Bethe-Peierls approximation for the triangular Ising antiferromagnet in a field

We perform a Bethe-Peierls approximation to calculate the phase diagram of a nearest-neighbor triangular Ising antiferromagnet in a uniform field. The conjectured ground state is taken into account by the consideration of three interpenetrating sublattices. We find a set of self-consistent equations of state and an associated thermodynamic potential. Besides a disordered paramagnetic solution, at higher temperatures and fields, there are also low-temperature antiferromagnetic solutions. We show that the associated free energy is always smaller for the paramagnetic solution, which indicates that the antiferromagnetic ordering is just a local minimum of the thermodynamic potential. Similar results can still be obtained from a reaction-field approximation.