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?
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).