Symmetry Topological Orders (symTOs)

• Symmetry Topological Orders (symTOs) are a higher-dimensional framework that categorically encodes conventional, anomalous, and higher symmetries in quantum matter.
• They relate boundary operator algebras to (d+1)D braided fusion categories, enabling the classification of SPT, SET, and emergent symmetry phases.
• Practical methodologies like symmetry flux insertion and tensor network analysis extract anomaly data and reveal phase transitions in quantum systems.

Symmetry Topological Orders (symTOs) refer to the formalism and physical structures that encode not just conventional symmetries (as described by groups or higher groups), but their full generalized, potentially anomalous, non-invertible, and higher-categorical extensions via quantum topological orders in one higher dimension. In this sense, the symTO provides a unifying, operational, and classifiable framework for all generalized symmetry phenomena occurring in quantum phases of matter, encompassing symmetry-protected topological (SPT) phases, symmetry-enriched topological (SET) orders, subsystem symmetries, anomalous symmetries, soft symmetries, and emergent or emanant symmetries in condensed matter systems. The essential insight is that the algebra of local symmetric operators (including both point-like and extended operators) in a d-dimensional quantum system organizes into a braided fusion n-category, matching the data of a (d+1)D topological order with gappable boundary—thus encoding the ‘symmetry’ in the more general, non-group-theoretic sense (Chatterjee et al., 2022, Chatterjee et al., 2022, Inamura et al., 2023, Chen et al., 10 Sep 2025).

1. Formal Definition and Scope

A symTO is a topological order in one spatial dimension higher than the quantum system under consideration that, via a holographic correspondence, fully encodes the operational and anomaly data of generalized symmetry—including invertible, non-invertible, higher-form, and higher-categorical symmetries. Mathematically, for a system of dimension $d$, the symTO is a (d+1)-dimensional braided fusion (or, if necessary, fusion higher) category.

Key conceptual points:

• Ordinary global/group symmetries are included as a special case (via their representation categories).
• Non-invertible, anomalous (“algebraic”), and subsystem symmetries are natively described.
• The bulk topological order (symTO) is determined by the algebra of local (and transparent patch or commutant patch) symmetric operators in the boundary theory. This greatly generalizes the group-theoretic approach, replacing the group by a categorical structure (Chatterjee et al., 2022, Inamura et al., 2023).

Physically, this framework unifies the description of:

• SPT phases, which are short-range entangled states protected by symmetry and classified, for finite-land abelian symmetry, by group cohomology $H^{d+1}(G, U(1))$.
• SET orders, where intrinsic topological order is “enriched” by symmetry fractionalization, defects, domain walls, and permutation actions on anyons (Huang et al., 2013, Williamson et al., 2017, Lan et al., 2023, Ding et al., 24 Jun 2024).
• Gapped and gapless phases whose low-energy symmetry is non-invertible or emergent (Chen et al., 10 Sep 2025, Chatterjee et al., 2022).
• Fracton and subsystem-symmetry orders, for which the standard group-theoretic symmetry description fails (1803.02369).

2. symTOs as Categorical Symmetry: Operator Algebra and Holography

The construction of symTOs begins from the algebra of local symmetric operators, including special classes of “transparent patch operators” or “commutant patch operators,” extended operators built as products and sums of local operators with a prescribed transparency property—they commute with all local symmetric operators away from their boundary (Chatterjee et al., 2022, Inamura et al., 2023).

• These operators realize fusion, associativity (F-symbols), and braiding relations, and the algebra generated is a non-degenerate braided fusion (n-)category.
• The key principle is the symmetry/topological order (Symm/TO) correspondence: every generalized symmetry admits a unique associated bulk topological order whose excitations correspond to the “selection sectors” of the symmetry algebra (Chatterjee et al., 2022, Inamura et al., 2023).
• Boundary uniquely determines the bulk: the full (topological) operator algebra on the boundary determines the (d+1)-dimensional bulk topological order via center functor, in the Atiyah–Segal axiomatic TQFT sense (Inamura et al., 2023).

For example, in 1d spin chains with Z₂ symmetry, the algebra of local and nonlocal twist/string operators forms a fusion category whose Drinfeld center gives the corresponding 2d symTO. For anomalous symmetries or those with mixed anomaly, the associated bulk is a nontrivial topological order (e.g., double semion, double Ising, etc.) (Chatterjee et al., 2022, Inamura et al., 2023).

3. Detection and Characterization Methods

The operational extraction of symTO data (e.g., cohomology class, anomaly data, modular invariants) from ground states or low-energy spectra involves threading symmetry fluxes, examining nonlocal responses, and analyzing the action of symmetry on the entanglement spectrum or Schmidt states.

Notable methodologies include:

• Symmetry flux insertion: Introducing symmetry defects (e.g., by locally twisting the Hamiltonian or ground state with a group element) and examining the response—especially Berry phases under translation (momentum polarization)—to detect SPT invariants (Zaletel, 2013).
• Projective representations on entanglement cuts: Extracting the quantum numbers and projective symmetry class ($[\chi_g]$) of boundary modes via MPS/Schmidt decomposition (Zaletel, 2013).
• Cohomological invariants: Mapping physical measurements (Berry phases, projective representations) to elements of $H^{d+1}(G, U(1))$, fully determining the “type” of SPT or SET order in relevant cases (Zaletel, 2013, Lan et al., 2023).
• Tensor network and environment tensor approaches: Using iterative eigenvector procedures on PEPS or MPS to extract the “order parameter” in phases without conventional local signatures; a sharp transition in the symmetry structure of the environment tensor signals phase boundaries, including those between SPT, symmetry-breaking, and intrinsic topological orders (Liu et al., 2015).
• Commutant patch operators and their contractions: Computing contracted products of extended symmetric operators to obtain 2+1D topological invariants, even those beyond modular data (such as Borromean/Whitehead link invariants), thereby reconstructing the underlying symTO from 1+1D data (Inamura et al., 2023).

4. Group Cohomology, Anomalies, and Classification

SymTOs naturally unify the anomaly-theoretic and cohomological classification of quantum phases:

• For finite abelian symmetries in 2d, SPT orders are classified by $[ω]\in H^{3}(G, U(1))$ extracted via the slant product from measurements of spin and projective representations of symmetry defects (Zaletel, 2013).
• SET orders and more general symmetry-enrichments are formalized via G-crossed (or G-graded) braided fusion categories, where symmetry fractionalization, permutation, and higher-defect data are encoded categorically (Williamson et al., 2017, Lan et al., 2023).
• For crystalline (spatial) symmetries, folding techniques translate the enrichment problem into onsite layer-exchange symmetries with gapped boundary/junction classification, where H² and H¹ obstruction functions diagnose anomalous states and decorate defects with abelian anyons or SPT phases (Ding et al., 24 Jun 2024, Ding et al., 16 Feb 2025).
• Anomalous symmetries or mixed anomalies are embodied as nontrivial bulk symTOs; the correspondence between, for example, Z₂×Z₂ symmetry with mixed anomaly and Z₄ symmetry is encoded at the level of their shared bulk braided fusion category (Chatterjee et al., 2022).

5. symTOs in Gapless Phases, Emergent/Emanant Symmetries, and Soft Symmetry

SymTOs provide a canonical framework for describing the generalized (even emergent or emanant) symmetries of gapless and critical phases. In such contexts:

• The low-energy spectrum, resolved under various symmetry-twisted boundary conditions, is organized by the symTO; for the spin-½ antiferromagnetic Heisenberg chain, for example, the D₈ quantum double is identified as the symTO encoding all emergent and anomalous symmetry data (Chen et al., 10 Sep 2025).
• Both emergent continuous symmetries (e.g., SO(4)) and discrete (projective) symmetries are “holographically” captured by the categorical structure of the symTO—formally realized as a boundary of a (d+1)D topological order.
• “Soft symmetries,” corresponding to braided autoequivalences that act trivially on torus (anyon) labels but nontrivially on higher-genus surfaces, are realized physically by decorating submanifolds with gauged SPT defects that modify fusion junctions or the structure of gapped boundaries without permuting anyon types (Kobayashi et al., 6 Jan 2025). Such distinctions have consequences for boundary classification and noninvertible spontaneous symmetry breaking.

6. Practical Implications: SETs, Gapped Boundaries, and Phase Transitions

The categorical machinery underpinning symTOs directly informs the paper of SET orders, phase transitions, and gapped/gapless boundaries:

• SymTOs enable a complete classification of SET phases by prescribing when symmetry is preserved upon anyon condensation (existence of G-equivariant étale algebra structures) and when spontaneous symmetry breaking must occur (Lan et al., 2023, Schatz, 20 Aug 2024).
• The structure of gapped boundaries and symmetric junctions is governed by the presence or absence of obstruction classes ($H^1$, $H^2$, etc.), with explicit correspondences to the classification of onsite and crystalline SETs (Ding et al., 24 Jun 2024, Ding et al., 16 Feb 2025).
• Duality transformations, automorphism hierarchies (fixed charge/algebra/symmetry), and commutativity of gauging and condensation are all systematically controlled by the categorical framework (Lan et al., 2023, Aksoy et al., 27 Mar 2025).

7. Extensions: Noninvertible, Subsystem, and Higher Symmetry

The symTO formalism generalizes to accommodate:

• Noninvertible categorical gauge symmetries (e.g., the Fibonacci fusion 2-category associated with the doubled Fibonacci topological order), where symmetries act categorically on the gauge space of non-Abelian anyons without permutation (Zhao et al., 5 Aug 2024).
• Subsystem-protected topological orders (SSPTs), where symmetries act along lower-dimensional subsystems and their classification and phenomena are manifest in the patterns of domain wall and membrane operators (1803.02369).
• Higher-form and higher-group symmetries (as in emergent U(1) symmetries and their higher categorical generalizations), via their representation as higher braided fusion categories.

Conclusion

Symmetry Topological Orders (symTOs) formalize and operationalize the most general conception of symmetry in quantum matter via a systematic correspondence between the algebra of local/extended symmetric operators and topological field theory in one higher dimension. The symTO encodes all features of invertible, non-invertible, higher, and anomalous symmetries, and provides a foundational structure for classifying and detecting SPT, SET, SSPT, and other quantum phases—uniting the perspectives of group theory, tensor networks, algebraic topology, and categorical quantum field theory (Chatterjee et al., 2022, Chatterjee et al., 2022, Chen et al., 10 Sep 2025, Inamura et al., 2023, Zaletel, 2013, Williamson et al., 2017, Ding et al., 24 Jun 2024, Lan et al., 2023, Kobayashi et al., 6 Jan 2025, Zhao et al., 5 Aug 2024, Schatz, 20 Aug 2024, 1803.02369).