Phase transition in a linear chain of classical spins with a logarithmic nearest neighbour pair potential

We present the complete calculation of the partition function and correlation functions of a linear array of classical spins coupled by a nearest neighbour logarithmic pair potential. In the case of a ferromagnetic coupling there occurs a phase transition at Tc > 0. The critical exponents of the specific heat C and the magnetic susceptibility χ (in the absence of an external field) are shown to have the non-classical value α = 2 and classical value γ = 1 respectively. The underlying mathematical mechanism of the phase transition is the complete degeneracy of all the eigenvalues of the corresponding integral equation (Kac's mechanism). Below Tc the partition function becomes complex. For antiferromagnetic coupling the free energy is analytic in the whole temperature range and so no phase transition occurs in this case.