Symmetry-enforced quantum spin Hall insulators in $\pi$-flux models

We prove a Lieb-Schultz-Mattis theorem for the quantum spin Hall effect (QSHE) in $\pi$-flux models. In the presence of time reversal, U(1) charge conservation and magnetic translation ($\pi$-flux per unit cell) symmetries, if a gapped Hamiltonian has a unique symmetric ground state at half filling (one electron per unit cell), it can only be a QSH insulator i.e. a trivial Mott insulator ground state is forbidden. We further show that such a symmetry-enforced QSHE can be realized in cold atoms, by shaking optical lattices and applying an oscillating Zeeman field.