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Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions

机译：SPT和SET状态的对称覆盖界面：系统构造

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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 + 1 D 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 + 1 D boundary of 2 + 1 D 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.
机译：对称保护拓扑（SPT）国家有边界特霍夫特异常是阻碍有效边界理论与UV完成自己的尺寸，并与现场摹-symmetry实现。在这项工作中，但我们表明，沿边界一定反常非现场ģ-symmetry通过合适的基团延长1→ķ→H→G→1作为扩展ħ-symmetry观察时，现场变。即，G中的非触发全球（仪表或引力）异常在h中自由变为异常。这引导我们构造适用于任何有限对称组的任何SPT状态的对称覆盖边界的完全可溶性的晶格路径积分和Hamiltonian，包括现场整体和难以逆转的时间反转对称。将所得的对称跳空边界既可以由H -symmetry扩展边界中任何时空尺寸或更自然地与全局对称G于一个3 + 1 d散装或如上所述，通过拓扑射ķ-Gauge理论。这种对称拓扑上有序边界的激发可以承载由H表示描述的对称G的分数量子数。 （将我们的方法应用于2 + 1 d散装的1 + 1 d边界，发现解构的仪表边界确实具有远程顺序的自发对称性。解缠绕的对称阶段在没有相反。）与通过对称性的已知的撕次边界或接口（用于仪表理论的全球对称性破坏或Anderson-Higgs机制），我们的方法基于对称扩展。更一般地，将我们的方法应用于SPT状态，拓扑有序的仪表理论和富集拓扑有序（设定）状态的对称性导致具有对称性断裂，对称延伸和动态测量的混合的通用边界或接口。

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