For a zero-temperature Landau symmetry-breaking transition in $n$-dimensional space that completely breaks a finite symmetry $G$, the critical point at the transition has the symmetry $G$. In this paper, we show that the critical point also has a dual symmetry—a $(n-1)$-symmetry described by a higher group when $G$ is Abelian or an algebraic $(n-1)$-symmetry beyond a higher group when $G$ is non-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of $G$-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk $G$-gauge theory gives rise to the symmetry and the dual symmetry, respectively. So any $G$-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry-breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3 + 1-dimensional $Z_2$ gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a $Z_2$ 0-symmetry and its dual, a $Z_2$ 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a $Z_2$ 1-symmetry and its dual, another $Z_2$ 1-symmetry.

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对称破缺和拓扑相变中的绝对对称性和不可逆异常

对于零温度的Landau对称破坏跃迁 $ñ$完全打破有限对称性的三维空间 $G$，过渡的临界点具有对称性 $G$。在本文中，我们表明临界点还具有双重对称性-a$（ñ-\mathrm{1个}）$-较高的群体描述的对称性$G$是阿贝尔语或代数$（ñ-\mathrm{1个}）$对称性超出更高的群体时$G$是非阿贝尔文的。实际上，任何$G$对称系统可以看作是 $G$量规理论的一个更高维度。散装中规整电荷和规整通量的守恒$G$规范理论分别引起对称和对偶对称。所以任何$G$对称系统实际上有一个较大的对称性，称为类别对称性，它是对称性和对偶对称性的组合。但是，分类对称的一部分（只有一部分）必须在系统的任何间隙阶段都自发地破坏，但是存在无间隙状态，其中分类对称性不会自发地破坏。这样的无间隙状态对应于Landau对称破坏转变的通常临界点。即使我们将对称性概念扩展为包括更高的对称性和代数的更高对称性，以上结果仍然有效。因此，我们的结果也适用于物质拓扑阶段之间转换的临界点。特别是，我们表明从3 + 1维过渡可能存在几个关键点${ž}_2$规范理论发展到一个微不足道的阶段。希格斯凝聚的临界点具有由${ž}_2$ 0对称及其对偶 ${ž}_2$ 2对称，而限制过渡的临界点具有由 ${ž}_2$ 1-对称及其对偶 ${ž}_2$ 1个对称。