  We study the set of infinite volume ground states of Kitaev's quantum double
model on $\mathbb{Z}^2$ for an arbitrary finite abelian group $G$. It is known
that these models have a unique frustration-free ground state. Here we drop the
requirement of frustration freeness, and classify the full set of ground
states. We show that the ground state space decomposes into $|G|^2$ different
charged sectors, corresponding to the different types of abelian anyons (also
known as superselection sectors). In particular, all pure ground states are
equivalent to ground states that can be interpreted as describing a single
excitation. Our proof proceeds by showing that each ground state can be
obtained as the weak$^*$ limit of finite volume ground states of the quantum
double model with suitable boundary terms. The boundary terms allow for states
which represent a pair of excitations, with one excitation in the bulk and one
pinned to the boundary, to be included in the ground state space.
