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When we study the two-dimensional isotropic Heisenberg Model using Mean Field Theory or by Monte Carlo simulation we observe a phase transition at a temperature not equal to zero. This is opposed to Mermin Wagner theorem. Interestingly this ordering happens only in z direction. Can anybody explain why we observe such a transition?

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  • $\begingroup$ @JamalS OP is not asking about Ising model, but the isotropic Heisenberg. From mentioning the z direction I conclude that it is 2D as well, as required. $\endgroup$ – mikuszefski Nov 3 '16 at 9:30
  • $\begingroup$ Are you working in $3d$ or $2d$? $\endgroup$ – valerio Nov 3 '16 at 9:31
  • $\begingroup$ BTW...mean field is neglecting fluctuations (check also local mean field), MC is problematic with spin wave excitations. $\endgroup$ – mikuszefski Nov 3 '16 at 9:42
  • $\begingroup$ I am working 2d. I understand MFT is ignoring fluctuations but why is preffering z direction specifically? $\endgroup$ – ayushi singhania Nov 3 '16 at 11:27
  • $\begingroup$ Then this is the so-called XY model, if I understand correctly. MFT is not good in $2d$, and the ordering you are witnessing must be a simulation artifact. There is no phase transition in the XY model, except for the BKT transition. $\endgroup$ – valerio Nov 3 '16 at 12:26
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If you simulate with angles you shouldn't take theta and phi uniform random numbers. In here

http://mathworld.wolfram.com/SpherePointPicking.html

explained why. If you take random theta and phi uniformly, you oversampled in poles and your spins are a lot more up an down (like Ising).

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