Two-Dimensional Ising Model in an Annealed Random Field

The critical behavior of the two-dimensional Ising model (square lattice, exchange constant J) in an uniform field, and in an annealed random field is considered. The random field is generated by decorating the horizontal and vertical bonds of the lattice, and it satisfies an arbitrary distribution which is imposed by introducing a pseudo-chemical potential. By decimating the decorating variables the model can be mapped onto a homogeneous Ising model with effective exchange constant J′ and effective external field h′, dependent on the temperature. These parameters, which satisfy a set of coupled equations, depend on the spin average and nearest-neighbor two-spin correlation, and are obtained numerically. For the symmetric field distribution [Formula: see text] the mapping of the critical frontier on the (K′=βJ′,H′=βh′) plane onto the (K=β J,H=βh) plane is determined and, as in the model introduced by Essam and Place, there is a region on the (K, H) plane which cannot be reached from any real values of (K′, H′). The critical exponents are determined numerically, and it is shown that they do not satisfy renormalization relations obtained for their model.