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A state sum construction on closed manifolds \'{a} la Kuperberg can be usedto construct the partition functions of $3D$ lattice gauge theories based oninvolutory Hopf algebras, $\mathcal{A}$, of which the group algebras,$\mathbb{C}G$, are a particular case. Transfer matrices can be obtained bycarrying out this construction on a manifold with boundary. VariousHamiltonians of physical interest can be obtained from these transfer matricesby playing around with the parameters the transfer matrix is a function of. The$2D$ quantum double Hamiltonians of Kitaev can be obtained from such transfermatrices for specific values of these parameters. A initial study of suchmodels has been carried out in \cite{p1}. In this paper we study other regionsof this parameter space to obtain some new and known models. The new modelcomprise of Hamiltonians which "partially" confine the excitations of thequantum double Hamiltonians which are usually deconfined. The state sumconstruction allows for parameters depending on the position in obtaining thetransfer matrices and thus it is natural to expect disordered Hamiltonians fromthem. Thus one set of known models consist of the disordered quantum doubleHamiltonians. Finally we obtain quantum double Hamiltonians perturbed bymagnetic fields which have been considered earlier in the literature to studythe stability of topological order to perturbations.

Toric code-like models from the parameter space of $3D$ lattice gauge theories