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We study the Kitaev model on regular hyperbolic trivalent tilings. Depending on the length p of the elementary polygons, we examine two distinct tri-colorings of the tiling. Using a recent conjecture on the ground-state flux sector, we compute the phase diagram via exact diagonalizations and derive analytical expressions for the effective Hamiltonians in the isolated-dimer limit which are valid for all values of p. Our results interpolate between the Euclidean honeycomb lattice and the trivalent Bethe lattice (p=) for which we derive the exact solution of the phase boundaries.
Vidal et al. (Fri,) studied this question.