Lattice non-invertible symmetry from non-commuting transfer matrices

Abstract: We establish a direct connection between Onsager symmetry, duality defects, and quantum integrability in the XXZ spin chain at roots of unity, $Δ=(q+q^{-1})/2$ with $q^N=\pm1$. Using a non-Abelian algebra of transfer matrices governed by an unbalanced version of the Yang--Baxter/RLL relation, we construct an explicit lattice realization of the Onsager algebra and its duality automorphism. The duality is represented by a matrix product operator related to the transfer matrices of the $τ_2$ model. We show that this operator obeys $\mathbb{Z}_N$ Tambara--Yamagami fusion rules and therefore realizes on the lattice the topological defect lines of the free compactified boson conformal field theory. Our results identify non-Abelian integrability as a natural framework for the emergence of the Onsager symmetry and categorical dualities in lattice models.

• The paper introduces a framework where non-invertible symmetries emerge from non-commuting transfer matrices in the XXZ spin chain at roots of unity.
• It demonstrates how the Onsager algebra and duality operators derived from unbalanced RLL relations lead to exponentially degenerate spectra and superintegrability.
• The analysis bridges integrable models with lattice fusion categories, connecting non-Abelian integrability to categorical dualities in quantum many-body systems.

Lattice Realization of Non-Invertible Symmetries via Non-Commuting Transfer Matrices

Introduction

This essay summarizes the primary contributions and technical advances presented in "Lattice non-invertible symmetry from non-commuting transfer matrices" (2606.25660). The paper provides a rigorous framework for the emergence and explicit construction of lattice non-invertible symmetries, grounding them in the context of quantum integrable systems with non-Abelian symmetries. Focusing on the XXZ spin chain at roots of unity, the authors elucidate the interplay between Onsager symmetry, duality defects, and integrability, extending the scope of symmetry-constrained dynamics and categorical dualities in quantum many-body lattices.

Non-Abelian Generalization in Integrable Models

The XXZ chain at $\Delta = (q + q^{-1})/2$ with $q^N = \pm 1$ exhibits enhanced symmetry properties due to the non-semisimplicity of $U_q(\mathfrak{sl}_2)$ at roots of unity. The authors exploit transfer matrices derived from higher-dimensional cyclic and nilpotent representations to construct families of commuting and non-commuting charges. In contrast to generic $q$, these auxiliary transfer matrices do not mutually commute, resulting in a non-Abelian algebra that induces exponentially large degeneracies in the spectrum and signals superintegrability.

The central technical tool is the formulation of an unbalanced RLL relation—a non-Abelian generalization of the Yang–Baxter equation. This relation encapsulates quadratic identities between products of transfer matrices, providing a robust train argument analogous to the standard commutativity proof for abelian charges. The result is an explicit lattice realization of the Onsager algebra, including its infinite family of generators and duality automorphism.

Onsager Algebra Structure and Factorization

The authors demonstrate that the XXZ spin chain at roots of unity possesses a symmetry governed by the Onsager algebra, originally defined in solving the 2D Ising model. While previous constructions (e.g., $\mathrm{L}(\mathfrak{sl}_2)$) act only on subspaces, the Onsager generators constructed here commute with the Hamiltonian across the entire Hilbert space. The explicit factorization of non-commuting transfer matrices reveals elegant exponential forms in terms of Onsager generators, with central elements $T(0)$ and generating functions for abelian and non-abelian quasi-local charges:

• $$T_{\pm}(u) = T(0)\exp(\pm \frac{1}{2} \mathcal{G}(u) + \frac{i}{2} \mathcal{H}(u))$$
• $$\mathcal{G}(u) = \sum_{m\geq 1} \frac{\tanh^m(Nu)}{m} \mathbf{G}_m$$

All Dolan–Grady relations (the defining identities of Onsager algebra) are proved via detailed auxiliary-space manipulations and Baker–Campbell–Hausdorff expansions, confirming that these generators fully capture extended non-Abelian integrability.

Duality Operators and Lattice Fusion Categories

The duality automorphism—an extension of Kramers–Wannier duality—has a matrix product operator realization related to the transfer matrices of the $\tau_2$ model. The explicit construction of duality operators $D_\pm$ as certain limits of $q^N = \pm 1$0 transfer matrices enables the representation of non-invertible symmetries at the lattice level.

These operators satisfy fusion rules characteristic of Tambara–Yamagami categories:

• $q^N = \pm 1$1
• $q^N = \pm 1$2, with $q^N = \pm 1$3 generating a $q^N = \pm 1$4 subgroup of $q^N = \pm 1$5 (magnetization symmetry).

This provides a lattice realization of the topological defect lines studied in free compactified boson CFTs at rational radii, concretely linking integrable spin chains to categorical symmetry structures and non-invertible defect fusion.

Implications and Theoretical Advances

The identification of non-Abelian integrability as a natural framework for lattice categorical dualities advances both practical and conceptual understanding:

• The explicit construction of the Onsager algebra demonstrates that non-invertible symmetries can be derived from integrable structures, rather than being merely emergent or postulated.
• Enhanced symmetry leads to highly degenerate energy spectra and constraints on dynamics, pertinent for quantum scars and symmetry-protected phases.
• The lattice realization of Tambara–Yamagami fusion categories offers direct connections between integrable models at roots of unity and rational CFTs, opening pathways for classification of lattice phases beyond the Landau paradigm.

These results suggest that Onsager-type structures and categorical dualities may be generic features in integrable systems with superintegrability, especially at roots of unity or with non-semisimple quantum groups.

Future Directions

Several extensions are indicated:

• Classification and construction of fusion categories from non-Abelian integrability in higher-rank, higher-spin, and other root-of-unity models.
• Extraction of full categorical data (including associators/F-symbols) via tensor network approaches, as fusion algebra alone does not specify the fusion category uniquely.
• Connections to three-dimensional Chern–Simons theory, topological-holomorphic four-dimensional theories, and broader string-theoretic frameworks, especially regarding quantum groups at roots of unity and topological defects.

A critical question is whether all non-invertible symmetry fusion categories admit lattice realizations as descendants of integrable models, and how integrability interacts with categorical symmetries in more general settings.

Conclusion

The paper provides a comprehensive lattice framework for non-invertible symmetries, grounded in non-Abelian integrability and explicit algebraic constructions. By connecting transfer matrix algebras, Onsager symmetry, and fusion categories, it establishes a rich interplay between integrability and categorical dualities in quantum lattice systems. This approach paves the way for deeper classification schemes and analytic tools for symmetry and dynamics in quantum many-body physics, with promising avenues for theoretical exploration in higher-dimensional QFTs and quantum topology.