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

TL;DR

The paper develops a lattice realization of anomaly-free non-invertible symmetries described by ${ m Rep}(H)$ for finite-dimensional semisimple Hopf algebras ${H}$, using Hopf-qudit ZX calculus to construct Hopf Pauli operators and symmetry MPOs. It defines a generalized Kramers–Wannier duality, ${ m D}_{ m KW}$, via gauging ${ m Rep}(H^*)$ and a Hopf Hadamard gate that maps between ${ m Rep}(H)$ and ${ m Rep}(H^*)$, illustrating a lattice-level duality that flows to a $ frac{1}{2}$-graded extension of ${ m Rep}(H)$ in the IR. The authors instantiate this framework with the 8-dimensional Kac–Paljutkin algebra ${H_8}$, presenting a Hopf-Ising model whose phase diagram exhibits four gapped phases and Ising critical lines, including a multicritical point, analyzed by DMRG and iDMRG techniques. Beyond the ${ m Rep}(H)$ fixed points, they classify all ${ m Rep}(H)$-symmetric gapped phases via ${H}$-comodule algebras, yielding six phases for ${H_8}$ and connecting to module-category classifications. The work unifies non-invertible symmetries, dualities, and tensor-product lattice models in a Hopf-algebraic framework, paving the way for exploring broader self-dual Hopf structures and their IR fusion categories.

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.

Figures (9)

• Figure 1: Schematic phase diagram of the $H_8$ Ising model $\mathsf{H}_{H_8}$. The self-dual line corresponds to $J=0.5$.
• Figure 2: (a) Plot of the Binder cumulant $B_m$ computed in the ground state of $\mathsf{H}_{H_8}$ with ${L=40}$ qudits, ${J\in[0.4,0.5]}$, ${K\in[0.9,2.5]}$, and open boundary conditions. (b) Phase boundary obtained from the crossing of Binder cumulant as a function of $K$ and $L$ for different values of ${J\in[0.475,0.5]}$ (blue circles), along with a nonlinear fit (red line).
• Figure 3: Correlation length $\xi$, defined by the second largest eigenvalue of the transfer matrix associated with the iMPS ground state, plotted along two different constant $K$ slices: (a) ${K=0}$ and (b) ${K=10}$. Different MPS bond dimensions ($D$) are shown in different colors.
• Figure 4: (a) Bipartite entanglement entropy in the ground state MPS of a chain with ${L=200}$ qudits, ${J = 0.4}$, ${K = 2.42}$ and open boundary conditions. The horizontal axis plots $\log d(l)$, where $l$ is the number of qudits in the subsystem and ${d(l) = \frac{2 L}{\pi} \sin\left ( \frac{\pi l }{L} \right ) }$. The red line shows a linear fit to the Calabrese-Cardy formula ${S(l) = \frac{c}{6}\log d(l) + c_1}$ with the value of $c$ extracted in the inset. (b) Binder cumulant for ${L=20,30,40}$ with $J=0.4$ and ${K\in[2.35,2.5]}$.
• Figure 5: (a) Bipartite entanglement entropy and Calabrese-Cardy fit similar to \ref{['fig:ccfit-Ising1']} with ${L=200}$, ${J=0.5}$, ${K=10}$ and open boundary conditions. (b) Smallest 10 energy eigenvalues, normalized as ${e_n = \frac{E_n}{|E_0|}}$, for ${J=0.5}$, ${K=2}$, and ${L=6,7,8,9,15}$ qudits with periodic boundary conditions. The lines are guides to the eye.

Theorems & Definitions (4)

• Definition B.1: Left $H$-module
• Definition B.2: Right $H$-comodule
• Definition B.3: Morphisms of comodules
• Definition B.4: Hopf module