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Infinite volume ground states for Kitaev's abelian quantum double models

Student-Run Applied & Math Seminar

Speaker: Matthew Cha, UC Davis
Location: 2112 MSB
Start time: Wed, Feb 15 2017, 12:10PM

We study the set of infinite volume ground states of Kitaev's quantum double model on $\ZZ^2$ for an arbitrary finite abelian group $G$. In the finite volume, the ground state space is frustration-free and the low-lying excitations correspond to abelian anyons. The ribbon operators act on the ground state space to create pairs of single excitations at their endpoints. It is known that in the infinite volume these models have a unique frustration-free ground state. We show that the complete set of ground states decomposes into $|G|^2$ different charged sectors, corresponding to the different types of abelian anyons (or superselection sectors). In particular, all pure ground states are equivalent to the single excitation states. Our proof proceeds by showing that each ground state can be obtained as the weak$*$-limit of the finite volume ground states of the quantum double model with a suitable boundary term. The boundary terms allow for states which represent an excitation pair with one excitation in the bulk and one pinned to the boundary to be included in the ground state space. This is joint work with P. Naaijkens and B. Nachtergaele.



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