We consider various aspects of Kitaev's toric code model on a plane in the C^*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher-Haag-Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of D(Z_2), i.e., Drinfel'd's quantum double of the group algebra of Z_2.
The category of anyon sectors for non-abelian quantum double models
Institute for Theoretical Physics, ETH Zürich（苏黎世联邦理工学院理论物理研究所）；School of Mathematics, Cardiff University（卡迪夫大学数学学院）；Department of Physics, University of California, Davis（加州大学戴维斯分校物理系）
Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；Center of Mathematical Sciences and Applications, Harvard University（哈佛大学数学科学应用中心）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）
