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We introduce an algebraic methodology for designing exactly-solvable Liemodel Hamiltonians. The idea consists in looking at the algebra generated bybond operators. We illustrate how this method can be applied to solve numerousproblems of current interest in the context of topological quantum order. Theseinclude Kitaev's toric code and honeycomb models, a vector exchange model, anda Clifford gamma model on a triangular lattice.

Bond Algebras and Exact Solvability of Hamiltonians: Spin S=1/2 Multilayer Systems and Other Curiosities