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

Abstract

We propose a diagnostic tool for detecting nontrivial 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 as the in-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability, i.e., the ability to find a consistent boundary state, and gappability, i.e., the ability to gap out a CFT, in the presence of G. We study several cases including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 ×Z2 symmetric topological superconductors. CONTENTS

Intrinsic and emergent anomalies at deconfined critical points

“…While anomalies in this example have been studied at length before [19,32], our interpretation of the "emergent anomaly" and its consequences is somewhat different from that in the literature.…”

“…Reference [31] discusses LSM-like anomalies at deconfined critical points using less formal methods. Reference [32] discusses LSM-like anomalies in a number of gapless systems, including the 1+1D S = 1/2 chain. Reference [25] provides a field-theoretic analysis of anomalies of the CP 1 model describing deconfined critical points in 1+1D and 2+1D; we give a slightly different derivation of these anomalies here and provide a physical interpretation.…”

Generalized Wen-Zee terms

“…3) to boson / spin systems [4]. Especially, internal symmetries, which simultaneously act on internal space (e.g., spin space of spin models) of each lattice site leaving spatial coordinates unaltered, have been systematically studied through different approaches in bosonic systems, such as group cohomology [4], cobordism groups [5,6], non-linear sigma models [7,8], topological field theories [9][10][11][12][13], conformal field theories [14][15][16][17], decoration picture [18], topological response / gauged theory [19][20][21][22][23][24], and projective / parton construction [25][26][27][28][29], braiding statistics approach [30][31][32][33] in different spatial dimensions.…”

Mixed global anomalies and boundary conformal field theories

“…Second motivation is that in some cases we can detect (2 + 1)-dimensional SPT phases ((1+1)-dimensional 't Hooft anomalies) from boundary states [20,21]. According to [20,21], if we find Cardy states invariant under symmetry transformations, the symmetry is anomaly free and they do not corresponds to the edge theory of non-trivial SPT phases. On the other hand, if we cannot construct such boundary states, the symmetry has 't Hooft anomalies.…”