Mean field theory in the Heisenberg model

[Jeff Chen]

I have a problem about Heisenberg model.When applying mean field approximation,why does the average of spin flip terms be zero(<S+>=<S-> = 0 )? Thanks

 

[king vitamin]

It's just a convention - you take the "mean field" such that the system magnetizes in the z direction, and take the magnetization in the other directions to zero.

More generally, in an arbitrary coordinate system, you assume that the system has a uniform magnetization given by
$$
\mathbf{M} = \langle \mathbf{S}_i \rangle = M \hat{e}
$$
where $\hat{e}$ is a unit vector in some direction. Now just choose your coordinate system such that the z direction points along $\hat{e}$, and then you have $\langle S^+ \rangle = \langle S^- \rangle = 0$ in that coordinate system.