While the Mermin–Wagner theorem prevents any spontaneous symmetry breaking on a global scale, ordering transitions of Kosterlitz–Thouless–type may be allowed. This is the case for the XY model where the continuous (internal) O (2) symmetry on a spatial lattice of dimension d ≤ 2, i.e. the (spin-)field's expectation value, remains zero for any finite temperature (quantum phase transitions remain unaffected). However, the theorem does not prevent the existence of a phase transition in the sense of a diverging correlation length ξ. To this end, the model has two phases: a conventional disordered phase at high temperature with dominating exponential decay of the correlation function G ( r ) ∼ ∼ exp ⁡ ⁡ ( − − r / ξ ξ ) for r / ξ ξ ≫ ≫ 1 , and a low-temperature phase with quasi-long-range order where G (r) decays according to some power law for "sufficiently large", but finite distance r (a ≪ r ≪ ξ with a the lattice spacing).