We systematically examine how long-range hopping and its synergy with extended interactions leads to light bound pairs. Pair properties are determined for a dilute extended Hubbard model with large on-site repulsion (U) and both near- and next-nearest neighbour hopping (t and t′) and attraction (V and V′), for cubic and tetragonal lattices. The presence of t′ and V′ promotes light pairs. For tetragonal lattices, t′<0 pairs can be lighter than non-interacting particles, and d-symmetric pairs form. Close packing transition temperatures, T⁎ are estimated for the Bose-Einstein condensation (BEC) of pairs to be kBT⁎∼t‾0.1, where t‾ is the geometric mean of the hoppings on the Cartesian axes. When pairs have d-symmetry, the condensate has d-wave character. Thus, the presence of both t′ and V′ leads ubiquitously to small strongly bound pairs with an inverse mass that is linear in hopping, which could lead to high temperature BECs.
We study the Bose-Einstein condensate in a honeycomb optical lattice within Bogoliubov theory and find that for a
k=0 condensate, the Dirac points appear in the Bogoliubov excitation spectrum when
0<β<2, which illustrates that the bose-bose interaction does not change the Dirac point structure but only give a modification of the velocity of the Dirac cone. When the bosons are driven to condense at
k=K, however, we find that the topology of the Dirac points will be altered by arbitrary weak interaction. Furthermore, we find that the next-nearest-neighbor hopping in an isotropic and an anisotropic lattice has different effects to the dynamics of the condensate and it should be taken into account when the lattice is not sufficiently deep.