Abstract:
Symmetry protected and symmetry enriched topological phases of matter are of great interest in condensed matter physics due to new materials such as topological insulators. The Levin-Wen model for spin/boson systems is an important rigorously solvable model for studying $2D$ topological phases. The input data for the Levin-Wen model is a unitary fusion category, but the same model also works for unitary multi-fusion categories. In this paper, we provide the details for this extension of the Levin-Wen model, and show that the extended Levin-Wen model is a natural playground for the theoretical study of symmetry protected and symmetry enriched topological phases of matter.

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:
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:
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:
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:
Recent interest in the Kashaev-Murakami-Murakami hyperbolic volume conjecture has made it seem important to be able to understand the asymptotic behaviour of certain special functions arising from representation theory -- for example, of the quantum 6j-symbols for SU(2). In 1998 I worked out the asymptotic behaviour of the classical 6j-symbols, proving a formula involving the geometry of a Euclidean tetrahedron which was conjectured by Ponzano and Regge in 1968. In this note I will try to explain the methods and philosophy behind this calculation, and speculate on how similar techniques might be useful in studying the quantum case.

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:
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:
The purpose of this paper is to provide definitions for, and proofs of, the asymptotic formulae given by Edmonds, which relate the 3j and 6j symbols to rotation matrices.

Abstract:
It is known that the Fourier transformation of the square of (6j) symbols has a simple expression in the case of su(2) and U_q(su(2)) when q is a root of unit. The aim of the present work is to unravel the algebraic structure behind these identities. We show that the double crossproduct construction H_1\bowtie H_2 of two Hopf algebras and the bicrossproduct construction H_2^{*}\lrbicross H_1 are the Hopf algebras structures behind these identities by analysing different examples. We study the case where D= H_1\bowtie H_2 is equal to the group algebra of ISU(2), SL(2,C) and where D is a quantum double of a finite group, of SU(2) and of U_q(su(2)) when q is real.