  The robustness of the topological color code, which is a class of error
correcting quantum codes, is investigated under the influence of an uniform
magnetic field on the honeycomb lattice. Our study relies on two high-order
series expansions using perturbative continuous unitary transformations in the
limit of low and high fields, exact diagonalization and a classical
approximation. We show that the topological color code in a single parallel
field is isospectral to the Baxter-Wu model in a transverse field on the
triangular lattice. It is found that the topological phase is stable up to a
critical field beyond which it breaks down to the polarized phase by a
first-order phase transition. The results also suggest that the topological
color code is more robust than the toric code, in the parallel magnetic field.
