# Phase transition in 2D Heisenberg model

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?

• @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. – mikuszefski Nov 3 '16 at 9:30
• Are you working in $3d$ or $2d$? – valerio Nov 3 '16 at 9:31
• BTW...mean field is neglecting fluctuations (check also local mean field), MC is problematic with spin wave excitations. – mikuszefski Nov 3 '16 at 9:42
• I am working 2d. I understand MFT is ignoring fluctuations but why is preffering z direction specifically? – ayushi singhania Nov 3 '16 at 11:27
• 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. – valerio Nov 3 '16 at 12:26