TY - JOUR

T1 - Symmetric Gapped Interfaces of SPT and SET States

T2 - Systematic Constructions

AU - Wang, Juven

AU - Wen, Xiao Gang

AU - Witten, Edward

N1 - Funding Information:
J. W. thanks Thomas Church, Tian Lan, Kantaro Ohmori, and Pavel Putrov for conversations. J. W. gratefully acknowledges the Corning Glass Works Foundation Fellowship and the NSF. X.-G. W. research is supported by NSF Grants No. DMR-1506475 and No. NSFC 11274192. J. W. work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation Grant No. PHY-1066293. The research at I. A. S. is supported by the NSF Grants No. PHY-1606531 and No. PHY-1314311.
Publisher Copyright:
© 2018 authors.

PY - 2018/8/22

Y1 - 2018/8/22

N2 - Symmetry-protected topological (SPT) states have boundary 't Hooft anomalies that obstruct the 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 nonperturbative global (gauge or gravitational) anomaly in G becomes anomaly free in H. This guides us to construct an exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries applicable to any SPT state of any finite symmetry group, including on-site unitary and antiunitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an H-symmetry extended boundary in any spacetime dimension 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. (Applying 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 or interfaces obtained via symmetry breaking (either global symmetry breaking or the 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 or interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

AB - Symmetry-protected topological (SPT) states have boundary 't Hooft anomalies that obstruct the 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 nonperturbative global (gauge or gravitational) anomaly in G becomes anomaly free in H. This guides us to construct an exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries applicable to any SPT state of any finite symmetry group, including on-site unitary and antiunitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an H-symmetry extended boundary in any spacetime dimension 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. (Applying 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 or interfaces obtained via symmetry breaking (either global symmetry breaking or the 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 or interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.

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U2 - 10.1103/PhysRevX.8.031048

DO - 10.1103/PhysRevX.8.031048

M3 - Article

AN - SCOPUS:85052686782

VL - 8

JO - Physical Review X

JF - Physical Review X

SN - 2160-3308

IS - 3

M1 - 031048

ER -