the decay of the correlations is only power law: Fröhlich and Spencer found the lower bound while McBryan and Spencer found the upper bound, for any Three and higher dimensions Independently of the range of the interaction, at low enough temperature the magnetization is positive. At high temperature, the spontaneous magnetization vanishes: . Besides, cluster expansion shows that the spin correlations cluster exponentially fast: for instance . At low temperature, infrared bound shows that the spontaneous magnetization is strictly positive: . Besides, there exists a 1-parameter family of extremal states, , such that but, conjecturally, in each of these extremal states the truncated correlations decay algebraically. Phase transition As mentioned above in one dimension the XY model does not have a phase transition, while in two dimensions it has the Berezinski-Kosterlitz-Thouless transition between the phases with exponentially and powerlaw decaying correlation functions. In three and higher dimensions the XY model has a ferromagnet-paramagnet phase transition. At low temperatures the spontaneous magnetization is nonzero: this is the ferromagnetic phase. As the temperature is increased, spontaneous magnetization gradually decreases and vanishes at a critical temperature. It remains zero at all higher temperatures: this is the paramagnetic phase. In four and higher dimensions the phase transition has mean field theory critical exponents (with logarithmic corrections in four dimensions). Three dimensional case: the critical exponents The three dimensional case is interesting because the critical exponents at the phase transition are nontrivial. Many three-dimensional physical systems belong to the same universality class as the three dimensional XY model and share the same critical exponents, most notably easy-plane magnets and liquid Helium-4. The values of these critical exponents are measured by experiments, Monte Carlo simulations, and can also be computed by theoretical methods of quantum field theory, such as the renormalization group and the conformal bootstrap. Renormalization group methods are applicable because the critical point of the XY model is believed to be described by a renormalization group fixed point. Conformal bootstrap methods are applicable because it is also believed to be a unitary three dimensional conformal field theory. Most important critical exponents of the three dimensional XY model are . All of them can be expressed via just two numbers: the scaling dimensions and of the complex order parameter field and of the leading singlet operator (same as in the Ginzburg–Landau description). Another important field is (same as ), whose dimension determines the correction-to-scaling exponent