Small perturbations of the $2d$ classical Heisenberg model

Source abstract

Polyakov predicted in 1975 ([Pol75) that the spin $O(3)$-model (or classical Heisenberg model) should have exponential decay of correlations at all positive temperatures on $\mathbb{Z}^2$. Our main result is as follows: if we replace the Euclidean scalar product $$\begin{align*} \langle σ_i , σ_j\rangle = σ_i^x σ_j^x + σ_i^y σ_j^y + σ_i^z σ_j^z \end{align*}$$ by the $λ$-modified scalar product $$\begin{align*} \langle σ_i , σ_j\rangle_λ:= σ_i^x σ_j^x + σ_i^y σ_j^y + λσ_i^z σ_j^z \end{align*}$$ we show that for all $λ\in(-1,1)$, the $\mathbb{S}^2$ valued spin system with Gibbs measure $\exp(β\sum_{i\sim j} \langle σ_i,σ_j\rangle_λ)$ has power-law decay of correlations if $β$ is chosen large enough. This is the first evidence that arbitrary small smooth modifications of the classical Heisenberg model produce a BKT phase in $2d$. It was proved in the 1970s and 80s that there exist arbitrary small smooth modifications of the classical Heisenberg model which produce instead an Ising phase. In these work, they considered the regime $λ>1$ while in this paper, we consider the regime $λ<1$ which cannot be analyzed via reflection positivity and chessboard estimate. The two main ingredients used in our proof are the stability of the BKT phase under percolation type disorder ([DG25]) and weak regularity type properties of the groundstates or $\mathbb{S}^2$-valued harmonic maps associated to our $λ$-energy functional. This analytic step is interesting on its own. Finally, the exponential and power-law decay regime should be separated by a curve $λ\mapsto β_c(λ)$. We prove a rigorous lower bound in the case of the $λ$-anisotropic hemisphere model $\mathbb{S}^2_+$ which matches with RG flow predictions.