Adatom diffusion on a square lattice: Comparison of real-space renormalization group and Monte Carlo approaches

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A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The most accurate results have been obtained for the largest cluster containing 34 sites. The minimal relative error in determining the critical values of interaction parameter is equal to 0.13%. It has been shown that the ``antiferromagnetic majority rule'' for the choice values of block spins gives much better results for the repulsion between the adsorbed particles, as compared to the ordinary ``majority rule.'' Using the RSRG method, we have explored phase diagrams of a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We have calculated: (a) adsorption isotherms for different temperatures, (b) the coverage and temperature dependences of both the pair correlation function for the nearest neighboring adparticles and the chemical diffusion coefficient, (c) the temperature dependence of the specific heat, and (d) the coverage dependences of the isothermal susceptibility for different temperatures. All these quantities have also been obtained by Monte Carlo simulations. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG method can be applied at least for the systems discussed here to characterize the thermodynamic and kinetic properties of interacting adsorbates.