A bstract We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category Rep( H ) of a semisimple weak Hopf algebra H , the fermionized TQFT has a superfusion category symmetry SRep(
H H u ), which is the supercategory of super representations of a weak Hopf superalgebra
H H u . The weak Hopf superalgebra
H H u depends not only on H but also on a choice of a non-anomalous ℤ 2 subgroup of Rep( H ) that is used for the fermionization. We derive a general formula for
H H u by explicitly constructing fermionic TQFTs with SRep(
H H u ) symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on F -symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs.
Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.
Abstract A ‐D lattice model inspired is proposed by a weak Hopf algebra generalization of the cluster state model. This construction serves as a candidate framework for systematically realizing the Haagerup fusion category symmetry on spin chains. It is shown that the model exhibits Haagerup fusion algebra symmetry and features a tensor product Hilbert space. The construction begins with a reconstruction of the Haagerup weak Hopf algebra from the Haagerup fusion category , ensuring that the representation category of is equivalent to . Utilizing the framework of symmetry topological field theory (SymTFT), an ultra‐thin weak Hopf quantum double model that has a smooth topological boundary condition is developed. It is shown that this model supports Haagerup fusion algebra symmetry. Finally, the ground state of the model is solved for in terms of a weak Hopf matrix product state, which serves as a natural generalization of the cluster state, embodying Haagerup fusion algebra symmetry. Since the Haagerup fusion category is an anomalous symmetry, the model presented realizes the gapped Haagerup spontaneous symmetry breaking (SSB) phase.
We investigate (1+1)d symmetry-protected topological (SPT) phases with fusion category symmetries. We emphasize that the UV description of an anomaly-free fusion category symmetry must include the fiber functor, giving rise to a local symmetry action, a charge category and a trivial phase. We construct an ``onsite'' matrix-product-operator (MPO) version of the Hopf algebra symmetry operators in a lattice model with tensor-product Hilbert space. In particular, we propose a systematic framework for classifying and constructing SPTs with non-invertible symmetries. An SPT phase corresponds to a Q-system in the charge category, such that the Q-system becomes a matrix algebra when the symmetry is forgotten. As an example, we provide an explicit microscopic realization of all three
Rep†(D8) SPT phases, including a trivial phase, and further demonstrate the
S3-duality among these three SPT phases.
We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category
C, an indecomposable semisimple right
C-module category
M, a Q-system
Q∈C specifying the bulk phase, and a right
Q-module
K∈MQ, regarded as an object of
MQop, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor
MQop→BCond is an equivalence, so simple boundary conditions are classified by simple objects of
MQ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to
(CM∨)rev, and the canonical action of the bulk DHR category on it agrees with the categorical action of
Z1(Crev). Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of
(CM∨)rev on
MQop. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.
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.