Ground States Of A Boson Quantum Lattice Model

. A boson system on Z d ; d 2; is considered, with the Hamiltonian H = \Gamma1=2 X y a + y (\Deltaa) y + X hy; y 0 i U(N y ; N y 0 ): Here a + y , a y are the creation and annihilation operators, and N y = a + y a y : The nearestneighbor potential U is given by: U(n; n 0 ) = 0 if n + n 0 m and U(n; n 0 ) = +1 if n + n 0 ? m (m is a fixed positive integer). The Gibbs ensemble is determined by exp (\Gammafi (H \Gamma ¯N)), where N = P N y . By using a polymer expansion technique, we prove that, if d 2 and ¯ ? 0 and fi ? 0 are large enough, the system has (precisely) two translation-periodic pure phases when m is odd and a single pure phase when m is even. We prove that the same is true of the ground states, and establish that the ground states are singular (degenerate) and correspond to `dim' chess board configurations. Key words and phrases: quantum lattice boson models, hard-core interaction, Gibbs states and measures, ground states, uniqueness and non-un..