Abstract:
We study properties of topological phases by calculating the ground state degeneracy (GSD) of the 2d Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatial topology of the system. Then we show that the ground state on a sphere is always non-degenerate. Moreover, we study an example associated with a quantum group, and show that the GSD on a torus agrees with that of the doubled Chern-Simons theory, consistent with the conjectured equivalence between the LW model associated with a quantum group and the doubled Chern-Simons theory.

Abstract:
It is known that every ribbon category with unimodality allows symmetrized $6j$-symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category $\mathcal{E}$. We define the mirror conjugate symmetry of $6j$-symbols instead and show that $6j$-symbols of any unitary spherical category can be normalized to have this property. As an application, we discuss an exactly soluble model on a honeycomb lattice. We prove that the Levin-Wen Hamiltonian is exactly soluble and hermitian on a unitary spherical category.

Abstract:
We construct quantum circuits for measuring the commuting set of vertex and plaquette operators that appear in the Levin-Wen model for doubled Fibonacci anyons. Such measurements can be viewed as syndrome measurements for the quantum error-correcting code defined by the ground states of this model (the Fibonacci code). We quantify the complexity of these circuits with gate counts using different universal gate sets and find these measurements become significantly easier to perform if n-qubit Toffoli gates with n = 3,4 and 5 can be carried out directly. In addition to measurement circuits, we construct simplified quantum circuits requiring only a few qubits that can be used to verify that certain self-consistency conditions, including the pentagon equation, are satisfied by the Fibonacci code.

Abstract:
We remark on the claim that the string-net model of Levin and Wen is a microscopic Hamiltonian formulation of the Turaev-Viro topological quantum field theory. Using simple counterexamples we indicate where interesting extra structure may be needed in the Levin-Wen model for this to hold (however we believe that some form of the correspondence is true). In order to be accessible to the condensed matter community we provide a brief and gentle introduction to the relevant concepts in category theory (relying heavily on analogy with ordinary group representation theory). Likewise, some physical ideas are briefly surveyed for the benefit of the more mathematical reader. The main feature of group categories under consideration is Turaev's unimodality. We pinpoint where unimodality should fit into the Levin-Wen construction, and show that the simplest example fails to be unimodal. Unimodality is straightforward to compute for group categories, and we provide a complete classification at the end of the paper.

Abstract:
We review the key steps of the construction of Levin-Wen type of models on lattices with boundaries and defects of codimension 1,2,3 in a joint work with Alexei Kitaev. We emphasize some universal properties, such as boundary-bulk duality and duality-defect correspondence, shared by all these models. New results include a detailed analysis of the local properties of a boundary excitation and a conjecture on the functoriality of the monoidal center.

Abstract:
Levin and Wen [Phys. Rev. B 71, 045110 (2005)] have recently given a lattice Hamiltonian description of doubled Chern-Simons theories. We relate the partition function of these theories to an expectation of Wilson loops that form a link in 2+1 dimensional spacetime known in the mathematical literature as Chain-Mail. This geometric construction gives physical interpretation of the Levin-Wen Hilbert space and Hamiltonian, its topological invariance, exactness under coarse-graining, and how two opposite chirality sectors of the doubled theory arise.

Abstract:
In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two dimensional topological phases, it is relatively easy to describe single fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological phases, a generalization of the LW models is proposed in this paper. We first enlarge the Hilbert space with adding a tail on one of the edges of each trivalent vertex, to describe its internal charge degrees of freedom. Then we study the full dyon spectrum of generalized LW models, including both quantum numbers and wave functions for dyonic quasiparticle excitations. The local operators associated with the dyonic excitations are shown to form the so-called Tube algebra, whose representations (modules) form the quantum double (categoric center) of the input data (unitary fusion category). In physically relevant cases, the input data is from a finite or quantum group (with braiding $R$-matrices), we find that the elementary excitations (or dyon species), as well as any localized/isolated excited states, are characterized by three quantum numbers: charge, fluxon type, and twist. They provide a "complete basis" for many-body states in the enlarged Hilbert space. Concrete examples are presented and the relevance of our results to the electric-magnetic duality is discussed.

Abstract:
Quantum graphity is a background independent model for emergent locality, spatial geometry and matter. The states of the system correspond to dynamical graphs on N vertices. At high energy, the graph describing the system is highly connected and the physics is invariant under the full symmetric group acting on the vertices. We present evidence that the model also has a low-energy phase in which the graph describing the system breaks permutation symmetry and appears to be ordered, low-dimensional and local. Consideration of the free energy associated with the dominant terms in the dynamics shows that this low-energy state is thermodynamically stable under local perturbations. The model can also give rise to an emergent U(1) gauge theory in the ground state by the string-net condensation mechanism of Levin and Wen. We also reformulate the model in graph-theoretic terms and compare its dynamics to some common graph processes.

Abstract:
We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows to carry over to these string net models the representation-theoretic classification of the excitations in quantum double models, as well as define them in arbitrary lattices, and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalising the map to broader classes of string nets is considered.

Abstract:
In this paper we study the symmetry protected Majorana edge states for the Z2 topological order of the Wen-plaquette model and the toric-code model and calculate the dispersion of the Majorana edge states. For the system with translational symmetry, the Majorana edge states are gapless and have the nodal points at k=0 and k=pi. For the edge states of the toric-code model without translational symmetry, the edge modes become gapped.