Bond algebras and exact solvability of Hamiltonians: Spin S=1/2 multilayer systems

We present an algebraic methodology for designing exactly solvable Lie model Hamiltonians. The idea consists in looking at the algebra generated by bond operators. We illustrate how this method can be applied to solve numerous problems of current interest in the context of topological quantum order. These include Kitaev's well-known toric code and honeycomb models as well as new models: a vector-exchange model and a Clifford gamma model in a triangular lattice.