Editing Classical XY model (section)

== Phase transition ==
As mentioned above in one dimension the XY model does not have a phase transition, while in two dimensions it has the [[Berezinskii-Kosterlitz-Thouless transition|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 [[Lambda point|Helium-4]]. The values of these [[critical exponent]]s 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 exponent]]s of the three dimensional XY model are <math>\alpha,\beta,\gamma,\delta,\nu,\eta</math>. All of them can be expressed via just two numbers: the scaling dimensions <math>\Delta_\phi</math> and <math>\Delta_s</math> of the complex order parameter field <math>\phi</math> and of the leading singlet operator <math>s</math> (same as<math>|\phi|^2</math> in the [[Ginzburg–Landau theory|Ginzburg–Landau]] description). Another important field is <math>s'</math>(same as <math>|\phi|^4</math>), whose dimension <math>\Delta_{s'}</math> determines the correction-to-scaling exponent <math>\omega</math>. According to a conformal bootstrap computation,<ref>{{Cite journal|last1=Chester|first1=Shai M.|last2=Landry|first2=Walter|last3=Liu|first3=Junyu|last4=Poland|first4=David|last5=Simmons-Duffin|first5=David|last6=Su|first6=Ning|last7=Vichi|first7=Alessandro|title=Carving out OPE space and precise O(2) model critical exponents|url=http://link.springer.com/10.1007/JHEP06(2020)142|journal=Journal of High Energy Physics|year=2020|language=en|volume=2020|issue=6|pages=142|arxiv=1912.03324|doi=10.1007/JHEP06(2020)142|bibcode=2020JHEP...06..142C|s2cid=208910721|issn=1029-8479}}</ref> these three dimensions are given by: 
{| class="wikitable"
|-
|<math>\Delta_{\phi}</math>
|0.519088(22)
|-
|<math>\Delta_s</math>
|1.51136(22)
|-
|<math>\Delta_{s'}</math>
|3.794(8)
|}
This gives the following values of the critical exponents:
{| class="wikitable"
|-
! 
!general expression (<math>d=3</math>)
! numerical value
|-
|{{math|<var>α</var>}}
|<math>2-d/(d-\Delta_s)</math>
|  -0.01526(30)
|-
|{{math|<var>β</var>}}
|<math> \Delta_\phi/(d-\Delta_s)</math>
| 0.34869(7)
|-
|{{math|<var>γ</var>}}
|<math>(d-2\Delta_\phi)/(d-\Delta_s) </math>
| 1.3179(2)
|-
|{{math|<var>δ</var>}}
|<math> (d-\Delta_\phi)/\Delta_\phi</math>
| 4.77937(25)
|-
|{{math|<var>η</var>}}
|<math>2\Delta_\phi - d+2</math>
| 0.038176(44)
|-
|{{math|<var>ν</var>}}
|<math>1/(d-\Delta_s)</math>
| 0.67175(10)
|-
|{{math|<var>ω</var>}}
|<math>\Delta_{s'}-d</math>
| 0.794(8)
|}

Monte Carlo methods give compatible determinations:<ref>{{Cite journal|last=Hasenbusch|first=Martin|date=2019-12-26|title=Monte Carlo study of an improved clock model in three dimensions|url=https://link.aps.org/doi/10.1103/PhysRevB.100.224517|journal=Physical Review B|language=en|volume=100|issue=22|pages=224517|arxiv=1910.05916|doi=10.1103/PhysRevB.100.224517|bibcode=2019PhRvB.100v4517H|s2cid=204509042|issn=2469-9950}}</ref> <math>\eta=0.03810(8),\nu=0.67169(7), \omega=0.789(4)</math>.