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.
Abstract We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum is an H-simple left H-comodule algebra, where H is a finite dimensional semisimple Hopf algebra. We show that the actions of fusion category symmetries
C C on the boundary conditions of these state sum TQFTs are represented by module categories over
C C . This agrees with the classification of gapped phases with symmetry
C C . We also find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and hence provide lattice realizations of gapped phases with fusion category symmetries. As an application, we discuss the edge modes of SPT phases based on these commuting projector Hamiltonians. Finally, we mention that we can extend the construction of topological field theories to the case of anomalous fusion category symmetries by replacing a semisimple Hopf algebra with a semisimple pseudo-unitary connected weak Hopf algebra.
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.