$v$ or $w$, then since $h_z=0$, the only possible way to get from the insulating phase described by the case $v<w$ to the insulating phase described by the case $v>w$ (or the other way round), is via crossing the origin! What this means is that somewhere in the middle we have to get to the solution of eigenstates corresponding to a trajectory, $h(\vec{k})$, in the $h_x-h_y$ space, that passes through the origin in that space. Therefore, a smooth transition from one insulating phase to the other is not possible without closing the energy gap (or crossing the metallic phase) at least once. This is the hallmark of a **topological phase transition** [@kane]. An analogy can be drawn here with a 3D topological insulator (TI) like Bi$_2$Se$_3$. In a 3D TI, the bulk is an insulator which is topologically distinct (has different winding number) from the vacuum (outside of the TI). Note that vacuum can be thought of like a band insulator with the particle (like electron) and the antiparticle (like hole) states separated by a “band gap". In order to smoothly transition from the TI phase to the band insulator phase, one must cross a point (the interface between the 