Development of a Phase Transition for a Rigorously Solvable Many-Body System

The ferromagnetic transition is considered in detail for the Heisenberg Hamiltonian with all spins coupled equally. For this special model the statistical problem is solved exactly. The transition develops very slowly as the number $N$ of particles in the system is increased; the spin order at the nominal Curie temperature is proportional to $N^{-\frac{1}{4}}$. The order in the most probable state differs from the mean value of the order, except for $N\to \infty$. The connections are studied with the molecular-field, spin-wave, and self-consistent approximations. The failure of the self-consistent approximation is relatively severe as compared with the nearest-neighbor problem. The antiferromagnetic ground state is very close in energy to the Néel state.