A many-body generalization of the $Z_2$ topological invariant for the quantum spin Hall effect

We propose a many-body generalization of the $Z_2$ topological invariant for the quantum spin Hall insulator, which does not rely on single-particle band structures. The invariant is derived as a topological obstruction that distinguishes topologically distinct many-body ground states on a torus. It is also expressed as a Wilson-loop of the SU(2) Berry gauge field, which is quantized due to the time-reversal symmetry.