---
title: Annihilable Symmetries in Non-Semisimple Models
url: https://www.emergentmind.com/topics/annihilable-symmetries
type: topic
---

# Annihilable Symmetries in Non-Semisimple Models

Searching arXiv for the cited paper and closely related work on annihilable/non-invertible symmetries.
{"query":"id:2412.19635 OR \"A non-semisimple non-invertible symmetry\""}
{"query":"id:2507.22394 OR \"Sequential Circuit as Generalized Symmetry on Lattice\""}
{"query":"id:2208.05973 OR \"Universal Non-Invertible Symmetries\""}
Annihilable symmetries are a class of generalized symmetries for which the action of a symmetry defect can collapse, vanish under repeated application, or be trivialized by an allowed operation. In the spin-chain realization based on the Taft Hopf algebra of dimension \(4\), the notion is tied to a non-semisimple, non-invertible symmetry whose defect generator is nilpotent, so that repeated action annihilates states or line insertions; in other frameworks, the term is used for symmetries whose defects are removed by gauging, screening, or condensation, or for lattice symmetries whose fusion with an appropriate dagger contains the identity channel [2412.19635].

## 1. Definitions across generalized-symmetry frameworks

In the spin-chain models built from \(\mathrm{Mod}(\mathbb{T}_4)\), an annihilable symmetry is implied to be a non-invertible symmetry whose defect generators admit nilpotent actions or fusion, for example \(D^2=0\). The paper’s basic physical instance is the non-local operator \((\hat\omega_0)^1{}_0\), which is nonzero but satisfies \(((\hat\omega_0)^1{}_0)^2=0\); it creates the \(W\) state from the product state, while its second application vanishes. The same analysis ties annihilation-like behavior to projective covers and to maps factoring through simple objects in a non-semisimple module category [2412.19635].

In the lattice sequential-circuit formulation, a symmetry is called annihilable if its fusion with an appropriate Hermitian conjugate contains the trivial operator as one of the outcomes. For a simple symmetry object \(a\), there exists a unique \(a^\dagger\) such that
\[
a^\dagger \times a
=
\mathbf{1}
+
\sum_{c\neq \mathbf{1}} N_{a^\dagger a}^{\ \ c}\, c,
\qquad
N_{a^\dagger a}^{\ \ \mathbf{1}}=1.
\]
In \(1\)D, generalized symmetries constructed from sequential circuits are annihilable in this sense, whereas the \(2\)D Cheshire-string symmetry is presented as unannihilable, with \(C^\dagger\times C\sim C\) [2507.22394].

In the field-theoretic lecture-note formulation, a generalized symmetry is “annihilable” if there exists an allowed QFT operation that removes its symmetry defects as protected topological constraints. The mechanisms listed are gauging when there is no ’t Hooft anomaly, anomaly cancellation by stacking or extension/higher-group mixing, and introducing dynamical charged matter so that symmetry defects can end. This usage is explicitly distinguished from spontaneous breaking, where the defect algebra persists [2307.07547].

A further categorical formulation appears in the universal-duality framework. There, a symmetry is annihilable by condensation or higher-gauging if there exists a condensation defect whose module action trivializes the symmetry on the operators of interest. In the \(0\)-form case, the criterion is
\[
\mathrm{Hom}_{\Gamma^{(0)'},\alpha}\big(\mathbf{1},\, \mathrm{Res}^{\Gamma^{(0)}}_{\Gamma^{(0)'}} R\big)\neq 0,
\]
so that Wilson lines can end on the condensation defect and the charged sector is screened [2208.05973].

## 2. Non-semisimple realization from the Taft algebra

The basic algebraic realization uses the finite tensor category \(\mathrm{Mod}(\mathbb{T}_4)\), the category of left modules over the Taft Hopf algebra of dimension \(4\), also known as Sweedler’s Hopf algebra. Its algebra, coalgebra, and antipode are
\[
\mathbb{T}_4=\mathbb{C}\langle x,g\mid x^2=0,\ g^2=1,\ xg=-gx\rangle,
\]
\[
\Delta(g)=g\otimes g,\qquad
\Delta(x)=x\otimes 1+g\otimes x,\qquad
\epsilon(g)=1,\qquad
\epsilon(x)=0,
\]
\[
S(g)=g,\qquad S(x)=xg.
\]
This is the minimal non-commutative, non-cocommutative Hopf algebra of dimension \(4\); \(S^2\neq \mathrm{id}\). It can be viewed as a non-semisimple extension of \(\mathbb{C}[\mathbb{Z}/2\mathbb{Z}]\) by \(\mathbb{C}[x]/(x^2)\) [2412.19635].

Its representation theory contains two simples and two projectives. The simples are
\[
S_0=\mathbb{C}\{w_1\},\qquad x\cdot w_1=0,\qquad g\cdot w_1=w_1,
\]
\[
S_1=\mathbb{C}\{v_1\},\qquad x\cdot v_1=0,\qquad g\cdot v_1=-v_1.
\]
The projective covers are
\[
P_0=\mathbb{C}\{v_0,v_1\},\qquad x\cdot v_0=v_1,\quad x\cdot v_1=0,\quad g\cdot v_0=v_0,\quad g\cdot v_1=-v_1,
\]
\[
P_1=\mathbb{C}\{w_0,w_1\},\qquad x\cdot w_0=w_1,\quad x\cdot w_1=0,\quad g\cdot w_0=-w_0,\quad g\cdot w_1=w_1.
\]
Their Loewy structures are encoded by the short exact sequences
\[
0\to S_1\to P_0\to S_0\to 0,\qquad
0\to S_0\to P_1\to S_1\to 0.
\]
Thus \(P_0\) has Loewy layers \(S_0\) on top and \(S_1\) in the socle, while \(P_1\) has \(S_1\) on top and \(S_0\) in the socle [2412.19635].

The tensor products already exhibit the non-semisimple fusion structure:
\[
S_1\otimes S_1\cong S_0,
\]
\[
P_0\otimes S_1\cong P_1,\qquad
P_1\otimes S_1\cong P_0,\qquad
S_1\otimes P_0\cong P_1,\qquad
S_1\otimes P_1\cong P_0,
\]
and
\[
P_0\otimes P_0\cong P_0\oplus P_1,\qquad
P_0\otimes P_1\cong P_0\oplus P_1,\qquad
P_1\otimes P_0\cong P_0\oplus P_1,\qquad
P_1\otimes P_1\cong P_0\oplus P_1.
\]
Rigidity takes the form \(S_1^\vee\cong S_1\), \(P_0^\vee\cong P_1\), and \(P_1^\vee\cong P_0\). These data supply the categorical source of the annihilation-like phenomena seen in the spin chains [2412.19635].

## 3. Defect operators, nilpotency, and annihilation

The symmetry is implemented in the spin chain by matrix product operators built from \(\mathrm{Mod}(\mathbb{T}_4)\). If \(K\) is a left \(\mathbb{T}_4\)-comodule with coaction \(\lambda:K\to \mathbb{T}_4\otimes K\), \(\lambda(k)=k_{(-1)}\otimes k_{(0)}\), and \(\rho:\mathbb{T}_4\to \mathrm{End}(V)\) is an indecomposable module, the basic interchanger is
\[
\omega(k\otimes v)=(k_{(-1)}\cdot v)\otimes k_{(0)}.
\]
Choosing \(K=\mathbb{C}\{1,y\}\) with \(\lambda(y)=x\otimes 1+g\otimes y\) and \(\lambda(1)=1\otimes 1\), together with \(V=P_0\), yields local tensors \(\omega_0\) with components
\[
(\omega_0)^0{}_0=I,\qquad
(\omega_0)^1{}_0=\sigma^{-},\qquad
(\omega_0)^0{}_1=0,\qquad
(\omega_0)^1{}_1=\sigma^z.
\]
For open boundaries, one obtains the non-local symmetry operator \(\hat\omega_0\), whose distinguished components are
\[
(\hat\omega_0)^0{}_0=\mathrm{identity},\qquad
(\hat\omega_0)^1{}_1=\prod \sigma^z,
\]
and, crucially,
\[
(\hat\omega_0)^1{}_0\neq 0,\qquad
\big((\hat\omega_0)^1{}_0\big)^2=0.
\]
The nilpotency of \((\hat\omega_0)^1{}_0\) is the hallmark of the non-invertible, non-semisimple symmetry defect in this model [2412.19635].

Fusion is mediated by local junction tensors \(\phi^{\alpha_1\alpha_2}_{\alpha_3}\) and their adjoints \(\bar\phi^{\alpha_1\alpha_2}_{\alpha_3}\), with orthogonality
\[
\bar\phi^{\alpha_1\alpha_2}_{\alpha_4}\circ \phi^{\alpha_1\alpha_2}_{\alpha_3}
=
\delta_{\alpha_3,\alpha_4} I.
\]
Their non-vanishing entries reproduce the tensor products in \(\mathrm{Mod}(\mathbb{T}_4)\). Physically, the nilpotent MPO can map a state non-trivially to another state but annihilates upon repeated application. This is the paper’s explicit realization of “defect annihilation” [2412.19635].

The lattice sequential-circuit framework provides a complementary criterion. There, annihilable \(1\)D symmetries are reconstructed as sequential MPOs, and the canonical example is Kramers–Wannier symmetry \(\mathcal D\), which satisfies
\[
\mathcal D^\dagger\times \mathcal D=\mathbf{1}+\eta.
\]
The general result is that for a simple symmetry \(\mathcal D_\alpha\),
\[
\mathcal D_\alpha^\dagger\times \mathcal D_\alpha=\mathbf{1}+\cdots,
\qquad
N_{\alpha^\dagger\alpha}^{\ \ \mathbf{1}}=1,
\]
with a unique dagger object. This suggests a structural parallel between nilpotent MPO defects in the non-semisimple Taft-chain realization and identity-channel annihilability in the sequential-circuit formulation [2507.22394].

## 4. Symmetric Hamiltonians and inequivalent symmetry actions

The spin-chain realization sacrifices Hermiticity in order to construct several symmetric, frustration-free, gapped Hamiltonians with real spectra. One family consists of commuting projectors
\[
H(\xi)=-\sum h(\xi)_{i,i+1},
\qquad
h(\xi)_{i,i+1}
=
\frac12
\big[
I_i\otimes I_{i+1}
+
\sigma^x(\xi)_i\otimes \sigma^x(\xi)_{i+1}
\big],
\]
with
\[
\sigma^x(\xi)=\sqrt{\xi}
\begin{bmatrix}
0 & 1/\xi\\
1 & 0
\end{bmatrix},
\qquad
\xi\in U(1).
\]
This model has a \(\mathbb{Z}/2\mathbb{Z}\) symmetry \(\prod \sigma^z\) and two ground product states
\[
|\pm \xi\rangle^{\otimes |\Lambda|},
\qquad
|\pm \xi\rangle=|0\rangle \pm \sqrt{\xi}\,|1\rangle.
\]
The family \(\{H(\xi)\}\) represents the same \(\mathbb{Z}/2\) symmetry-broken gapped phase [2412.19635].

The second family is non-Hermitian but remains symmetric and frustration-free:
\[
\hat H(\xi)=-\sum_i \hat h(\xi)_{i,i+1},
\qquad
\hat h(\xi)_{i,i+1}
=
I_i\otimes (\sigma^+\sigma^-)_{i+1}
+
\sqrt{\xi}\,\sigma^x(\xi)_i\otimes \sigma^-_{i+1}.
\]
Although the local projectors do not commute, the model is frustration-free, gapped, and its eigenvalues can be verified to be all real negative. It carries both the invertible \(\mathbb{Z}/2\) generator \((\hat\omega_0)^1{}_1=\prod \sigma^z\) and the non-invertible nilpotent defect \((\hat\omega_0)^1{}_0\) [2412.19635].

For each \(\xi\in U(1)\), the ground space of \(\hat H(\xi)\) is spanned by
\[
|+\xi\rangle^{\otimes |\Lambda|},
\qquad
|-\xi\rangle^{\otimes |\Lambda|}.
\]
All these states lie in the same \(\mathbb{Z}/2\)-broken gapped phase because \(\prod \sigma^z\) exchanges them. However, under the full non-semisimple symmetry they transform inequivalently. The action is encoded by tensors \(\phi^\alpha:\mathbb{C}^2\otimes (\mathbb{C}\oplus\mathbb{C})\to (\mathbb{C}\oplus\mathbb{C})\) whose non-vanishing components include
\[
\phi^0_{0,0}=\phi^0_{1,1}=1,\qquad
\phi^0_{0,1}=2\sqrt{\xi},\qquad
\phi^0_{1,0}=-2\sqrt{\xi},
\]
\[
\phi^1_{1,0}=\phi^1_{0,1}=1,\qquad
\phi^1_{0,0}=2\sqrt{\xi},\qquad
\phi^1_{1,1}=-2\sqrt{\xi}.
\]
The corresponding module associator \(F\)-symbols contain \(\xi\)-dependent entries, for example
\[
(F^{\mathrm{act}})^{000}_0=
\begin{bmatrix}
1 & -1/(4\xi)\\
0 & -1
\end{bmatrix},
\qquad
(F^{\mathrm{act}})^{001}_1=
\begin{bmatrix}
0 & -1\\
1 & -1/(4\xi)
\end{bmatrix}.
\]
Gauge transformations cannot remove \(\xi\) from these matrices without changing other entries, so different \(\xi\in S^1\) define inequivalent \(\mathrm{Mod}(\mathbb{T}_4)\)-module structures even though they occupy the same invertible-symmetry phase [2412.19635].

## 5. Spontaneous breaking, the \(W\) state, and infrared collapse

A particularly sharp realization of annihilation appears in the \(\xi\to 0\) limit of the non-Hermitian family. The local terms become
\[
\hat h(0)_{i,i+1}
=
I_i\otimes (\sigma^+\sigma^-)_{i+1}
+
\sigma_i^+\otimes \sigma^-_{i+1},
\]
and the ground space is spanned by the product state and the \(W\) state,
\[
|\psi_{\mathrm{prod}}\rangle=|0\rangle^{\otimes L},
\qquad
|W_L\rangle=\sum_{i=1}^L \sigma_i^- |0\rangle^{\otimes L}.
\]
The nilpotent non-invertible defect acts as
\[
(\hat\omega_0)^1{}_0\,|0\rangle^{\otimes L}=|W_L\rangle,
\qquad
\big((\hat\omega_0)^1{}_0\big)^2=0.
\]
By contrast, the invertible subsymmetry acts by
\[
\prod \sigma^z\,|0\rangle^{\otimes L}=|0\rangle^{\otimes L},
\qquad
\prod \sigma^z\,|W_L\rangle=-|W_L\rangle.
\]
Thus the product state is even and the \(W\) state is odd under \(\mathbb{Z}/2\), while both ground states spontaneously break the non-semisimple symmetry down to \(\mathbb{Z}/2\) [2412.19635].

In the thermodynamic limit, these two finite-volume ground states become locally indistinguishable. For any local operator \(O\) supported on finitely many sites,
\[
\langle W_L|O|W_L\rangle
-
\langle 0^{\otimes L}|O|0^{\otimes L}\rangle
=
O(1/L)\to 0.
\]
The reason is that \( |W_L\rangle \) contains exactly one down-spin among \(L\), so the local density of the excitation vanishes. The two states therefore define the same limiting expectations of local observables and a unique infrared vacuum [2412.19635].

The paper relates this collapse directly to non-semisimplicity. The product state corresponds to a simple object \(S\) in \(\mathrm{Mod}(A(1,0))\), while the \(W\) state corresponds to its projective cover \(P\), fitting into
\[
0\to S\to P\to S\to 0.
\]
There are non-zero maps \(P\to S\) and \(S\to P\) factoring through this sequence, furnishing local topological operators that map \(|W_L\rangle\) locally to \(|0\rangle^{\otimes L}\). This suggests that the infrared indistinguishability of the two ground states is the physical manifestation of a simple object and its projective cover becoming locally indistinguishable in a non-semisimple module category [2412.19635].

The generalized-symmetry lecture notes make a related distinction in different language: spontaneous symmetry breaking leaves the defect algebra intact, whereas annihilation in the sense of gauging or screening removes the topological generators themselves. The Taft-chain example is unusual because the nilpotent defect persists at finite volume, yet the projective-cover structure collapses local distinctions in the infinite-volume limit [2307.07547].

## 6. Broader constructions, categorical mechanisms, and open problems

The broader generalized-symmetry literature supplies several distinct mechanisms by which a symmetry may be annihilated. One mechanism is gauging: for a finite or continuous \(p\)-form symmetry with background \(B_{p+1}\), one replaces the background by a dynamical field and sums over gauge-inequivalent classes,
\[
Z_{T/G^{(p)}}\propto \sum_{[B_{p+1}]} Z_T[B_{p+1}]
\quad \text{or} \quad
Z_{T/G^{(p)}}\propto \int D[B_{p+1}]\, Z_T[B_{p+1}],
\]
so that the original symmetry defects become invisible as global operators. A second mechanism is screening by dynamical charged matter, which allows defects to end. A third is anomaly cancellation by stacking or symmetry extension, after which gauging becomes possible [2307.07547].

A categorical version of the same idea is supplied by universal non-invertible symmetries. Gauging a finite \(0\)-form symmetry \(\Gamma^{(0)}\) produces a dual higher-categorical symmetry whose \(2\)-categorical piece is
\[
C_2^{\Gamma^{(0)}}=2Rep(\Gamma^{(0)}).
\]
Its simple objects are the minimal \(2\)D \(\Gamma^{(0)}\)-symmetric TQFTs \(T_{\Gamma^{(0)'},\alpha}\), with \(\Gamma^{(0)'}<\Gamma^{(0)}\) and \(\alpha\in H^2(\Gamma^{(0)'},U(1))\). In this setting, a Wilson line in representation \(R\) can end on the condensation defect when
\[
\mathrm{Hom}_{\Gamma^{(0)'},\alpha}\big(\mathbf{1},\, \mathrm{Res}^{\Gamma^{(0)}}_{\Gamma^{(0)'}} R\big)\neq 0.
\]
This is the screening or trivialization criterion for annihilation by condensation [2208.05973].

A field-theoretic realization with explicitly non-invertible projector-like behavior appears in \(2\)D sigma models based on half-space gauging and T-duality. There the T-duality defect \(D_T\) has fusion
\[
D_T\otimes D_T=\sum_{a=0}^{p-1}\eta^a
\]
for \(\mathbb{Z}_p\) half-space gauging at the self-dual radius, and more generally
\[
X\otimes X=\bigoplus_{h\in H} h
\]
for a subgroup \(H\). The defect averages over discrete holonomies or gauge orbits and can kill non-invariant components; the paper describes this as projector-like action, sector erasure, and annihilability [2503.20865].

The sequential-circuit construction on the lattice imposes a sharp \(1\)D/ \(2\)D contrast. In \(1\)D, the bulk sweep already determines the full symmetry action, and annihilable symmetries reconstructed as sequential MPOs always contain an identity channel in dagger fusion. In \(2\)D, the Cheshire-string symmetry is unannihilable: its full action requires both a \(2\)D sweep circuit and a \(1\)D generation/annihilation circuit, and the resulting operator obeys
\[
\mathcal C^\dagger\times \mathcal C\sim \mathcal C,
\]
rather than containing \(\mathbf 1\) [2507.22394].

Within the Taft-chain setting, several open problems remain explicit. The paper lists conditions ensuring real spectra of non-Hermitian symmetric parent Hamiltonians built from comodule algebra morphisms \(\Delta\circ \mu\), the precise role of modified traces and quantum dimensions of projectives for periodic boundary conditions, the systematic identification of annihilable defects via categorical nilpotents or idempotents and their physical signatures, and the extension of the \(S^1\)-family phenomenon—inequivalent non-semisimple symmetry actions within the same invertible symmetry phase—to other non-semisimple tensor categories [2412.19635].

Taken together, these constructions show that annihilable symmetry is not a single universal algebraic condition but a family of closely related mechanisms. In non-semisimple spin chains it appears through nilpotent symmetry operators and projective-cover physics; in gauging frameworks it appears through the disappearance of topological generators; in condensation form it is encoded by module categories and endpoint conditions; and on the lattice it can be characterized by whether the identity channel appears in dagger fusion. The Taft-chain models provide a particularly explicit realization because they combine all of the following in one setting: non-semisimplicity, non-invertibility, nilpotent defect action, spontaneous symmetry breaking, and infrared collapse of simple and projective sectors [2412.19635].

Source: https://www.emergentmind.com/topics/annihilable-symmetries