# The argument for a mass gap for the $O(3)$ Heisenberg ferromagnet

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?

I asked this question in physics overflow but I thought it might be useful to ask here too.

• Not an answer, but for context: the review arxiv.org/abs/1708.00058 summarizes what is and isn't currently known about the $O(N)$ generalization of the Heisenberg model ($N=3$) in all numbers of dimensions, including two. It cites Polyakov's 1975 paper, but it doesn't mention instantons or the paper by Belavin and Polyakov. For some cautionary remarks about asymptotic freedom, see arxiv.org/abs/hep-th/0312015. – Chiral Anomaly Feb 21 at 14:34

I think the statement in the paper is not quite right. A better summary of what Polyakov understood about the $$d=2$$ $$O(3)$$ model is his text book "Gauge fields and strings".
The first important observation is that the $$O(n>2)$$ model is asymptotically free $$\beta(e^2)=-\frac{n-2}{2\pi}e^4,$$ see (2.49) in the book. This has nothing to do with instantons, it is the result of a purely perturbative calculation involving spin waves. In the language of statistical mechanics it suggests that the only critical point is $$T=0$$, and the theory is disordered at any non-zero temperature.
The $$n=3$$ theory has instantons. It is also scale invariant, so instantons come in all sizes, and as in QCD we should integrate over the size distribution $$Z \sim \int d^2x \frac{d\rho}{\rho^3} \exp(-4\pi^2/e^2)$$ where the form of the $$\rho$$ integration follows from scale invariance (or an explicit calclation). But the charge runs, so $$Z \sim \int d^2x \frac{d\rho}{\rho}$$ which is both UV and IR divergent (see 6.13 in the book). So the role of instantons is unclear, and the topological susceptibilty is not well defined. This divergence has been verified on the lattice, including some pretty recent results.