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

TL;DR

This paper reveals that symmetry-breaking and topological phase transitions possess a dual categorical symmetry structure, which influences the nature of critical points and phase transitions in various quantum systems.

Contribution

It introduces the concept of categorical symmetry combining symmetry and dual symmetry at critical points, extending to higher and algebraic symmetries in topological phases.

Findings

Critical points have dual symmetries described by higher groups or algebraic structures.

Systems can be viewed as boundaries of higher-dimensional gauge theories, revealing larger categorical symmetries.

Different critical points in 3+1D $Z_2$ gauge theory exhibit distinct categorical symmetry structures.

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.