Geometric bulk-edge correspondence for $\mathbb{Z}_2$-topological insulators

Fermionic time-reversal-invariant insulators in two dimensions--class AII in the Kitaev table--come in two topological phases.

These phases are characterized by a $\mathbb{Z}_2$-valued invariant, the Fu-Kane-Mele index.

We prove a geometric bulk-edge correspondence for curved interfaces: if two such insulators occupy complementary regions separated by a curved boundary, then the $\mathbb{Z}_2$ edge index of the interface system is the product, modulo two, of the difference of the two bulk $\mathbb{Z}_2$ indices and a geometric intersection number associated with the boundary and the measurement region.

The argument is a $\mathbb{Z}_2$ analogue of the curved-interface connection formula proved for Hall insulators in \cite{DZ24}.