1+1d SPT phases with fusion category symmetry: interface modes and non-abelian Thouless pump

Kansei Inamura, Shuhei Ohyama
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Abstract

We consider symmetry protected topological (SPT) phases with finite non-invertible symmetry $\mathcal{C}$ in 1+1d. In particular, we investigate interfaces and parameterized families of them within the framework of matrix product states. After revealing how to extract the $\mathcal{C}$-SPT invariant, we identify the algebraic structure of symmetry operators acting on the interface of two $\mathcal{C}$-SPT phases. By studying the representation theory of this algebra, we show that there must be a degenerate interface mode between different $\mathcal{C}$-SPT phases. This result generalizes the bulk-boundary correspondence for ordinary SPT phases. We then propose the classification of one-parameter families of $\mathcal{C}$-SPT states based on the explicit construction of invariants of such families. Our invariant is identified with a non-abelian generalization of the Thouless charge pump, which is the pump of a local excitation within a $\mathcal{C}$-SPT phase. Finally, by generalizing the results for one-parameter families of SPT phases, we conjecture the classification of general parameterized families of general gapped phases with finite non-invertible symmetries in both 1+1d and higher dimensions.
具有融合类对称性的 1+1d SPT 相:界面模式和非阿贝尔无苏泵
我们考虑 1+1d 中具有有限子不可逆对称性 $\mathcal{C}$ 的对称保护拓扑(SPT)相。特别是,我们在矩阵产物态的框架内研究了它们的界面和参数化族。在揭示了如何提取 $\mathcal{C}$-SPT 不变量之后,我们确定了作用于两个 $\mathcal{C}$-SPT 相界面的对称算子的代数结构。通过研究这个代数的表示理论,我们证明在不同的$\mathcal{C}$-SPT相之间一定存在一个退化的界面模。这一结果概括了普通 SPT 相的边界对应关系。然后,我们基于对$\mathcal{C}$-SPT态不变式的明确构造,提出了单参数族的分类。我们的不变量与Thouless电荷泵的非阿贝尔广义化是一致的,Thouless电荷泵是$\mathcal{C}$-SPT相内局部激发的泵。最后,通过归纳 SPT 相单参数族的结果,我们推测出在 1+1d 和高维度中具有有限不可逆对称性的一般参数族的分类。
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