The bulk of my question relates to a two-level system, but I have some questions about the Chern number in general as well.

The Chern number of a gapped periodic system (free fermions or mean field) is given by
\begin{equation}
\mathcal C = \frac{1}{2\pi} \sum_n \int_{\textrm{BZ}} d^2\mathbf k \, \mathcal F_n(\mathbf k),
\end{equation}
where the sum runs over the occupied bands. Here $\mathcal F_n(\mathbf k)$ is the Berry curvature of the $n$-th Bloch band which can be calculated from the corresponding Bloch eigenstates.

- Why is the Chern number only defined for gapped systems? Or is it just that it is an integer only for gapped systems?
- Why is the Chern number of a gapped continuum model a half integer? Or does this only apply to massive Dirac fermions?

As an example, consider the Berry curvature of the ground state of a two-level system with Bloch Hamiltonian $H(\mathbf k)=d_i(\mathbf k) \sigma_i$:\begin{equation}
\mathcal F(\mathbf k) = \frac{1}{2} \hat{\mathbf d} \cdot \left( \partial_{k_x} \hat{\mathbf d} \times \partial_{k_y} \hat{\mathbf d} \right).
\end{equation}
Up to a constant, this is the Jacobian of the map $X \rightarrow Y : \mathbf k \mapsto \hat{\mathbf d}(\mathbf k)$ which is a parameterization of the unit 2-sphere. Therefore, the flux through the image $Y$ in $S^2$ is given by
\begin{equation}
\int_Y \hat{\mathbf d} \cdot d\mathbf S = 2 \int_X d^2\mathbf k \, \mathcal F(\mathbf k).
\end{equation}
If the mapping covers the sphere exactly ones, i.e. it wraps around the sphere one time, then the integral equals $4\pi$ and therefore the Chern number (or wrapping number) is one.

Is this correct? Also, the unit vector $\hat{\mathbf d}(\mathbf k)$ is not defined if the gap closes ($\frac{0}{0}$). But this is just a single point, why does the integral make no sense in this case?

Why does the mapping wrap around the 2-sphere an integer amount of times if the domain $X$ is compact (e.g. Brillouin zone torus)? Or is this statement incorrect?

What if a band is only partially filled? In this case, the domain of the mapping is compact (or not?), but the integral is not quantized?

Thanks!

This post imported from StackExchange Physics at 2015-07-15 14:47 (UTC), posted by SE-user Praan