  We study systems of staggered boson Hamiltonians in a one dimensional lattice
and in particular how the translation symmetry by one unit in these systems is
in reality a non-invertible symmetry closely related to T-duality. We also
study the simplest systems of clock models derived from these staggered boson
Hamiltonians. We show that the non-invertible symmetries of these lattice
models together with the discrete ${\mathbb Z}_N$ symmetry predict that these
are critical points with a $U(1)$ current algebra at $c=1$ and radius
$\sqrt{2N}$ whenever $N>4$.
