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Ashkin-Teller universality in a quantum double model of Ising anyons

We study a quantum double model whose degrees of freedom are Ising anyons. The terms of the Hamiltonian of this system give rise to a competition between single and double topologies. By studying the energy spectra of the Hamiltonian at different values of the coupling constants, we find extended gapless regions which include a large number of critical points described by conformal field theories with central charge c=1. These theories are part of the Z_2 orbifold of the bosonic theory compactified on a circle. We observe that the Hilbert space of our anyonic model can be associated with extended Dynkin diagrams of affine Lie algebras which yields exact solutions at some critical points. In certain special regimes, our model corresponds to the Hamiltonian limit of the Ashkin-Teller model, and hence integrability over a wide range of coupling parameters is established.