Non-Invertible Symmetry Breaking

• Non-invertible symmetry breaking is the disruption of global symmetries characterized by fusion algebra rather than invertible group operations.
• It employs categorical symmetry frameworks that integrate ordinary and dual higher-form symmetries to constrain phase diagrams and critical behavior.
• Operator algebra techniques, including patch and Wilson operators, reveal nontrivial mutual statistics and anomaly-induced constraints in quantum systems.

Non-invertible symmetry breaking refers to the spontaneous or explicit breaking of global symmetries whose operator algebra is governed not by a group, but by more general categorical or fusion-algebraic structures. Unlike conventional (invertible) symmetry operations, which possess inverses and obey group multiplication, non-invertible symmetry generators lack such inverses and frequently fuse according to rules of a fusion category or related algebraic structure. This framework significantly broadens the modern understanding of symmetries in quantum many-body systems, quantum field theories, and statistical systems, generating new constraints on phase diagrams, critical points, and the organization of excitations.

1. Categorical Symmetry and Dual/Higher Symmetries

A central development in non-invertible symmetry breaking is the introduction of "categorical symmetry," which generalizes group symmetry by combining ordinary ("0-form") symmetries with "dual" or higher-form symmetries. For a system with a finite on-site symmetry group $G$, the Hilbert space and operator content secretly encode a larger symmetry: the categorical symmetry $\mathcal{C} = G \vee G^{(n-1)}$, where $G^{(n-1)}$ is a dual algebraic (higher-form or higher-group) symmetry, depending on whether $G$ is Abelian or non-Abelian. The fusion rules reflect the interplay between representations of $G$ (charges) and dual representations associated with extended excitations (fluxes) or Wilson operators.

• In Abelian cases, $G^{(n-1)}$ can be realized as a higher-form symmetry, with charges living on submanifolds of codimension $(n-1)$ (e.g., lines, membranes).
• For non-Abelian $G$, the dual symmetry is algebraic and generally not a group, but rather a fusion $n$-category determined by the bulk topological order in one higher dimension.

Mathematically, the categorical symmetry is defined through a braided fusion category or $n$-category, often realized as the Drinfeld center of the representation category $n(G)$. In the holographic (bulk-boundary) viewpoint, a $G$-symmetric system can be viewed as a boundary of a $G$ gauge theory in one higher dimension, with symmetry and dual symmetry arising from bulk charge and flux conservation, respectively (Ji et al., 2019).

2. Emergence, Anomaly, and Partial Breaking

Categorical symmetries inherently possess a non-invertible gravitational anomaly: the fusion algebra cannot be implemented by local, invertible operators on the Hilbert space without extending the system (e.g., introducing a topological order in one higher dimension). In gapped phases, only one component of the categorical symmetry—either the $G$ symmetry or its dual—is spontaneously broken, but never both simultaneously.

• In a conventional symmetry-broken phase, the $G$ symmetry is broken and the dual symmetry remains unbroken (or vice versa).
• At gapless (critical) points, such as the Landau critical point or a transition between distinct topological orders, the entire categorical symmetry is preserved.
• The anomaly enforces that in any proximate gapped phase, some part of the categorical symmetry must be spontaneously broken (Ji et al., 2019).

This anomaly provides a mechanism for classifying and constraining possible infrared (IR) dynamics, including the impossibility of unique, gapped, symmetric ground states in systems with self-dual, non-invertible symmetry unless stringent arithmetic or topological conditions are met (Apte et al., 2022). In several dimensions, the critical point exhibits a richer symmetry than any single phase, which can restrict or enforce gapless behavior.

3. Patch Operators, Mutual Statistics, and Operator Algebras

The implementation of categorical symmetry in concrete models employs "patch" or "string" operators—generalized symmetry generators defined over regions or submanifolds. These operators obey algebraic relations encoding the non-invertible (mutually non-local) nature of the symmetry, often manifest as non-trivial commutation (or monodromy) between charge and dual operators.

For example, in $1+1$D Ising-type models:

• The on-site $Z_2$ symmetry generator $U_{Z_2}(i,j) = \prod_{k=i}^j X_k$ creates a pair of domain walls at the patch endpoints.
• The dual symmetry, represented by a Wilson loop $$W_q(S^1) = \mathrm{Tr} \{\prod_{l \in S^1} R_q(g)\}$$, creates extended excitations (fluxes).
• These satisfy a mutual algebra $$U_{Z_2}(i,j) W_q(S^1) = e^{i\pi} W_q(S^1) U_{Z_2}(i,j)$$ when the patch and loop overlap in a prescribed way.

Such mutual $\pi$-statistics generalizes “order-disorder” duality, and obstruction to simultaneous condensation underlies the impossibility of preserving the full categorical symmetry in a gapped phase. More generally, these operators can be organized into a fusion category with non-invertible fusion rules, whose simple objects correspond to topological defects or symmetry sectors (Ji et al., 2019).

4. Critical Points, Gapless States, and Conformal Field Theory

At phase transitions associated with non-invertible symmetry breaking, the critical state preserves the entire categorical symmetry. In $1+1$d, for instance, the critical Ising model at $B=J$ features both $Z_2$ and dual $Z_2$ symmetries; the critical theory’s partition function decomposes into modular-invariant combinations of conformal blocks, reflecting preservation of both symmetries:

$$Z(\tau) = |\chi_0(\tau)|^2 + |\chi_{1/2}(\tau)|^2 + |\chi_{1/16}(\tau)|^2.$$

Gapless critical points correspond to states invariant under the full categorical symmetry, while any neighboring gapped phase must spontaneously break at least one factor.

In higher dimensions, categorical symmetry places strict constraints. For instance, in self-dual $\mathbb{Z}_N^{(1)}$ gauge theory in $3+1$d, a non-invertible duality symmetry arising from gauging the one-form symmetry either enforces a gapless phase at self-duality or leads to spontaneous duality symmetry breaking with multiple vacua—except in cases where $N=k^2\ell$ and $-1$ is a quadratic residue modulo $\ell$, in which case a unique gapped symmetric ground state can exist (Apte et al., 2022).

5. Examples and Universal Implications

Topological Phase Transitions

• In the transition between $3+1$D $Z_2$ gauge theory and the trivial phase, the critical points correspond to distinct ways of partially breaking the categorical symmetry:
  • The Higgs transition (condensation of "electric" charges) has a categorical symmetry made of a $Z_2$ 0-symmetry and a dual $Z_2$ 2-symmetry.
  • The confinement transition (condensation of "magnetic" fluxes) features a $Z_2$ 1-symmetry and a dual $Z_2$ 1-symmetry (Ji et al., 2019).

Non-Abelian Generalizations

• For non-Abelian $G$, the dual symmetry is not a higher-form symmetry but an algebraic symmetry describable only by a fusion category structure. In $S_3$ gauge theories in $2+1$D, point-like excitations include pure charges, pure fluxes, and their composites, with non-integer quantum dimensions and fusion coefficients related by a detailed table (spin, fusion, and quantum dimension data) (Ji et al., 2019).

Wilson/Patch Operator Algebra

• In all these examples, explicit operator formulations are given for the symmetry generators, and their algebraic relations (often encoded by fusion, associator, and braiding data) can be systematically constructed and lead to observable signatures in spectrum degeneracies or scattering amplitude selection rules.

6. Connections to Broader Frameworks and Dynamics

Categorical symmetry and non-invertible symmetry breaking underlie a new paradigm generalizing Landau’s theory of phase transitions:

• The notion of “order parameter” is expanded to include non-local, extended operators whose condensation or absence signals symmetry breaking in the categorical sense.
• Spontaneous breaking of a categorical symmetry leads to representations of the observable algebra that are unitarily inequivalent, as rigorously established via algebraic quantum theory and the theory of superselection sectors (Sinha et al., 17 Feb 2025).
• The anomaly structure encoded in non-invertible symmetry defects can force anomalous, protected gaplessness, especially when the anomaly cannot be cancelled by extending the symmetry algebra. This enforces the presence of robust gapless edges or bulk critical points (Zotto et al., 25 Apr 2025).

Furthermore, the perspective of categorical symmetry fits naturally into the construction of extended topological quantum field theories, where the modular data and fusion categories arising from higher-dimensional gauge theory and their boundaries holographically encode the possible symmetry structures and symmetry-breaking scenarios (Ji et al., 2019, Zotto et al., 25 Apr 2025).

7. Summary Table: Key Features of Non-Invertible Symmetry Breaking

Concept	Abelian $G$	Non-Abelian $G$
Dual symmetry	Higher-form symmetry	Algebraic (fusion category)
Symmetry algebra	Group × group (partial)	Fusion category (no group)
Order parameter	String/membrane operator	Non-local operator
Gapped phase	Partial categorical SSB	Partial categorical SSB
Critical point	Full categorical symmetry	Full categorical symmetry
Anomaly (obstruction)	1D/2D bulk top order	Fusion category anomaly

This framework demonstrates that non-invertible symmetry breaking is a natural and robust extension of symmetry-based classification of phases and criticality. It unifies and generalizes conventional group-based symmetry breaking, highlighting new algebraic structures governing the organization of Hilbert space sectors, anomaly-induced constraints on ground states, and universal features at quantum critical points (Ji et al., 2019).