Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

Abstract

Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and with an on-site G-symmetry. In this work, yet we show that a certain anomalous non-on-site G symmetry along the boundary becomes on-site when viewed as an extended H symmetry, via a suitable group extension 1 → K → H → G → 1. Namely, a non-perturbative global (gauge/gravitational) anomaly in G becomes anomaly-free in H. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, always existent for any SPT state in any spacetime dimension d ≥ 2 of any finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an H-symmetry extended boundary of bulk d ≥ 2, or more naturally by a topological emergent K-gauge theory with a global symmetry G on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry G, described by representations of H. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped boundaries/interfaces obtained via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT states, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

Quantum symmetries in orbifolds and decomposition

“…This paper is motivated by renewed recent interest in orbifolds and their anomalies, see e.g. [1][2][3][4][5][6][7][8][9]. Specifically, the need for this work arose while studying proposals for anomalyresolution in [2,4].…”

“…[1][2][3][4][5][6][7][8][9]. Specifically, the need for this work arose while studying proposals for anomalyresolution in [2,4]. As outlined in e.g.…”

“…Technically, the set of genus-one twisted sectors in the Γ orbifold forms a torsor under K 2. Note that the relations are almost but not quite symmetric under an S transformation.…”

Anomaly resolution via decomposition

“…This paper is devoted to a study of anomalous orbifolds and their resolutions, a subject that has been of renewed interest, see e.g. [1][2][3][4][5][6][7][8][9]. In particular, this paper utilizes decomposition (which relates e.g.…”

Global Anomalies and Bordism Invariants in One Dimension

“…This idea can be applied to topological quantum field theories (TQFTs) [15]. Anomalies in d-dimensional TQFT and its relations to (d + 1)-dimensional symmetry-protected topological (SPT) phases or (d + 1)-dimensional bordism invariants have been studied in [13,[16][17][18][19][20][21][22][23][24][25]. For each (d + 1)-dimensional bordism invariant, there exists essentially unique d-dimensional TQFT whose partition function is determined by the bordism invariant [26][27][28].…”