Fusion surface models generalize the concept of anyon chains to 2+1 dimensions, utilizing fusion 2-categories as their input. We investigate bond-algebraic dualities in these systems and show that distinct module tensor categories \mathcal{M} ℳ over the same braided fusion category \mathcal{B} ℬ give rise to dual lattice models. This extends the 1+1d result that dualities in anyon chains are classified by module categories over fusion categories. We analyze two concrete examples: (i) a \text{Rep}(S_3) Rep ( S 3 ) model with a constrained Hilbert space, dual to the spin- \tfrac{1}{2} 1 2 XXZ model on the honeycomb lattice, and (ii) a bilayer Kitaev honeycomb model, dual to a spin- \tfrac{1}{2} 1 2 model with XXZ and Ising interactions. Unlike regular \mathcal{M}=\mathcal{B} ℳ = ℬ fusion surface models, which conserve only 1-form symmetries, models constructed from \mathcal{M} ≠ \mathcal{B} ℳ ≠ ℬ can exhibit both 1-form and 0-form symmetries, including non-invertible ones.
Fusion surface models, as introduced by Inamura and Ohmori, extend the concept of anyon chains to 2+1 dimensions, taking fusion 2-categories as their input. In this work, we construct and analyze fusion surface models on the honeycomb lattice built from braided fusion 1-categories. These models preserve mutually commuting plaquette operators and anomalous 1-form symmetries. Their Hamiltonian is chosen to mimic the structure of Kitaev’s honeycomb model, which is unitarily equivalent to the Ising fusion surface model. In the anisotropic limit, where one coupling constant is dominant, the fusion surface models reduce to Levin-Wen string-nets. In the isotropic limit, they are described by weakly coupled anyon chains and are likely to realize chiral topological order. We focus on three specific examples: (i) Kitaev’s honeycomb model with a perturbation breaking time-reversal symmetry that realizes chiral Ising topological order, (ii) a \mathbb{Z}_N ℤ N generalization proposed by Barkeshli et al., which potentially realizes chiral parafermion topological order, and (iii) a novel Fibonacci honeycomb model featuring a non-invertible 1-form symmetry.
It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In Ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin- 12 Heisenberg XXZ model. One model has symmetry operators forming the fusion category Rep(S3) of representations of the group S3 . We find that the mapping between its topological sectors and those of the XXZ model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of Rep(S3) . Published by the American Physical Society 2024
We construct (2+1)-dimensional lattice systems, which we call fusion surface models. These models have finite non-invertible symmetries described by general fusion 2-categories. Our method can be applied to build microscopic models with, for example, anomalous or non-anomalous one-form symmetries, 2-group symmetries, or non-invertible one-form symmetries that capture non-abelian anyon statistics. The construction of these models generalizes the construction of the 1+1d anyon chains formalized by Aasen, Fendley, and Mong. Along with the fusion surface models, we also obtain the corresponding three-dimensional classical statistical models, which are 3d analogues of the 2d Aasen-Fendley-Mong height models. In the construction, the “symmetry TFTs” for fusion 2-category symmetries play an important role.
The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.
Until recently, a careful derivation of the fusion structure of anyons from some underlying physical principles has been lacking. In Shi et al. (2020), the authors achieved this goal by starting from a conjectured form of entanglement area law for 2D gapped systems. In this work, we instead start with the principle of exchange symmetry, and determine the minimal prescription of additional postulates needed to make contact with unitary ribbon fusion categories as the appropriate algebraic framework for modelling anyons. Assuming that 2D quasiparticles are spatially localised, we build a functor from the coloured braid groupoid to the category of finite-dimensional Hilbert spaces. Using this functor, we construct a precise notion of exchange symmetry, allowing us to recover the core fusion properties of anyons. In particular, given a system of n quasiparticles, we show that the action of a certain n-braid βn uniquely specifies its superselection sectors. We then provide an overview of the braiding and fusion structure of anyons in the usual setting of braided 6j fusion systems. By positing the duality axiom of Kitaev (2006) and assuming that there are finitely many distinct topological charges, we arrive at the framework of ribbon categories.
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
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
Z2 extension of
Rep(H). Specializing to the Kac-Paljutkin algebra
H8, the smallest self-dual Hopf algebra beyond abelian group algebras, we numerically study the phase diagram and identify four of the six
Rep(H8)-symmetric gapped phases, separated by Ising critical lines and meeting at a multicritical point. We also realize all six
Rep(H8)-symmetric gapped phases on the lattice via the
H-comodule algebra formalism, in agreement with the module-category classification of
Rep(H8). Our results provide a unified Hopf-algebraic framework for non-invertible symmetries, dualities, and the tensor product lattice models that realize them.