A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are “physically feasible.” We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

functions for ranks of pre-modular categories

Abstract. From the moment of its discovery, the Jones polynomial of a knot has been linked to quantum physics. The main discovery, made by E. Witten, was that it is related to quantum field theory, which unfortunately lacks a mathematical foundation. But already in Witten’s work it was noted that the Jones polynomial is related to quantum mechanics. In this paper we discuss progress made in the study of the Jones polynomial from the point of view of quantum mechanics. This study reduces to the understanding of the quantization of the moduli space of flat SU(2)-connections on a surface with the Chern-Simons lagrangian. We outline some background material, then present the particular example of the torus, in which case it is known that the quantization in question is the Weyl quantization. The paper concludes with a possible application of this theory to the study of the fractional quantum Hall effect, an idea originating in the works of Moore and Read. 1. Resumen En este artículo de exposición se presenta una introducción al estudio del polinomio de Jones de un nudo desde un punto de vista físico basado en la mecánica cuántica. 1 El polinomio de Jones de un nudo fue descubierto por V.F.R. Jones [14] como una consecuencia de sus investigaciones en la teoría de álgebras de operadores. La definición del polinomio es muy sencilla y nos permite calcularlo facilmente, pero para entender y estudiar sus propiedades se necesita un punto de vista mas profundo y mas geométrico. Witten explicó [24] que el polinomio de Jones esta relacionado a la teoría de campos cuánticos. Pero la teoría de campos cuánticos requerida no tiene un fundamento matemático riguroso al momento de escribir esto.

Abstract. In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from a geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of flat SU(2)-connections on the torus. Two results are emphasized: the reconstruction from Weyl quantization of the restriction to the torus of the modular functor, and a description of a basis of the space of quantum observables on the torus in terms of colored curves, which answers a question related to quantum computing. 1.