The Mermin-Wagner theorem has pointed out that there should be no spontaneous magnetization at finite temperature in 2D-isotropic Heisenberg model, or there will be no long-range ferromagnetic order. However, under classical Monte Carlo method with Metropolis configuration-updating strategy, I have found the equilibrium state at a low temperature exhibits the configuration shown in the attachment, in which a vortex structure similar to that in the XY model appears. Is this a normal result? Or is there something wrong with my program?
The configuation has been shown in the attachment. My code file has been added as well.
First of all, equilibration can take a ``long'' time. Second, what the Mermin-Wagner theorem implies is that there's no spontaneous magnetization at any finite temperature, no matter how low, for the Heisenberg ferromagnet; it doesn't exclude the appearance of the Berezinskii-Kosterlitz-Thouless transition, for the O(2) Heisenberg ferromagnet, since, in both phases, the magnetization vanishes and the order parameter is non-local, not local. For the O(N) Heisenberg ferromagnet, for N>2, no such order parameter can be defined, that's why there's no Berezinkii-Kosterlitz-Thouless transition, when N>2. This doesn't imply that vortices can't appear, whose lifetime can be very ``long'' in Monte Carlo time, especially at ``low'' temperatures. Hence the usefulness of knowing the theory, before doing the simulations.
So the ground state configuration of the D=2, O(N) Heisenberg ferromagnet, when N>2, is characterized by the correlation functions of the spins as = 0, a=1,2,...,N, ~δabexp(-lx-y|m), with m finite for any finite temperature-there's a mass gap for any finite temperature. The other correlation functions satisfy, as for any field theory, the so-called Schwinger-Dyson identities.
In fact the same statements hold, also, for N=2, except, in that case, there's a phase transition at the Berezinskii-Kosterlitz-Thouless temperature, below which the vortices, which are non-local in terms of the spins, have power-law correlation functions, with exponent that is a function of the temperature.
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