Classical Heisenberg model at zero temperature

It is shown under a certain rather weak condition on the pair interaction potential: $|\Lambda|^{-1}\sum_{r,r'\in\Lambda}|u_{r,r'}|^2<K$, that the free energy of the classical Heisenberg model in the ordered as well as disordered case in the limit $T\to0$ converges to the expression corresponding to the self-consistent field approximation. As a by-product, the free energy for the Eguchi–Kawai $U(N)$ gauge model is found for arbitrary dimension $d$ of the lattice.