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}.