Generalized Kramers-Wannier Self-Duality in Hopf-Ising Models

• URL: http://arxiv.org/abs/2602.10183v1
• Date: Tue, 10 Feb 2026 19:00:00 GMT
• Title: Generalized Kramers-Wannier Self-Duality in Hopf-Ising Models
• Authors: Da-Chuan Lu, Arkya Chatterjee, Nathanan Tantivasadakarn,
• Abstract summary: We build a generalized 1+1d Ising model based on a finite-dimensional semisimple Hopf algebra $H$.<n>We extend such self-duality symmetries beyond gauging of abelian groups in tensor-product Hilbert spaces.<n>Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Kramers-Wannier transformation of the 1+1d transverse-field Ising model exchanges the paramagnetic and ferromagnetic phases and, at criticality, manifests as a non-invertible symmetry. Extending such self-duality symmetries beyond gauging of abelian groups in tensor-product Hilbert spaces has, however, remained challenging. In this work, we construct a generalized 1+1d Ising model based on a finite-dimensional semisimple Hopf algebra $H$ that enjoys an anomaly-free non-invertible symmetry $\mathrm{Rep}(H)$. We provide an intuitive diagrammatic formulation of both the Hamiltonian and the symmetry operators using a non-(co)commutative generalization of ZX-calculus built from Hopf-algebraic data. When $H$ is self-dual, we further construct a generalized Kramers-Wannier duality operator that exchanges the paramagnetic and ferromagnetic phases and becomes a non-invertible symmetry at the self-dual point. This enlarged symmetry mixes with lattice translation and, in the infrared, flows to a weakly integral fusion category given by a $\mathbb{Z}_2$ extension of $\mathrm{Rep}(H)$. Specializing to the Kac-Paljutkin algebra $H_8$, the smallest self-dual Hopf algebra beyond abelian group algebras, we numerically study the phase diagram and identify four of the six $\mathrm{Rep}(H_8)$-symmetric gapped phases, separated by Ising critical lines and meeting at a multicritical point. We also realize all six $\mathrm{Rep}(H_8)$-symmetric gapped phases on the lattice via the $H$-comodule algebra formalism, in agreement with the module-category classification of $\mathrm{Rep}(H_8)$. Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.

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