Localized endomorphisms in Kitaev's toric code on the plane

Details Localized endomorphisms in Kitaev's toric code on the plane Localized endomorphisms in Kitaev's toric code on the plane

2010-12-17

arxiv.org/abs/1012.3857v2

Rev. Math. Phys. 23 (2011), 347-373

Physics

Mathematical Physics

26 pages, 5 figures. v2: proof of Prop. 2.2 fixed, other minor corrections

Scientific paper

10.1142/S0129055X1100431X

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.

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