  Quantum double models, such as the toric code, can be constructed from
transfer matrices of lattice gauge theories with discrete gauge groups and
parametrized by the center of the gauge group algebra and its dual. For general
choices of these parameters the transfer matrix contains operators acting on
links which can also be thought of as perturbations to the quantum double model
driving it out of its topological phase and destroying the exact solvability of
the quantum double model. We modify these transfer matrices with perturbations
and extract exactly solvable models which remain in a quantum phase, thus
nullifying the effect of the perturbation. The algebra of the modified vertex
and plaquette operators now obey a deformed version of the quantum double
algebra. The Abelian cases are shown to be in the quantum double phase whereas
the non-Abelian phases are shown to be in a modified phase of the corresponding
quantum double phase. These are illustrated with the groups $\mathbb{Z}_n$ and
$S_3$. The quantum phases are determined by studying the excitations of these
systems namely their fusion rules and the statistics. We then go further to
construct a transfer matrix which contains the other $\mathbb{Z}_2$ phase
namely the double semion phase. More generally for other discrete groups these
transfer matrices contain the twisted quantum double models. These transfer
matrices can be thought of as being obtained by introducing extra parameters
into the transfer matrix of lattice gauge theories. These parameters are
central elements belonging to the tensor products of the algebra and its dual
and are associated to vertices and volumes of the three dimensional lattice. As
in the case of the lattice gauge theories we construct the operators creating
the excitations in this case and study their braiding and fusion properties.
