Monte Carlo and renormalization-group studies of surface structural phase transitions

The ${\phi }^4$ model for structural phase transitions in two dimensions is studied using the Migdal renormalization-group method and a Monte Carlo coarse-graining calculation. The methods employed are useful for more general models of surface reconstructive transitions. Particular emphasis is placed on accuracy of phase diagrams and the nature of the "crossover" from order to disorder behavior above (in temperature) but close to the phase transition and displacive behavior at higher temperatures. The Monte Carlo coarse-graining calculation is found to give the fullest information on this crossover and the short-range order above the transition and also verifies the accuracy of the more economical Migdal calculation of the phase diagram. The results are discussed with reference to the surface reconstruction phase transition of the clean W(001) surface.