Notice that the topological invariant $g$ which does not change with small deformations on the surface of the Manifold is analogous to the winding number. Small changes in the Berry Potential (or in the Hamiltonian via the parameters $v$ or $w$) will indeed change the Berry potential locally, but when it is integrated over the entire Brillouin zone, the winding number remains invariant! In other words, small changes to $h(k)$ will change the trajectory of the vector $\vec{h}(k)$ and the trajectory will no longer remain a perfect circle but the winding number will still be defined and remain unchanged.

Now we have seen that the two seemingly equivalent insulating cases are ‘topologically’ different which is concluded by exploiting the information of wavefnuctions in these two cases. But how do these different behaviours of the eigenvectors make the two insulating cases different physically? In other words, we are looking for a physical consequence of the winding number in the seemingly similar insulating cases. But before answering that in its full glory, let us note one physical consequence that can be drawn from this distinct behaviour. If we were to smoothly transform the Hamiltonian, in other words change any of the parameters 