  We perform the state-of-the-art tensor network simulations directly in the
thermodynamic limit to clarify the critical properties of the $q$-state clock
model on the square lattice. We determine accurately the two phase transition
temperatures through the singularity of the classical analog of the
entanglement entropy, and provide extensive numerical evidences to show that
both transitions are of the Berezinskii-Kosterlitz-Thouless (BKT) type for
$q\ge 5$ and that the low-energy physics of this model is well described by the
$\mathbb{Z}_q$-deformed sine-Gordon theory. We also determine the
characteristic conformal parameters, especially the compactification radius,
that govern the critical properties of the intermediate BKT phase.
