The replica symmetric solution for orthogonally constrained Heisenberg model on Bethe lattice

In this paper, we study the thermodynamic properties of a system of D-components classical Heisenberg spins lying on the vertices of a random regular graph, with an unconventional first neighbor non-random interaction J( S_i.S_k)^2. We can consider this model as a continuum version of antiferromagnetic D-states Potts model. We compute the paramagnetic free energy, using a new approach, presented in this paper for the first time, based on the replica method. Through the linear stability analysis, we obtain an instability line on the temperature-connectivity plane that provides a bound to the appearance of a phase transition. We also argue about the character of the instability observed.