Dirac Point Structure in a Bose-Einstein Condensate in a Honeycomb Optical Lattice

We study the Bose-Einstein condensate in a honeycomb optical lattice within Bogoliubov theory and find that for a ${\bf k} = 0$ condensate, the Dirac points appear in the Bogoliubov excitation spectrum when $0 < \beta < 2$, 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 ${\bf k} = {\bf K}$, 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.