We study a family of non-Abelian topological models in a lattice that arise by modifying the Kitaev model through the introduction of single-qudit terms. The effect of these terms amounts to a reduction in the discrete gauge symmetry with respect to the original systems, which corresponds to a generalized mechanism of explicit symmetry breaking. The topological order is either partially lost or completely destroyed throughout the various models. The systems display condensation and confinement of the topological charges present in the standard non-Abelian Kitaev models, which we study in terms of ribbon operator algebras.
We study a family of non-Abelian topological models in a lattice that arise by modifying the Kitaev model through the introduction of single-qudit terms. The effect of these terms amounts to a reduction in the discrete gauge symmetry with respect to the original systems, which corresponds to a generalized mechanism of explicit symmetry breaking. The topological order is either partially lost or completely destroyed throughout the various models. The systems display condensation and confinement of the topological charges present in the standard non-Abelian Kitaev models, which we study in terms of ribbon operator algebras.
We introduce the concept of nested topological order in a class of exact quantum lattice Hamiltonian models with non-Abelian discrete gauge symmetry. The topological order present in the models can be partially destroyed by introducing a gauge symmetry reduction mechanism. When symmetry is reduced in several islands only, this imposes boundary conditions on the rest of the system, giving rise to topological ground-state degeneracy. This degeneracy is related to the existence of topological fluxes in between islands or, alternatively, hidden charges at islands. Additionally, island deformations give rise to extension of topological quantum computation beyond quasiparticles.
Condensed matter physics is the study of the macroscopic and microscopic properties of condensed phases of matter. For quite some time, Landau’s symmetry breaking theory was believed to describe and explain the nature of any phase transition. However, since the late 1980s, it has become apparent that it is necessary to introduce some new kind of order, named topological order, that transcends the traditional symmetry description. In this thesis we will study the Kitaev model, which is a Hamiltonian lattice model that allows one to incorporate the concept of topological order, as well as the corresponding operators and algebras. First, we consider the model on an infinite lattice, and show how to relate local and global degrees of freedom of the anyons/quasi-particles living on sites to the ribbon operators. Afterwards, we introduce holes and an external boundary to the lattice, and examine the ramifications of this generalization in terms of the ground state degeneracy. Lastly, we verify that the algebra formed by boundary site operators has the structure of a quasi-Hopf algebra.
Transitions out of non-Abelian topological order are difficult to describe in microscopic quantum models with numerical methods that remain tractable at large scales. We develop a sign-free path-integral framework for Kitaev quantum doubles D(G) by organizing single-link perturbations in terms of non-invertible electric and magnetic 1-form symmetries that proliferate distinct anyon species. For any finite group G, an exact matterization isometry introduces vertex degrees of freedom and maps the link-only model onto a G gauge--Higgs theory. Furthermore, the 1-form symmetries of the fixed point allow us to define generalized Fredenhagen--Marcu order parameters that become finite when the corresponding anyons condense. For G=S3, quantum Monte Carlo simulations show that proliferating a non-Abelian electric anyon drives a first-order transition in which all nontrivial electric anyons condense. In the purely magnetic limit, the model reduces to a (2+1)D pure G gauge theory; for G=S3, it exhibits a first-order confinement transition, diagnosed by the onset of a Wilson-loop area law and the restoration of an emergent magnetic 1-form symmetry. These results provide a unified numerical framework for non-Abelian anyon condensation, confinement, and generalized symmetry breaking.
The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.
We present a wide class of partially integrable lattice models with two-spin interactions which generalize the Kitaev honeycomb model. These models have a conserved quantity associated with each plaquette, conserved large loop operators on the torus, and topological degeneracy. We introduce a "slave-genon" approach which generalizes the Majorana fermion approach in the Kitaev model. The Hilbert space of our spin model can be embedded in an enlarged Hilbert space of non-Abelian twist defects, referred to as genons. In the enlarged Hilbert space, the spin model is exactly reformulated as a model of non-Abelian genons coupled to a discrete gauge field. We discuss in detail a particular Z_{3} generalization, and we show that in a certain limit the model is analytically tractable and produces a non-Abelian topological phase with chiral parafermion edge states.
Higher-form symmetries generalize conventional global symmetries and act on lower-dimensional submanifolds of a quantum system. While Abelian topological phases can be organized by 1-form symmetries that form a group, non-Abelian topological phases based on finite groups require 1-form symmetry operators governed by non-invertible fusion algebras. In this work, we make this statement precise in quantum double lattice models D(G) for finite non-Abelian groups G. We construct the electric, magnetic, and dyonic 1-form operators directly at the lattice fixed point and show that together they form a complete nonlocal diagnostic algebra for the topological Hilbert space. Using these operators, we explicitly determine the cylinder and torus ground-state subspaces for arbitrary finite G. Furthermore, we calculate the microscopic fusion and gluing of the 1-form symmetries and show that their topological deformation properties emerge after projection to the defect-free topological subspace. Our results establish ground states of non-Abelian quantum double models as a concrete microscopic realization of spontaneous non-invertible 1-form symmetry breaking and provide an operator language that may be useful for characterizing such states in quantum processors.