MATHPH In the literature: open

Exponential decay of correlations in the 2D O(N) model at all temperatures

In plain words

Prove that in a flat grid of three-component (or more) arrows, the correlation between two arrows dies off exponentially with their distance at every temperature above zero. For two-component arrows this is false at low temperature, which is why the claim is subtle.

Precise statement

Classical $O(N)$ model on $Z^{2}$, $N \ge 3$, spins $s_{x}$ in $S^{N-1}$, Gibbs weight exp(beta sum_<xy> s_x . s_y). Prove that for every $\beta < \infty$ there are $C, m(\beta) > 0$ with $\langle s_{0} . s_{x}\rangle \le C\ \operatorname{exp}(-m(\beta)\ \mid x\mid)$ uniformly in the volume. For $N = 2$ the opposite holds at large $\beta$ (power-law decay, Frohlich and Spencer, 1981). Answer: a proof, or a proof of a massless low-temperature phase.

What would settle it

A proof of a positive mass $m(\beta)$ for all finite $\beta$, or a proof of power-law decay at some finite $\beta$, for $N = 3$.

Status in the literature

Exponential decay is proven only at small $\beta$ by high-temperature expansion; Patrascioiu and Seiler argued for a massless low-temperature phase, and percolation properties of the $2D$ Heisenberg model were studied by Aru, Garban and Sepulveda (arXiv 2212.06767, 2022).