  Two-dimensional (2D) model of a weak topological insulator with $N$-stacked
Su-Schrieffer-Heeger (SSH) chain is studied. This study starts with a basic
model with all the fundamental symmetries (chiral, time-reversal, and
particle-hole) preserved. Different topological phases are introduced in this
model by systematically breaking the system's symmetries. The symmetries are
broken by introducing different bonds (hopping terms) in the system. First, the
chiral symmetry is broken by introducing hopping within each sub-lattice or
intra-sub-lattice hopping, where the hopping strengths of the sub-lattices are
equal in magnitudes but opposite in sign. Then, following Haldane, the
time-reversal (TR) symmetry is broken by replacing the real intra-sub-lattice
hopping strengths with imaginary numbers without changing the magnitudes. We
find that breaking chiral and TR symmetries are essential for the weak
topological insulator to be a Chern insulator. These models exhibit nontrivial
topology with the Chern number $C = \pm 1$. The preservation of the
particle-hole (PH) symmetry in the system facilitates an analytical calculation
of $C$, which agrees with the numerically observed topological phase transition
in the system. An interesting class of topologically nontrivial systems with
$C=0$ is also observed, where the non-triviality is identified by quantized and
fractional 2D Zak phase. Finally, the PH symmetry is broken in the system by
introducing unequal amplitudes of intra-sub-lattice hopping strengths, while
the equal intra-sub-lattice hopping strengths ensures the preservation of the
inversion symmetry. We investigate the interplay of the PH and the inversion
symmetries in the topological phase transition. A discussion on the possible
experimental realizations of this model is also presented.
