  We show that the Davies generator associated to any 2D Kitaev's quantum
double model has a non-vanishing spectral gap in the thermodynamic limit. This
validates rigorously the extended belief that those models are useless as
self-correcting quantum memories, even in the non-abelian case. The proof uses
recent ideas and results regarding the characterization of the spectral gap for
parent Hamiltonians associated to Projected Entangled Pair States in terms of a
bulk-boundary correspondence.
