Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions

TL;DR

The paper establishes that zero-temperature symmetry breaking critical points possess a categorical symmetry that combines the ordinary $G$ symmetry with an emergent dual $(n-1)$-symmetry, realized as an algebraic higher symmetry or beyond higher groups depending on whether $G$ is Abelian or non-Abelian. It develops both patch-operator and holographic (bulk-boundary) frameworks to characterize these categorical symmetries, and demonstrates their role in constraining gapless critical states and neighboring gapped phases. Through concrete 1+1D Ising and 2+1D Ising/Z2 gauge examples, as well as boundary theories of topological orders such as the double-semion model, the work shows how the emergent symmetry governs dualities, anomalies, and the organization of phase transitions, including Higgs and confinement scenarios in higher dimensions. The results suggest a universal organizational principle for critical points and phase transitions in finite-group symmetric systems, with maximal categorical symmetries offering a promising route to classify and understand gapless conformal field theories via their bulk topological data.

Abstract

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 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+1D $Z_2$ 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.

Figures (7)

• Figure 1: The same Ising model can be describe by $H^\text{Is}$ or by $H^\text{DW}$. The $Z_2$ symmetry is explicit in the $H^\text{Is}$ description, while the $\t Z_2$ dual symmetry is explicit in the $H^\text{DW}$ description. The Ising model has both the $Z_2$ symmetry and $\t Z_2$ dual symmetry. The ground state usually spontaneously breaks one of the symmetries, except at the $B=J$ critical point, where both the symmetry and the dual symmetry ( the full $Z_2\vee \t Z_2$ categorical symmetry) are not spontaneously broken.
• Figure 2: The reduced lattice, where spin-$\frac{1}{2}$ degrees of freedom live on the links.
• Figure 3: A $Z_2^{(1)}$ neutral charge, a $s$ string on the 2d boundary (red dashed loop) is created by a $Z_2$ membrane operator Z23dmembrane in the 3d bulk (red surface). It is only neutral when on the boundary. If we translate this membrane to the bulk, there are $U_{Z_2}(S^1)$ operators in the bulk that anticommute the membrane operator, justifying it as the $Z_2$ vortex topological excitation.
• Figure 4: The conservation (the fusion rule) of the $Z_2$ point-like charge and $\t Z^{(1)}_2$ loop-like flux in the 3+1D $Z_2$ gauge theory give rise to the categorical symmetry of the 2+1D lattice model $H_1$lmG. The mod-2 conservation of $Z_2$ charge $e$ gives rise to the $Z_2$ 0-symmetry. The mod-2 conservation of $Z_2^{(1)}$ flux $s$ gives rise to the $Z_2^{(1)}$ 1-symmetry. $e$ and $s$ has a mutual $\pi$ statistics between them.
• Figure 5: The phase diagram of FS.