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We review the nature of superfluid ground states and the universality of their properties with emphasis to the Bose Einstein condensate (BEC) systems in atomic physics. We then study the superfluid Mott transition in such systems. We find that there could be two types of Mott transitions and phases. One of them was described long ago and corresponds to the suppression of Josephson tunneling within superfluids sitting at each well. On the other hand, the conditions of optical lattice BEC experiments are such that either the coherence length is longer than the interwell separation, or there is too small a number of bosons per well. This vitiates the existence of a superfluid order parameter within a well, and therefore of Josephson tunneling between wells. Under such conditions, there is a transition to a Mott phase which corresponds to the suppression of individual boson tunneling among wells. This last transition is in general discontinuous and can happen for incommensurate values of bosons per site. If the coherence length is small enough and the number of bosons per site is large enough, the transition studied in the earlier work will happen.

We study the properties of a spin-density-wave antiferromagnetic mean-field ground state with d-wave superconducting correlations. This ground state always gains energy by Cooper pairing. It would fail to superconduct at half-filling due to the antiferromagnetic gap although its particle-like excitations would be Bogolyubov-BCS quasiparticles consisting of coherent mixtures of electrons and holes. More interesting and relevant to the superconducting cuprates is the case when antiferromagnetic order is turned on weakly on top of the superconductivity. This would correspond to the onset of antiferromagnetism at a critical doping. In such a case a small gap proportional to the weak antiferromagnetic gap opens up for nodal quasiparticles, and the quasiparticle peak would be discernible. We evaluate numerically the absorption by nodal quasiparticles and the local density of states for several ground states with antiferromagnetic and d-wave superconducting correlations.

We analyze the dynamics between $1/\ensuremath{\lambda}$-fractional statistics particles in an exact three-body solution of the Sutherland Hamiltonian. We show that they interact by means of a short-range attraction. This attraction appears to be a general feature of fractional statistics particles in one dimension. The interaction dictates important features of the spectral function for a charge-1 particle.