Topological order and the vacuum of Yang-Mills theories

We study, for $SU(2)$ Yang-Mills theories discretized on a lattice, a non-local topological order parameter, the center flux ${{z}}$. We show that: i) well defined topological sectors classified by $\pi_1(SO(3))=\mathbb{Z}_2$ can only exist in the ordered phase of ${{z}}$; ii) depending on the dimension $2 \leq d\leq 4$ and action chosen, the center flux exhibits a critical behaviour sharing striking features with the Kosterlitz-Thouless type of transitions, although belonging to a novel universality class; iii) such critical behaviour does not depend on the temperature $T$. Yang-Mills theories can thus exist in two different continuum phases, characterized by an either topologically ordered or disordered vacuum; this reminds of a quantum phase transition, albeit controlled by the choice of symmetries and not by a physical parameter.