Categorical Symmetries in Quantum Systems

• Categorical symmetries are advanced generalizations of conventional symmetries that use fusion categories to systematically encode non-invertible and higher-form defects.
• They enable a unified classification and realization of phases, anomalies, and topological field theories across quantum field theories and condensed matter systems.
• Fusion categories, module structures, and SymTFT frameworks provide actionable tools for experimentally probing these symmetries in systems like spin chains and atom arrays.

Categorical symmetries are an advanced generalization of conventional symmetry principles in quantum field theory, condensed matter systems, and mathematical physics. They extend the group-theoretic framework underlying the Landau paradigm to a categorical, fusion-based structure, thereby enabling the classification and realization of phases, anomalies, and critical phenomena not otherwise accessible to group symmetries. The implementation of categorical symmetries systematically incorporates non-invertible, higher, and duality symmetries through the machinery of fusion categories, higher fusion categories, module categories, and associated algebraic structures such as the Drinfeld center and tube algebras.

1. Mathematical Definition and Structural Properties

Categorical symmetries are modeled by fusion categories (or higher fusion categories), $(\mathcal{C},\otimes,\mathbf{1},F)$, where:

• Objects are topological defects (lines, surfaces, or higher codimension objects).
• Fusion rules: $a\otimes b = \sum_c N_{ab}^c\,c$, with $N_{ab}^c\in\mathbb{Z}_{\ge0}$.
• Associators (F-symbols): $$\alpha_{a,b,c}:(a\otimes b)\otimes c\to a\otimes (b\otimes c)$$, satisfying the pentagon axiom.
• Rigidity: each simple object has a dual.

The lack of invertibility ($\exists~a$ s.t. no $a^{-1}$ with $a\otimes a^{-1}\cong \mathbf{1}$) distinguishes these symmetries from group-like cases. Non-invertible defect lines or operators encode condensation, duality, or topological transformations unaccounted for by group symmetries (Warman et al., 19 Dec 2024, Bhardwaj et al., 2023, Costa et al., 13 Nov 2024). The Drinfeld center $Z(\mathcal{C})$ is a braided modular tensor category labeling topological bulk excitations and characterizing possible gapped boundaries and anomalies.

Fusion categories naturally generalize to higher fusion $n$-categories, encoding higher-form symmetries and more complex defect algebras with $k$-morphisms for $k\leq n$ (Kong et al., 2020, Bartsch et al., 2023).

2. Classification of Phases and Symmetry Topological Field Theory (SymTFT)

SymTFT formalism encodes categorical symmetry in a $(d+1)$-dimensional topological field theory whose bulk defects constitute the Drinfeld center $\mathcal{Z}(\mathcal{C})$ (Bhardwaj et al., 2023). The slab $M_{d+1} = M_d \times [0,1]$ is equipped with distinct topological boundaries:

• UV boundary $$\mathcal{A}_\mathrm{UV} \subset\mathcal{Z}(\mathcal{C})$$ encodes the categorical symmetry via a Lagrangian algebra.
• IR boundary $\mathcal{A}_\mathrm{IR}$ classifies the gapped phase.

Phases correspond to module categories over $\mathcal{C}$ (in $1+1$d, semisimple module categories), or to Lagrangian algebras in the modular category. The overlap determines vacua; defects ending on IR but not UV boundaries yield order parameters. The same SymTFT framework applies in arbitrary dimensions, providing a unification of SSB, SPT, and topological orders in the categorical Landau paradigm.

For explicit models, e.g., the $\mathsf{Rep}(D_8)$ spin chain (Warman et al., 19 Dec 2024), all gapped phases and second-order transitions (including SSB and SPT) can be enumerated (see Section 4 below).

3. Anomalies, Gauging, and Boundaries

Anomalies obstruct the consistent coupling of QFTs to background categorical symmetry: the existence of multiple, distinct gapped boundaries in $Z(\mathcal{C})$ is a hallmark of anomaly-free categorical symmetry (Zhang et al., 2023). In $1+1$ dimensions, anomaly detection reduces to the existence of a magnetic Lagrangian algebra $\mathcal{L}_m$ in $Z(\mathcal{C})$, orthogonal to the electric algebra $\mathcal{L}_e$; its absence signals an anomaly and enforces spontaneous symmetry breaking in any gapped realization. This test generalizes the group-cohomology obstruction to the full fusion category context.

Gauging categorical symmetries is realized via categorical condensation (promotion of anyons to Frobenius algebra objects and passage to local module categories (Yu, 2021)), or, in higher-dimensional settings, by folding module categories and taking Morita duals, encoding dual symmetries in spin systems (Delcamp et al., 2023).

4. Practical Realizations: Spin Chains, Atom Arrays, and Quantum Simulations

Realistic physical implementations are shown using spin chains and programmable atom arrays (Warman et al., 19 Dec 2024):

• Local Hilbert spaces, e.g., $V_i\cong\mathbb{C}[D_8]$ for $\mathsf{Rep}(D_8)$, embedded as three qubits per site.
• Benchmark Hamiltonians constructed from commuting projectors labeled by subgroups $F\subset D_8$ and cocycles $\beta\in H^2(F,U(1))$.
• Order parameters are generalized charges in twisted sectors, realized as Pauli strings or non-local operators.
• Critical points and transitions map to universal models such as the Ising or Ashkin-Teller chain.
• Digital quantum simulation protocols using Rydberg atom arrays with dual species and asymmetric blockade enable efficient realization of multi-qubit interaction terms, with dominant errors arising from Rydberg decay and Doppler dephasing.

This methodology generalizes to other fusion categories (e.g., $\mathsf{Rep}(S_3)$) and higher-dimensional systems. Experimental signatures include ground-state degeneracies under symmetry twists and extraction of non-local string correlators detecting SPT order.

5. Categorical Symmetry at Criticality, Gapless Phases, and Order Diagnostics

Gapless phases with categorical symmetry exhibit nontrivial critical behavior, distinct from adjacent gapped phases (Wu et al., 2020, Bhardwaj et al., 16 Mar 2025). At criticality, the fusion algebra rigidly governs defect operators, but expectation values (e.g., of order-diagnosis operators) exhibit scaling laws such as power, corner, or double log, distinguishing critical universality classes of SSB/SPT type enriched by categorical symmetry. These phenomena are systematically constructed by the "club sandwich" SymTFT formalism, encoding interface and phase transitions via generalized gauging and Kennedy-Tasaki transformations. Intrinsically gapless SPTs (igSPTs) appear when no gapped boundary preserves the full categorical symmetry, as for twisted $D_8$ DW theories.

6. Entanglement and Information-Theoretic Diagnostics

Categorical symmetries sharply affect entanglement measures. Symmetry-resolved entanglement entropy (SREE) generalizes to the categorical case (CaT-SREE) via projectors onto fusion-category charge sectors (Saura-Bastida et al., 9 Feb 2024). Equipartition of entanglement holds at leading order among categorical sectors when the symmetry is non-anomalous, breaking via quantum dimension corrections and subleading terms. Quantum relative entropy between symmetrized and original states serves as a universal order parameter for the presence and breaking of categorical symmetry, depending only on the set of quantum dimensions of the symmetry lines (Molina-Vilaplana et al., 20 Sep 2024).

7. Influence on Renormalization, Quantum Dynamics, and Physical Constraints

Categorical symmetries can protect couplings from renormalization in QFT. For instance, in $2$d NLSMs with WZ terms, self-duality defects implement discrete gauging and T-duality, enforcing exact protection of a coupling $\lambda$ that appears in the WZ term: $\beta(\lambda)=0$ up to two loops or non-perturbatively for group manifold targets (Arias-Tamargo et al., 24 Sep 2025). Such protection extends the familiar mechanisms of global symmetry conservation, furnishing new constraints on RG flows and possible quantum phase diagrams.

8. Outlook and Generalizations

Categorical symmetries unify invertible and non-invertible symmetry concepts, extend group-theoretic selection rules, explain robust IR constraints, and suggest new mechanisms for criticality, topological order, and dynamical constraints. Their mathematical formalism underpins advances in extended TQFT, anomaly matching, and lattice gauge theory. Future research directions include the full classification of categorical symmetries in high dimensions, explicit constructions for fermionic and chiral categories, and systematic experimental exploration in programmable quantum simulators. The conceptual toolkit provided by fusion categories, their centers, tube algebras, and associated diagnostics offers universal language and methods for analyzing and controlling symmetry properties in modern quantum many-body systems.