A controversial problem: modified Ising model in a random field

The effect of quenched impurities on a system having a commensurate-incommensurate (CI) phase transition is treated by real-space rescaling. A simplified form of the free energy is treated explicitly. New terms generated by the rescaling procedure do not modify the results. In D=3 dimensions, the Bragg peak shift (misfit) varies linearly with the chemical potential near the CI transition. The CI transition is killed for D<or=D c =2. At the critical dimension, the misfit decreases exponentially for large negative chemical potentials. The model treated here may be considered a modified Ising model in a random field. The modification is a strong anisotropy of the interfaces separating the up and down domains.