  For two-dimensional lattices in a tight-binding description, the intrinsic
spin-orbit coupling, acting as a complex next-nearest-neighbor hopping, opens
gaps that exhibit the quantum spin Hall effect. In this paper, we study the
effect of a real next-nearest-neighbor hopping term on the band structure of
several Dirac systems. In our model, the spin is conserved, which allows us to
analyze the spin Chern numbers. We show that in the Lieb, kagome, and T_3
lattices, variation of the amplitude of the real next-nearest-neighbor hopping
term drives interesting topological phase transitions. These transitions may be
experimentally realized in optical lattices under shaking, when the ratio
between the nearest- and next-nearest-neighbor hopping parameters can be tuned
to any possible value. Finally, we show that in the honeycomb lattice,
next-nearest-neighbor hopping only drives topological phase transitions in the
presence of a magnetic field, leading to the conjecture that these transitions
can only occur in multigap systems.
