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We construct a family of rotationally invariant, local, S=1/2 KleinHamiltonians on various lattices that exhibit ground state manifolds spanned bynearest-neighbor valence bond states. We show that with selected perturbationssuch models can be driven into phases modeled by well understood quantum dimermodels on the corresponding lattices. Specifically, we show that theperturbation procedure is arbitrarily well controlled by a new parameter whichis the extent of decoration of the reference lattice. This strategy leads toHamiltonians that exhibit i) $Z_2$ RVB phases in two dimensions, ii) U(1) RVBphases with a gapless ``photon'' in three dimensions, and iii) a Cantordeconfined region in two dimensions. We also construct two models on thepyrochlore lattice, one model exhibiting a $Z_2$ RVB phase and the other a U(1)RVB phase.Comment: 16 pages, 15 figures; 1 figure and some references added; some minor  typos fixe

physical review. b, condensed matter and materials physics

SU(2)-invariant spin-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mstyle scriptlevel="1"><mml:mfrac bevelled="false"><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle></mml:math>Hamiltonians with resonating and other valence bond phases

SU(2)-invariant spin-12Hamiltonians with resonating and other valence bond phases