(2 + 1)$d $G$-symmetric SPT phases and the corresponding CFTs are their edge theories.
• Figure 3: An illustration of the Cardy condition; a consistency condition for conformal boundary states. For boundary states that preserve conformal symmetry, the open channel partition function $Z_{\text{open}}:=\text{Tr}_{\mathcal{H}_{\text{open}}}\left[e^{-(2\pi i/\tau) H_{\text{open}}}\right]$ must equal the amplitude for a "Cardy" state $\mathcal{A}=\langle B|e^{2\pi i \tau H_{\text{closed}}}|B\rangle$