see FIG. \[windingfig\], that even though their dispersions are similar, these trajectories on the $h_x-h_y$ plane are very different!

![Trajectories of $\vec{h}(k)$ over the Brillouin zone in the $h_x-h_y$ space for different parameter values[]{data-label="windingfig"}](winding.pdf)

The vector $\vec{h}(k)$ that denotes the eigenstates will necessarily form a closed loop as a consequence of the periodicity of the Brillouin zone. For one of the insulating cases ($v<w$), the vector $\vec{h}(k)$ winds about the origin and for the other case ($v>w$) it does not. Notice that the origin is the point where $\vec{h}(k)=0$ which is the gap-less condition. Therefore a loop plotted for the $v=w$ case would pass through the origin indicating a metallic state. We can call this as a [*winding number*]{} $\nu$ that tells whether the trajectory of $\vec{h}(k)$ winds about the origin or not. Winding number distinguishes the two seemingly equivalent cases.

Notice that the bulk Hamiltonian we are looking at, $H(k)$, is also a function of some parameter, in this case $k$, which is known as the crystal momentum. Therefore, Berry potential for an energy band can be defined. The Brillouin zone is periodic and has a topology of a ring. If we consider the Brillouin zone as a parameter 