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Anomalies of Coset Non-Invertible Symmetries
by Po-Shen Hsin, Ryohei Kobayashi, Carolyn Zhang
Submission summary
| Authors (as registered SciPost users): | Po-Shen Hsin · Ryohei Kobayashi · Carolyn Zhang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202503_00040v2 (pdf) |
| Date submitted: | Nov. 5, 2025, 4:20 a.m. |
| Submitted by: | Ryohei Kobayashi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Anomalies of global symmetries provide important information on the quantum dynamics. We show the dynamical constraints can be organized into three classes: genuine anomalies, fractional topological responses, and integer responses that can be realized in symmetry-protected topological (SPT) phases. Coset symmetry can be present in many physical systems including quantum spin liquids, and the coset symmetry can be a non-invertible symmetry. We introduce twists in coset symmetries, which modify the fusion rules and the generalized Frobenius-Schur indicators. We call such coset symmetries twisted coset symmetries, and they are labeled by the quadruple $(G,K,\omega_{D+1},\alpha_D)$ in $D$ spacetime dimensions where $G$ is a group and $K\subset G$ is a discrete subgroup, $\omega_{D+1}$ is a $(D+1)$-cocycle for group $G$, and $\alpha_{D}$ is a $D$-cochain for group $K$. We present several examples with twisted coset symmetries using lattice models and field theory, including both gapped and gapless systems (such as gapless symmetry-protected topological phases). We investigate the anomalies of general twisted coset symmetry, which presents obstructions to realizing the coset symmetry in (gapped) symmetry-protected topological phases. We show that finite coset symmetry $G/K$ becomes anomalous when $G$ cannot be expressed as the bicrossed product $G=H\Join K$, and such anomalous coset symmetry leads to symmetry-enforced gaplessness in generic spacetime dimensions. We illustrate examples of anomalous coset symmetries with $A_5/\mathbb{Z}_2$ symmetry, with realizations in lattice models.
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List of changes
- We further explained our definition of Frobenius Schur indicator at the beginning of Sec.2.4.
- In Eq.2.6 and Eq.2.7 of the revised manuscript, we added the definitions of the slant products $i_g^A, i_g^B$.
- In Sec.3.6, we found that the statement of symmetry-enforced gaplessness should be separated into spacetime dimensions $D\ge 3$ and $D=2$; when $D=3$ the statement is shown in fully general, while at $D=2$ the argument is valid only for $G/K$ such that $H^2(BK',U(1))=0$ with any subgroup $K'\subset K$. We made edits of Sec.3.6 accordingly.
- We have replaced an example of a gapless phase with twisted coset symmetry in Sec.2.6.2.
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