  Motivated by the experiment [St-Jean {\it et al}., Nature Photon. {\bf 11},
651 (2017)] on topological phases with collective photon modes in a zigzag
chain of polariton micropillars, we study spinless $p$-orbital fermions with
local interorbital hoppings and repulsive interactions between $p_x$ and $p_y$
bands in zigzag optical lattices. We show that spinless $p$-band fermions in
zigzag optical lattices can mimic the interacting Su-Schrieffer-Heeger model
and the effective transverse field Ising model in the presence of local
hoppings. We analytically and numerically discuss the ground-state phases and
quantum phase transitions of the model. This work provides a simple scheme to
simulate topological phases and the quench dynamics of many-body systems in
optical lattices.
