  We study the $S_z$-conserving quantum spin Hall insulator in the presence of
Hubbard $U$ from a field theory point view. The main findings are the
following. (1) For arbitrarily small U the edges possess power-law correlated
antiferromagnetic XY local moments. Gapless charge excitations arise from the
Goldstone-Wilczek mechanism. (2) The decaying exponent of the XY correlation is
$\ge 2$ hence normally space-time vortices should proliferate. (3) For
thermodynamic samples, vortex proliferation is prohibited by the conservation
of edge charge. (4) For a sample with finite width electron tunneling between
opposite edges allows vortex proliferation hence is a strongly relevant
perturbation (more relevant than in the non-interacting theory). (5) The phase
transition between the antiferromagnetic and the topological insulating phases
is triggered by the condensation of magnetic excitons.
