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 $J(\mathbf{S}_i\cdot \mathbf{S}_k)^2$. We can consider this model as a continuum version of anti-ferromagnetic $D$-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.