Chern and Fu-Kane-Mele invariants as topological obstructions

The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in $\mathbb Z_2$. We illustrate how both the Chern number $c \in \mathbb Z$ and the Fu-Kane-Mele invariant $\delta \in \mathbb Z_2$ of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, $c$ quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, $\delta$ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.