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?

  • $\begingroup$ @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

If you simulate with angles you shouldn't take theta and phi uniform random numbers. In here


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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