Properties of some mean-field-like approximations for the triangular Ising antiferromagnet

Motivated by a recent proposal of a Bethe approximation for the triangular Ising antiferromagnet [Phys. Rev. B 56, 8241 (1997)], which seems to predict a disordered phase at any temperature in zero field, we analyze in some detail several mean-field-like approximations for this model, namely, the Bethe approximation itself, the cluster variation method, and the hard-spin mean-field theory. We show (i) that the disordered phase predicted by the Bethe approximation is unphysical at low enough temperature because of a negative entropy; (ii) how the results of the cluster variation method (namely, zero-temperature entropy and critical temperature of the spurious transition) converge to the exact ones for increasing cluster size; (iii) that it is possible to construct a cluster variation approximation which yields a disordered-phase which is stable down to zero temperature; (iv) a few, so far unknown, zero-temperature results (entropy and internal energy) of the hard-spin mean-field theory.