# The argument for asymptotic freedom for the Heisenberg model

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One possible argument for asymptotic freedom in the 2D O(3) ferromagnetic Heisenberg model is the existence of so-called instantons, discovered in the 1975 paper of Belavin and Polyakov. This is supposedly a simpler but similar-mechanism for asymptotic freedom in the the 4D SU(2) Yang-Mills gauge theory--also via instantons.

They write "we have thus proved that a ferromagnet has inhomogeneous metastable states. This apparently means that there is a finite correlation length in the system and there is no phase transition even at very low temperatures."

Can anyone explain a bit more about why the existence of these metastable solutions (i.e. classical solutions of the equations of motion, or local extrema of the action, which however have arbitrarily high energy) directly imply finite correlation length at all temperatures? I am aware of the 1975 paper by Polyakov where he lays an argument for that using renormalization group analysis, but if I understand correctly that uses an entirely different logic.

Also: the 2D $O(4)$ Heisenberg model supposedly has no such topological solutions, since $\pi_2(\mathbb{S}^3)=\{0\}$. But that apparently does not mean that it has a phase transition, right? I am not aware anyone claims the 2D $O(4)$ model has no mass gap. So is it the case that for $O(3)$ there happens to be, accidentally, an easier way to show asymptotic freedom using these instantons, but in principle the more robust argument that works for all $O(N)$, $N>2$ is the renormalization group argument?

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