We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the $1+1d$ case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors. An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as $1/L^{14}$. In addition to such new cases, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer—including a complete characterization of symmetry-enriched $1+1d$ Ising CFTs. Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched $2+1d$ CFT without gapped sectors.

中文翻译：

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无间隙拓扑相和对称性增强的量子临界性

我们为无间隙系统引入拓扑不变量并研究相关的边界现象。更一般地说，低能共形场理论 (CFT) 的对称特性提供了离散不变量，建立了对称性增强的量子临界性的概念。非局部缩放算子的电荷，或更一般地说，对称缺陷的电荷是拓扑的，暗示存在局部边缘模式。我们主要专注于$1+1d$边缘具有拓扑简并的情况，其有限大小的分裂在系统大小上可以是指数或代数的，这取决于附加间隙扇区的参与。通过将自旋 1 的海森堡链调整为破坏对称的 Ising 相位，给出了指数情况的一个示例。代数情况的一个例子出现在有间隙的 Ising 和集群阶段之间：这种对称性丰富的 Ising CFT 有一个边缘模式，其有限大小的分裂缩放为$1/{升}^{14}$. 除了这些新案例之外，我们的形式主义还统一了以前在文献中研究过的各种例子。与间隙对称保护拓扑相类似，给定的 CFT 可以分成几个不同的对称性丰富的 CFT。这就提出了分类的问题，对此我们给出了部分答案——包括对称性丰富的完整表征$1+1d$伊辛 CFT。非平凡的拓扑不变量也可以在更高的维度上构造，我们用对称性丰富的$2+1d$ CFT 没有缺口部门。