Title: Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries Show Chinese title

Title: 无能隙SPT相和具有非可逆对称性的自发对称破缺相的Hasse图

Abstract: We discuss (1+1)d gapless phases with non-invertible global symmetries, also referred to as categorical symmetries. This includes gapless phases showing properties analogous to gapped symmetry protected topological (SPT) phases, known as gapless SPT (or gSPT) phases; and gapless phases showing properties analogous to gapped spontaneous symmetry broken (SSB) phases, that we refer to as gapless SSB (or gSSB) phases. We fit these gapless phases, along with gapped SPT and SSB phases, into a phase diagram describing possible deformations connecting them. This phase diagram is partially ordered and defines a so-called Hasse diagram. Based on these deformations, we identify gapless phases exhibiting symmetry protected criticality, that we refer to as intrinsically gapless SPT (igSPT) and intrinsically gapless SSB (igSSB) phases. This includes the first examples of igSPT and igSSB phases with non-invertible symmetries. Central to this analysis is the Symmetry Topological Field Theory (SymTFT), where each phase corresponds to a condensable algebra in the Drinfeld center of the symmetry category. On a mathematical note, gSPT phases are classified by functors between fusion categories, generalizing the fact that gapped SPT phases are classified by fiber functors; and gSSB phases are classified by functors from fusion to multi-fusion categories. Finally, our framework can be applied to understand gauging of trivially acting non-invertible symmetries, including possible patterns of decomposition arising due to such gaugings. Show Chinese abstract

Abstract: 我们讨论具有非可逆全局对称性的(1+1)d无能隙相，也称为范畴对称性。 这包括表现出类似于能隙对称性保护拓扑（SPT）相性质的无能隙相，被称为无能隙SPT（或gSPT）相；以及表现出类似于能隙自发对称性破缺（SSB）相性质的无能隙相，我们称之为无能隙SSB（或gSSB）相。 我们将这些无能隙相以及能隙SPT和SSB相放入一个描述它们之间可能变形连接的相图中。 这个相图部分有序，并定义了一个所谓的Hasse图。 基于这些变形，我们识别出表现出对称性保护临界性的无能隙相，我们称之为内在无能隙SPT（igSPT）和内在无能隙SSB（igSSB）相。 这包括了第一个具有非可逆对称性的igSPT和igSSB相的例子。 这一分析的核心是对称拓扑场论（SymTFT），其中每个相对应于对称范畴Drinfeld中心中的凝聚代数。 从数学的角度来看，gSPT相由融合范畴之间的函子分类，推广了能隙SPT相由纤维函子分类的事实；而gSSB相由融合范畴到多融合范畴的函子分类。 最后，我们的框架可以应用于理解平凡作用的非可逆对称性的规范变换，包括由于这种规范变换可能产生的分解模式。