We propose a protocol to implement a quantum spin chain with non-invertible Rep(D8) symmetry using neutral atoms trapped in optical tweezer arrays.

We use the Symmetry Topological Field Theory (SymTFT) to study and classify gapped phases in (2+1)d for a class of categorical symmetries, referred to as being of bosonic type. The SymTFT has infinitely many topological boundary conditions obtainable from the canonical Dirichlet boundary condition by stacking a 3d TFT and gauging a sub-symmetry. these boundary conditions play a central organizational role.

We describe how the SymTFT information can be converted into an UV anyonic chain lattice model with fusion category symmetry, realizing in the IR limit, beyond-Landau gapped phases and transitions. In many cases, the Hilbert space of the anyonic chain is tensor product decomposable and the model can be realized as a quantum spin-chain Hamiltonian.

We study fermionic non-invertible symmetries in (1+1)d, which are generalized global symmetries that mix fermion parity symmetry with other invertible and non-invertible internal symmetries. These symmetries are described by fermionic fusion supercategories. The aim of this paper is to flesh out the categorical Landau paradigm for fermionic symmetries. We use the formalism of Symmetry Topological Field Theory (SymTFT) to study possible gapped and gapless phases for such symmetries, along with possible deformations between these phases, which are organized into a Hasse phase diagram. 

We present a framework to systematically investigate higher categorical symmetries in two-dimensional spin systems. Though exotic, such generalised symmetries have been shown to naturally arise as dual symmetries upon gauging invertible symmetries. Our framework relies on an approach to dualities whereby dual quantum lattice models only differ in a choice of module 2-category over some input fusion 2-category. Given an arbitrary two-dimensional spin system with an ordinary symmetry, we explain how to perform the (twisted) gauging of any of its sub-symmetries.

We study the gauging of 0-form symmetries in the presence of non-invertible symmetries. The starting point of our analysis is a theory with G 0-form symmetry, and we propose a description of sequential partial gaugings of sub-symmetries resulting in a network of gauging-related symmetry structures called a non-invertible symmetry web. The gauging implements the theta-symmetry defects of the companion paper. 

We study the gauging of 0-form symmetries in the presence of non-invertible symmetries. The starting point of our analysis is a theory with G 0-form symmetry, and we propose a description of sequential partial gaugings of sub-symmetries resulting in a network of gauging-related symmetry structures called a non-invertible symmetry web. The gauging implements the theta-symmetry defects of the companion paper.