  Algebraic contraction is proposed to realize mappings between models
Hamiltonians. This transformation contracts the algebra of the degrees of
freedom underlying the Hamiltonian. The rigorous mapping between the
anisotropic $XXZ$ Heisenberg model, the Quantum Phase Model, and the Bose
Hubbard Model is established as the contractions of the algebra $u(2)$
underlying the dynamics of the $XXZ$ Heisenberg model.
