Spin-ordering in a planar classical Heisenberg model

Abstract

We consider aD-dimensional system of classical spins rotating in a plane and interacting via a Heisenberg coupling. The spin-correlation functiong _D(r) is calculated for large distancesr in a low-temperature approximation (taking into account shortrange order):

$$\begin{gathered} g_1 (r) = \exp ( - C_1 Tr), \hfill \\ g_2 (r) \sim r^{ - C_2 T} , \hfill \\ \mathop {\lim }\limits_{r \to \infty } g_3 (r) = \exp ( - C_3 T). \hfill \\ \end{gathered}$$

In two dimensions this model exhibits infinite susceptibility

$$\chi = \frac{1}{{2T}}\sum\limits_r {g_2 (r)}$$

at low temperatures.

Comparison is made ofg ₁ with the exact result and ofg ₃ with a spinwave-treatment showing agreement of lng(r) within orderT Spontaneous magnetization forD=1 and 2 is ruled out exactly.