Dualities and the phase diagram of the p-clock model

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⩾5p⩾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⩾5p⩾5, is critical (massless) with decaying power-law correlations.