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.
Highlights
Consider a Landau symmetry-breaking transition [1,2] in a quantum system in n-dimensional space at zero temperature that completely breaks a finite on-site symmetry G
We would like to show that a similar symmetry “doubling” phenomenon appears for critical points of Landau symmetry-breaking phase transitions in all other dimensions
A state with the full categorical symmetry [i.e., both the G symmetry and the algebraic (n − 1)-symmetry] must be gapless. We show that such a gapless state describes the critical point of the Landau symmetry-breaking transition
Summary
Consider a Landau symmetry-breaking transition [1,2] in a quantum system in n-dimensional space at zero temperature that completely breaks a finite on-site symmetry G. The gapless state, at the critical point of G symmetry-breaking transition, has both the G zero symmetry and the algebraic (n − 1)-symmetry G(n−1) We call this combined symmetry a categorical symmetry. All possible gapless states in a Gsymmetric system are classified by gapless boundaries of G-gauge theory in one higher dimension This universal emergence of categorical symmetry at the critical point, and its origin from a noninvertible gravitational anomaly (i.e., topological order in one higher dimension), may help us to systematically understand gapless states of matter. We point out that the G-symmetric gapless critical theory at the (n + 1)D boundary has both the zero symmetry G and the dual algebraic (n − 1)-symmetry, which together form the categorical symmetry. We need to discuss the charge excitations of the symmetries
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