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XY Model

The XY model is the O(n) model at n=2, with a planar unit spin represented by an angle theta_x modulo 2pi. On a finite square lattice, the standard nearest-neighbor zero-field Gibbs measure is proportional to

exp(bsum_({x,y} in E)cos(theta_x-theta_y))product_(x)(dtheta_x)/(2pi),

where b>0 is inverse temperature and every bond is counted once. For the free-boundary infinite-volume state, let C_b(r)=Ecos(theta_0-theta_((r,0))). Put m(b)=-lim_(r->infty)r^(-1)lnC_b(r) and b_c=inf{b:m(b)=0}.

See also

Berezinskii-Kosterlitz-Thouless Transition, Correlation Length, Critical Exponent, Gibbs Measure, Imaginary Gaussian Multiplicative Chaos, O(n) Model, Villain Model

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References

Friedli, S. and Velenik, Y. Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction. Cambridge, England: Cambridge University Press, 2017. https://doi.org/10.1017/9781316882603.

Cite this as:

Weisstein, Eric W. "XY Model." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/XYModel.html

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