TKNN invariant

Remarks

The curvature $\mathcal{F}$ does not depend on the phase of $u_k$, so it is defined on all of $T$ even when no global smooth choice of $u_k$ exists. The number $C$ is an integer: it is the first Chern number of the line bundle over $T$ formed by the eigenvectors. Sign conventions for $\mathcal{F}$ and $C$ differ between sources. For a band insulator with the Fermi energy in a gap, the Hall conductance is $\sigma_{xy} = \frac{e^2}{h} \sum_{m \text{ filled}} C_m$.

Example: the two-band model $H(k) = d(k) \cdot (\sigma_x, \sigma_y, \sigma_z)$ with Pauli matrices $\sigma_j$ and $d(k) = (\sin k_1, \sin k_2, M + \cos k_1 + \cos k_2)$ on $\Lambda = \Z^2$. The lower band has $\abs{C}$ equal to the degree of $d/\abs{d} \colon T \to S^2$ in absolute value. For $0 < M < 2$, the direction $(0, 0, 1)$ is reached at $k = (0,0), (\pi, 0), (0, \pi)$ with local degrees $+1, -1, -1$, so $\abs{C} = 1$; for $M > 2$ it is also reached at $(\pi, \pi)$ with local degree $+1$, so $C = 0$.

Non-example: at $M = 2$ the vector $d(k)$ vanishes at $k = (\pi, \pi)$, the two bands touch, and $C$ is not defined.