Exponential decay for the two-dimensional Heisenberg model
In this work, we prove exponential decay for the spin correlation of the spin $O(\mathsf{N})$ model for $\mathsf{N}\geq3$ at every inverse temperature $ β>0 $. In particular, our work confirms Polyakov's prediction in 1975 on the Heisenberg model, i.e., the case of $\mathsf{N}=3$. To this end, we investigate a generalized $O(\mathsf{N})$ model where $ \mathsf{N} $ is a positive \emph{real number}, as follows: we sample a loop soup on the metric graph (say of a box) with intensity $ \tfrac{\mathsf{N}}{2} $ subject to the conditioning that the local time at each lattice vertex is $β$, and then for each loop we independently assign it to copies $1\leq \mathsf{i}\leq \lceil\mathsf{N}\rceil$ such that it is assigned to copy $\mathsf{i}$ with probability $ \tfrac{1}{\mathsf{N}} $ for $ 1\leq \mathsf{i}\leq \lfloor\mathsf{N}\rfloor $. Then, by Lupu's isomorphism theorem, we can construct a spin field by assigning a random sign to each loop soup cluster in each copy independently, such that the field coincides with the spin $O(\mathsf{N})$ model when $\mathsf{N}$ is an integer. We then establish exponential decay for every real-valued $\mathsf{N}>2$. Our result reveals an underlying mechanism for exponential decay, which emerges as soon as $\mathsf{N}>2$ (i.e., as soon as beyond the range of the XY model) and does not necessarily rely on the rotation symmetry in the sphere $ \mathbb{S}^{2} $.