  For the Su-Schrieffer-Heeger (SSH) model on the two-dimensional square
lattice, two third nearest neighbor hoppings which preserve chiral symmetry are
introduced. Like the case of one dimension, the longer-range hopping can drive
topological transitions and leads to the larger topological invariant (vectored
Berry phase for the two dimensional SSH model). We obtain the phase boundaries
and topological phase diagrams from the winding pattern of $\rho(k_l)$ of the
Bloch Hamiltonian. Effects of correlations on the extended two dimensional SSH
model are also investigated. The correlation shifts the phase boundaries and
leads to topological transitions. Several special points in non-interacting
phase diagrams are chosen to illustrate the different phase transitions and a
correlations-driven topologically non-trivial phase from the trivial one is
found. In this work, the slave-rotor mean field method is applied to the
interacting model and we focus on the case of physical electrons, i.e. the Mott
transition of charge degree of freedom is avoided in our investigations.
