of a staggering parameter. The staggering is introduced as a tunable parameter in the
manifestly supersymmetric Hamiltonian. We obtain analytic expressions for the ground
states in the limit of small and large staggering for the model on the class of doubly
decorated lattices. On this type of lattice there are two ground states, each with a different
density. In one limit we and these ground states to be a simple Wigner crystal and a valence
bond solid (VBS) state. In the other limit we and two types of quantum liquids. As a special
case, we investigate the quantum liquid state on the one dimensional chain in detail. It is
characterized by a massless kink that separates two types of order.
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