Non-invertible categorical symmetries have emerged as a powerful tool to uncover new beyond-Landau phases of matter, both gapped and gapless, along with second order phase transitions between them. The general theory of such phases in (1+1)d has been studied using the Symmetry Topological Field Theory (SymTFT), also known as topological holography. This has unearthed the infrared (IR) structure of these phases and transitions. In this paper, we describe how the SymTFT information can be converted into an ultraviolet (UV) anyonic chain lattice model realizing, in the IR limit, these 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 also describe operators acting on the lattice models that are charged under non-invertible symmetries and act as order parameters for the phases and transitions. In order to fully describe the action of non-invertible symmetries, it is crucial to understand the symmetry twisted sectors of the lattice models, which we describe in detail. Throughout the paper, we illustrate the general concepts using the symmetry category \mathsf{Rep}(S_3) 𝖱 𝖾 𝗉 ( S 3 ) formed by representations of the permutation group S_3 S 3 , but our procedure can be applied to any fusion category symmetry.
The discovery of generalized symmetries has led to powerful new insights into quantum matter. They have been used to classify new families of quantum phases, place constraints on phases realizable in a given physical system, and conceptually unify seemingly disparate phenomena. In many ways, they prove just as powerful as traditional symmetries at organizing and constraining the theories that describe quantum matter. In this thesis, we attempt a unification of such constraints by developing a holographic correspondence between (generalized) symmetries and topological orders, called the Sym/TO correspondence. For any (finite internal) symmetry of a quantum system in d (spatial) dimensions, we associate with it a unique topological order in d + 1 dimensions, called its Symmetry Topological Order (SymTO). We devise an operator algebraic recipe to compute the SymTO data for any lattice spin model, demonstrating it in a number of examples. We then use the SymTO to classify possible quantum phases allowed by the symmetry—we call this a generalized Landau paradigm. Besides classifying phases, we also identify constraints on the phase transitions between them using a SymTO-resolved modular bootstrap. We test this framework in a quantum spin chain with non-invertible symmetries. We discover a new Kramers-Wannier-like duality and a rich phase diagram including a noninvertible symmetry-enriched incommensurate phase. The translation symmetry of the spin chain has a nontrivial interplay with the lattice Kramers-Wannier duality, which matches the anomaly of the corresponding non-invertible symmetry in the low-energy effective field theory. Finally, we explore such unusual anomaly-matching mechanisms in more detail in the context of the chiral anomaly of a single massless Dirac fermion, demonstrating a novel lattice realization of chiral symmetries and their anomaly.
We extend non-invertible duality concepts from one-dimensional systems to two spatial dimensions by constructing a web of non-invertible dualities for lattice models with subsystem symmetries. For the
Z2×Z2 subsystem symmetry on the square lattice, we build two complementary dualities: a map that sends spontaneous subsystem symmetry-broken (SSSB) phases to the trivial phase (the analogue of the Kramers-Wannier (KW) duality in 1+1D), and a generalized subsystem Kennedy-Tasaki (KT) transformation that maps SSSB phases to subsystem symmetry-protected topological (SSPT) phases while leaving the trivial phase invariant. These dualities are boundary-sensitive. On open lattices, both subsystem KW and KT transformations act as unitary, invertible operators. In particular, the KT map not only matches the bulk Hamiltonians of the dual phases but also carries the spontaneous ground-state degeneracy of the SSSB phase onto the protected boundary degeneracy of the SSPT phase. On closed manifolds, however, both maps become intrinsically non-unitary and non-invertible when restricted to the original Hilbert space. We demonstrate this non-invertibility via ground-state degeneracy matching (in two copies of the Xu-Moore/Ising-plaquette model), analysis of symmetry-twist sectors mapping, and the fusion algebra of the duality operator. Enlarging the Hilbert space to include twisted sectors allows the subsystem KW map to be formulated as a projective unitary preserving quantum transition probabilities, consistent with generalized Wigner-theorem-based constructions. We also show that the KT map faithfully transmits the algebraic content of bulk and edge invariants diagnosing strong SSPT order: although strictly local SSPT repair operators map to highly nonlocal objects in the dual SSSB phase, the essential commutation algebra and the bulk-edge correspondence remain intact.
We explore 1+1 dimensional (1+1D) gapped phases in systems with non-invertible symmetries, focusing on symmetry-protected topological (SPT) phases (defined as gapped phases with non-degenerate ground states), as well as SPT orders (defined as the differences between gapped/gapless phases with identical bulk excitations spectrum). For group-like symmetries, distinct SPT phases share identical bulk excitations and always differ by SPT orders. However, for certain non-invertible symmetries, we discover novel SPT phases that have different bulk excitations and thus do not differ by SPT orders. Additionally, we also study the spontaneous symmetry-breaking (SSB) phases of non-invertible symmetries. Unlike group-like symmetries, non-invertible symmetries lack the concept of subgroups, which complicates the definition of SSB phases as well as their identification. This challenge can be addressed by employing the symmetry-topological-order (symTO) framework for the symmetry. The Lagrangian condensable algebras and automorphisms of the symTO facilitate the classification of gapped phases in systems with such symmetries, enabling the analysis of both SPT and SSB phases (including those that differ by SPT orders). Finally, we apply this methodology to investigate gapless phases in symmetric systems and to study gapless phases differing by SPT orders.
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.
Spontaneous symmetry breaking is a well-understood mechanism for generating distinct phases of matter. Recently, the notion of symmetry has been broadened to include operations without inverses, leading to the concept of non-invertible symmetries. How do symmetry-breaking phases associated with non-invertible symmetries differ from those arising from conventional invertible symmetries? We address this question using concrete lattice models of the gapped phases with non-invertible Rep(
H8) symmetry as an example. We find that, despite the symmetry being non-invertible, the symmetry-breaking phases can still be characterized by the long-range correlation of local order parameters, which obey a more general algebraic structure than in the invertible setting. Furthermore, via generalized gauging, certain non-invertible symmetry-breaking transitions can be mapped to deconfined quantum critical points of invertible symmetries, and vice versa. We establish precise conditions under which this duality holds and illustrate them with several families of examples, providing a systematic route to studying beyond-Landau phase transitions.