Quantum phase transitions of two-species bosons in square lattice

We investigate various quantum phase transitions of attractive two-species bosons in a square lattice. Using the algorithm based on the tensor product states, the phase boundaries of the pair superfluid states with nonzero pair condensate density \emph{and} vanishing atomic condensate density are determined. Various quantum phase transitions across the phase boundaries are characterized. Our work thus provides guides to the experimental search of the pair superfluid phase in lattice boson systems.