  A new "bond-algebraic" approach to duality transformations provides a very
powerful technique to analyze elementary excitations in the classical
two-dimensional XY and $p$-clock models. By combining duality and Peierls
arguments, we establish the existence of non-Abelian symmetries, the phase
structure, and transitions of these models, unveil the nature of their
topological excitations, and explicitly show that a continuous U(1) symmetry
emerges when $p \geq 5$. This latter symmetry is associated with the appearance
of discrete vortices and Berezinskii-Kosterlitz-Thouless-type transitions. We
derive a correlation inequality to prove that the intermediate phase, appearing
for $p\geq 5$, is critical (massless) with decaying power-law correlations.
