Is there an ordered phase for the Heisenberg model at d=2?

Question

Hey folks,

I am wondering whether there is an ordered phase (as for the XY model) in the Heisenberg model for dimension d = 2. On https://en.wikipedia.org/wiki/Classical_Heisenberg_model it is stated "... Polyakov has conjectured that, as opposed to the classical XY model, there is no dipole phase for any T > 0, i.e. at non-zero temperature the correlations cluster exponentially fast." Do you have any newer information than 1975?

Answer 1

in Lectures on the Spin and Loop O(n) Models , Ron Peled and Yinon Spinka say :

Polyakov predicted in 1975 that the spin O(n) model with n>=3 should exhibit exponential decay of correlations in two dimensions at any positive temperature. ... ... Giving a mathematical proof of this statement (or its analog in infinite volume) remains one of the major challenges of the subject. The best known results in this direction are by Kupiainen who performed a 1=n-expansion as n tends to infinity.

Antti J. Kupiainen Comm. Math. Phys. 73 (1980), no. 3, 273-294

Abstract. The ί/n expansion is considered for the π-component non-linear σ-model (classical Heisenberg model) on a lattice of arbitrary dimensions. We show that the expansion for correlation functions and free energy is asymptotic, for all temperatures above the spherical model critical temperature. Furthermore, the existence of a mass gap is established for these temperatures and n sufficiently large.

The § 2.2 Main results and conjectures in the first link is a good review and might complete the wiki page.