Unannihilable Symmetries in Quantum Theories

• Unannihilable symmetries are nontrivial symmetry structures that defy conventional trivialization by lacking an identity fusion channel and requiring extra sequential processes.
• They emerge in diverse frameworks such as lattice models, QCAs, and supergravity, manifesting through hidden conserved observables and robust topological defects.
• These persistent symmetries resist standard gauging and on-site representations, offering deep insights into dualities, anomaly obstructions, and higher-categorical structures.

Searching arXiv for papers on unannihilable/non-invertible symmetries and closely related frameworks. arXiv search query: "unannihilable symmetries non-invertible symmetry generalized symmetry" Unannihilable symmetries are symmetry structures whose defining data cannot be eliminated in the usual ways—by fusion to the identity, by rewriting the action as an on-site operator, by coupling to gauge fields, or by flowing to a featureless symmetric gapped phase. In recent arXiv literature the phrase does not denote a single universal formalism. Instead, it appears as a precise fusion-theoretic notion for lattice generalized symmetries, and as a closely related descriptive label for hidden conserved observables, anomaly-free but non-onsiteable quantum cellular automaton symmetries, non-invertible duality defects, and persistent low-energy remnants of apparently broken higher-form symmetries (Tantivasadakarn et al., 30 Jul 2025, Shirley et al., 28 Jul 2025, Gardas et al., 2013, Apte et al., 2022).

1. Terminological scope

The phrase is explicit in the sequential-circuit treatment of lattice generalized symmetry, whereas several other works describe the same persistence motif without adopting it as a formal term. In that broader usage, “unannihilable” refers to symmetry data that remains structurally nontrivial after the standard mechanisms of trivialization have failed or become unavailable (Tantivasadakarn et al., 30 Jul 2025, Shirley et al., 28 Jul 2025, Gardas et al., 2013, Apte et al., 2022, García-Valdecasas, 2023).

Setting	Meaning of “unannihilable”	Representative hallmark
Lattice generalized symmetry	No symmetry exists whose fusion with the given symmetry contains the identity channel	$$\mathcal{C}^\dagger \times \mathcal{C} \sim \mathcal{C}$$
2D QCA/FDQC symmetry	Cannot be consistently gauged or made on-site, yet is anomaly-free	Nontrivial $[\omega]\in H^2(G,\mathbb{Q}_+)$
Quantum Rabi model	A discrete symmetry survives even when the familiar local parity is broken	Hidden nonlocal $\mathbb{Z}_2$ involution
3+1D gauging duality defects	Cannot be absorbed into a trivial symmetric gapped vacuum except under arithmetic conditions	Gaplessness or spontaneous breaking
Supergravity higher-form remnants	Chern-Simons couplings do not fully destroy the symmetry, but reduce it to a rational non-invertible remnant	Topological defects dressed by auxiliary TQFT data

This plurality of meanings is not accidental. It reflects the fact that contemporary symmetry theory is organized not only by group actions on a fixed Hilbert space, but also by defect fusion, categorical composition, locality-preserving automata, module categories, and gauging operations. A plausible implication is that “unannihilability” functions less as a single definition than as a recurring structural property: symmetry content persists, but in a form that is no longer reducible to ordinary Wigner symmetry.

2. Fusion-theoretic definition on the lattice

A precise definition is given in the study of generalized lattice symmetries implemented by sequential quantum circuits. There, the central object is a symmetry twist that is moved across the system by a sequential circuit. The symmetry implementation is decomposed into three conceptual steps: create twists on the boundary of a subregion, sweep the twist through the bulk, and annihilate the twists at the opposite boundary. For annihilable symmetries, the sweep circuit already determines the full symmetry action, because the twist can be pair-created from the vacuum and the fusion with its Hermitian conjugate contains the identity channel. The canonical example is Kramers–Wannier, for which

$\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$

For unannihilable symmetries, by contrast, no symmetry exists whose fusion with the given defect contains the identity; the moving data is insufficient, and an additional one-dimensional sequential circuit is required to generate and annihilate the twist itself (Tantivasadakarn et al., 30 Jul 2025).

The Cheshire string in the $2+1$D toric code is the paper’s main example. The toric-code Hamiltonian is

$$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$

Its Cheshire-string symmetry is unannihilable because

$$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$

so the identity channel does not appear. The defect cannot be generated from the vacuum by a finite-depth $1$D circuit; instead, its creation requires a $1$D sequential circuit. This does not contradict the fact that a closed loop of the defect can be shrunk to the vacuum, because that shrinking itself requires a sequential circuit whose depth scales with loop size.

This definition isolates a stronger notion than mere non-invertibility. Many non-invertible symmetries are still annihilable because their fusion algebra contains $I$ as a channel. Unannihilability in this sense means that even the categorical analogue of “inverse times element equals identity” fails. The need for an extra $[\omega]\in H^2(G,\mathbb{Q}_+)$0D circuit shows that the full symmetry operator is not exhausted by the bulk sweep; it also contains irreducible defect-generation data. Matrix product operator and tensor network operator representations are important precisely because they separate these two ingredients.

3. Non-onsiteable, non-gaugeable, yet anomaly-free symmetries

A different but closely related use arises in two-dimensional lattice systems whose symmetry operators are QCAs. The central result is a family of finite-group internal symmetries that are not consistently gaugeable and not onsiteable, but are nevertheless anomaly-free in the paper’s precise sense: after possibly adding finite-dimensional ancillas that transform on-site, there exists a symmetric, gapped local Hamiltonian commuting with the symmetry and having a unique invertible ground state (Shirley et al., 28 Jul 2025).

The organizing invariant is a cohomology class

$[\omega]\in H^2(G,\mathbb{Q}_+)$1

where $[\omega]\in H^2(G,\mathbb{Q}_+)$2 is the multiplicative group of positive rational numbers labeling the GNVW index of one-dimensional QCAs. Choosing a restriction $[\omega]\in H^2(G,\mathbb{Q}_+)$3 of a global symmetry $[\omega]\in H^2(G,\mathbb{Q}_+)$4 to a large disk $[\omega]\in H^2(G,\mathbb{Q}_+)$5, one defines

$[\omega]\in H^2(G,\mathbb{Q}_+)$6

which is supported near $[\omega]\in H^2(G,\mathbb{Q}_+)$7 and therefore acts as an effectively one-dimensional QCA along the boundary. Its GNVW index gives

$[\omega]\in H^2(G,\mathbb{Q}_+)$8

and associativity implies the cocycle condition

$[\omega]\in H^2(G,\mathbb{Q}_+)$9

A nontrivial class $\mathbb{Z}_2$0 obstructs onsiteability; the same class also obstructs background gaugeability, and therefore dynamical gaugeability as well.

The explicit $\mathbb{Z}_2$1 example is defined on a honeycomb lattice with a plaquette qubit $\mathbb{Z}_2$2 and a vertex qubit $\mathbb{Z}_2$3. The symmetry operator is

$\mathbb{Z}_2$4

where $\mathbb{Z}_2$5 translates the $\mathbb{Z}_2$6-qubits along domain walls of the $\mathbb{Z}_2$7 configuration. Restricting $\mathbb{Z}_2$8 to a disk produces a boundary translation, and the cocycle is

$\mathbb{Z}_2$9

Since $\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$0 is not a square in $\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$1, the class is nontrivial. Yet the state

$\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$2

is a unique product ground state of the symmetric, gapped Hamiltonian

$\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$3

This use of “unannihilable” is therefore not anomaly-theoretic in the usual sense. The symmetry action is stuck in a non-onsite QCA form and cannot be consistently gauged, yet it does not force gaplessness or topological degeneracy. The paper further emphasizes that the $\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$4 obstruction is not preserved under renormalization group flow, which distinguishes it from a genuine ’t Hooft anomaly.

4. Hidden surviving symmetry in the quantum Rabi model

In the quantum Rabi model, “unannihilable symmetry” is an inferred description rather than the paper’s formal term. The model is defined by

$\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$5

For $\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$6, the familiar discrete $\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$7 symmetry is generated by

$\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$8

and satisfies $\mathcal{D}^{\dagger}\times \mathcal{D}= I+\eta .$9. The nonzero $2+1$0 regime had recently been conjectured to possess no nontrivial symmetry beyond total energy conservation. That conjecture is false. Under the conditions

$2+1$1

there exists a unique weak solution $2+1$2 of the operator Riccati equation

$2+1$3

with $2+1$4, yielding a self-adjoint involution $2+1$5 commuting with the Hamiltonian (Gardas et al., 2013).

The Hamiltonian is rewritten as the block operator matrix

$2+1$6

From the projection onto the graph $2+1$7, the symmetry generator is constructed as

$2+1$8

with

$2+1$9

The induced decomposition

$$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$0

gives two $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$1-invariant symmetry sectors, and the Hamiltonian block-diagonalizes as

$$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$2

The paper interprets $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$3 as a generalized parity and says that the model has an unbroken $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$4 symmetry even when the usual local parity $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$5 is broken. In that inferred sense, the symmetry is unannihilable: turning on $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$6 destroys the obvious parity, but not the existence of a conserved discrete involution.

5. Obstructions, remnants, and dynamical persistence

In $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$7 dimensions, non-invertible duality defects furnish a more stringent sense in which symmetry can resist annihilation. For theories with a $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$8 one-form symmetry that are invariant under gauging, half-space gauging produces a topological duality defect $$H = -\sum_p A_p - \sum_v B_v = -\sum_p \prod_{e\in p} X_e - \sum_v \prod_{v \in e} Z_e .$$9. Its fusion with its CPT conjugate is not the identity but a sum over one-form symmetry surface operators,

$$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$0

so the symmetry is manifestly non-invertible. Because these defects are topological, they are renormalization-group invariants. The main theorem states that a self-dual theory admits a symmetry-preserving gapped phase only if

$$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$1

and $$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$2 is a quadratic residue modulo $$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$3; otherwise the theory is gapless or spontaneously breaks the self-duality symmetry. The same framework extends to more general gauging operations, including triality, with condition

$$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$4

Here unannihilability means that the defect algebra cannot, in general, be absorbed into a unique trivial symmetric vacuum (Apte et al., 2022).

A low-energy effective-theory counterpart appears in 11d and 10d supergravities with Chern-Simons couplings or modified Bianchi identities. There, apparently broken higher-form symmetries survive as rational non-invertible defects after the naive Page operators are dressed by auxiliary topological degrees of freedom. In 11d supergravity, for example,

$$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$5

and for rational $$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$6 the defect

$$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$7

is topological only after stacking with the appropriate $$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$8d TQFT. These defects act invertibly on some probe branes and non-invertibly on others, often annihilating charged probes unless

$$\mathcal{C}^{\dagger}\times \mathcal{C}\sim \mathcal{C},$$9

The symmetry is genuine in supergravity but removed in the UV completion by dynamical branes, in line with the Completeness Hypothesis (García-Valdecasas, 2023).

The same persistence can acquire a directly dynamical form. In a one-dimensional spin chain with Rep$1$0 non-invertible symmetry, generated by $1$1, $1$2, and the Kennedy–Tasaki duality operator $1$3,

$1$4

symmetry-preserving disorder yields spectral degeneracies that can only be completely lifted at perturbative orders scaling with system size. The same symmetry organizes nontrivial string order in disordered eigenstates, protects interface edge modes whose autocorrelation is a temperature-weighted sum over effective chain lengths, and supports Floquet period-doubled edge dynamics at low effective temperature (Li et al., 19 Aug 2025). This is a particularly explicit form of unannihilability: the symmetry leaves measurable traces in spectra, eigenstates, and long-time dynamics even out of equilibrium.

6. Higher-categorical expansion, intrinsicity, and realizability

A broad conceptual generalization is that gauging a finite symmetry does not merely produce dual Wilson operators; it produces a universal $1$5-category of dual symmetries. For a $1$6-dimensional theory $1$7 with finite symmetry $1$8, the gauged theory $1$9 carries a higher-categorical symmetry sector $1$0 whose objects are lower-dimensional TQFTs with $1$1-symmetry, whose $1$2-morphisms are symmetry-preserving interfaces, and so on. In this framework, the familiar $1$3 obtained by gauging a finite $1$4-form symmetry is only a small piece of the full dual symmetry structure. The robust, non-group-like operators produced by gauging are therefore “unannihilable” in the sense that they are governed by higher-categorical fusion and condensation data rather than group inversion (Bhardwaj et al., 2022).

In class $1$5 theories, this categorical robustness is sharpened by the distinction between intrinsic and non-intrinsic non-invertibility. A non-invertible symmetry is non-intrinsic if it can be related, by discrete gauging or changing global form, to an invertible symmetry in another global variant. It is intrinsic if no such global form exists. Algebraically, the issue is whether the invariant-subspace condition

$1$6

admits a solution. The paper classifies these possibilities up to genus $1$7, finding many arithmetic classes in which enhanced modular symmetries are intrinsically non-invertible. In higher dimensions, the same distinction appears as the statement that the Symmetry TFT of intrinsically non-invertible cases cannot be obtained from $1$8d Chern–Simons theory alone, but only after coupling to topological gravity (Bashmakov et al., 2022).

A complementary development concerns the implementation of non-invertible symmetry on Hilbert space. Wigner’s theorem assumes a symmetry acts on a single fixed Hilbert space and hence must be implemented by an invertible unitary or antiunitary operator. For a unitary fusion category symmetry $1$9, however, the appropriate structure is a family of twisted-sector Hilbert spaces $I$0, and a defect $I$1 acts by transition channels

$I$2

assembled into an isometry

$I$3

The induced action on density matrices is a completely positive, trace-preserving quantum channel. In this formulation, non-invertible symmetries preserve probabilities not by acting invertibly on one Hilbert space, but by embedding states into a larger direct sum of outgoing twisted sectors. The construction requires $I$4 to be unitary and fails in non-unitary examples such as Yang–Lee (Bartsch et al., 6 Feb 2026).

Two limiting cases show that unannihilability does not imply a single physical outcome. In non-semisimple spin-chain symmetry based on the Taft algebra $I$5, one obtains frustration-free, gapped Hamiltonians with real spectra, a nilpotent non-invertible defect satisfying $I$6, an $I$7-family of symmetric states that remain in the same phase with respect to the invertible $I$8 subsymmetry yet transform inequivalently under the full non-semisimple symmetry, and a model where a product state and the $I$9 state spontaneously break the symmetry (Delcamp et al., 2024). By contrast, in the theory of “impossible symmetries,” a mixed anomaly between generalized electric and magnetic currents fixes correlators with a $[\omega]\in H^2(G,\mathbb{Q}_+)$00 pole that cannot arise from any local, unitary Lorentzian spectrum; the conformal scalar and linear conformal gravity are the principal examples. There, the symmetry pattern is not merely persistent but unrealizable in a unitary theory (Hinterbichler et al., 2024).

Taken together, these works indicate that unannihilability is best understood as a structural diagnosis rather than a single formal class. It may signal the absence of an identity fusion channel, an obstruction to onsiteability or gauging, a nonlocal conserved involution, an RG-invariant topological defect, a higher-categorical dual symmetry, a non-semisimple defect algebra, or even an impossible anomaly pattern. What is common across these cases is that the symmetry cannot be reduced to an ordinary invertible operator acting trivially on a featureless vacuum; some irreducible defect, boundary, categorical, or anomaly datum necessarily remains.