We did not denote the Berry connection as \(\mathbf{A}(\mathbf{k})\) just by chance. We picked that letter because this vector reminds us a lot of the vector potential \(\mathbf{A}(\mathbf{r})\) that is used in electromagnetism.

Just like the vector potential, the definition of \(\mathbf{A}(\mathbf{k})\) depends on a particular choice of the person making the calculation. If you decide to multiply the quantum state by a phase, \(\left|\psi(\mathbf{k})\right\rangle\,\to \exp\,[i\lambda(\mathbf{k})]\,\left|\psi(\mathbf{k})\right\rangle\), then you get that the Berry connection transforms as \(\mathbf{A}(\mathbf{k})\,\to\,\mathbf{A}(\mathbf{k})+\nabla_\mathbf{k} \,\lambda\). However, when you take the integral of \(\mathbf{A}(\mathbf{k})\) on a closed path, the result is independent of \(\lambda\). That’s why the Berry phase is only meaningful for closed paths.

Now that we have established an analogy with the vector potential, we cannot avoid the idea of taking the curl of the Berry connection, which is known as the Berry curvature:

\[\mathbf{\Omega}(\mathbf{k}) = \nabla_\mathbf{k} \times \mathbf{A}(\mathbf{k})=i\left[\left\langle \frac{\partial \psi(\mathbf{k})}{\partial k_x}\,\Bigg|\,\frac{\partial\,\psi(\mathbf{k})}{\partial k_y}\right\rangle-\left\langle \frac{\partial \psi(\mathbf{k})}{\partial k_y}\,\Bigg|\,\frac{\partial\,\psi(\mathbf{k})}{\partial k_x}\right\rangle\right]\,.\]

The Berry curvature is like a magnetic field in momentum space. Just like the magnetic field \(\mathbf{B}(\mathbf{r})=\nabla_\mathbf{r}\times\mathbf{A}(\mathbf{r})\) in electromagnetism, it is a local quantity which does not suffer from the ambiguities of the vector potential (it is gauge independent).

The main advantage of introducing the analogy with the magnetic field is that it motivates us to use Stokes theorem. The Brillouin Zone has the shape of a torus. Therefore the curve \(k_x=0\) and \(k_x=2\pi\) on the torus bounds the entire Brillouin zone. Using Stokes theorem on this curve we can conclude that

\[2\pi W=\gamma(2\pi)-\gamma(0)=\iint_{\textrm{BZ}} \mathbf{\Omega}(\mathbf{k})\,\cdot\,d\mathbf{S}\,,\]

where the integral extends over the entire Brillouin Zone.

As a result of this formalism, we have established two things. First, there is a Chern number which is defined entirely in terms of the momentum space wave functions. Second, the analogy with the magnetic field allows us to obtain an explicit expression for the Chern number in terms of derivatives of the wave functions.

Loosely speaking, a situation with a non-zero Chern number is a bit like having a magnetic monopole, because we have a finite flux coming out of a closed surface. Now, you probably know that experimentally a magnetic monopole was never observed. For our Chern number in the Brillouin zone the situation is more exciting, as situations where it is non-zero are realized in nature.

To see how this can happen, we first have to understand the following: if there is Berry curvature in the Brillouin zone, what are its sources?