Many-body Fu-Kane-Mele index

We define a ${\mathbb{Z}}_2$-valued index for stably short-range entangled states of two-dimensional fermionic lattice systems with charge conservation and time reversal symmetry. The index takes its non-trivial value precisely if the `fluxon’, the state obtained by inserting a $\pi$-flux through the system, transforms under time reversal as part of a Kramers pair. This index extends the Fu-Kane-Mele index of free fermionic topological insulators to interacting systems.