An analytical approach to quantum phase transitions of ultracold Bose   systems in bipartite optical lattices: Along the avenue of Green's function
  In this paper, we present a generalized Green's function method which can be used to investigate the quantum phase transitions analytically in a systematic way for ultracold Bose systems in bipartite optical lattices. As an example, to the lowest order, we calculate the quantum phase boundaries of the localized states (Mott insulator or charge density wave) of an ultracold Bose system in a $d$-dimensional hypercubic optical lattice with nearest-neighbor repulsive interactions. Due to the inhomogeneity of the system, in the generalized Green's function method, cumuants on different sublattices are calculated separately, together with re-summed Green's function technique, the analytical expression of the phase boundaries of the localized phases in the system is presented. 
Zhi Lin, Jun Zhang, Yan Chen, and Ying Jiang
cond-mat.stat-mech cond-mat.quant-gas