The replica symmetric solution for Orthogonally Constrained Heisenberg Model on Bethe lattice

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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(Sᵢ· Sₖ)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.