Renormalization-group study of crossover in structural phase transitions

The eigenvalues and eigenfunctions of the transfer operator are used to develop an exact renormalization-group (RG) transformation for the ${\phi }^4$ model of structural phase transitions in one dimension. The method we develop is applicable over the entire range from the displacive limit to the order-disorder limit. Analysis of the RG flow near the displacive limit and far from the fixed points enables us to identify a high-temperature Gaussian-like displacive region where phononlike excitations dominate. At lower temperatures a crossover to order-disorder behavior is driven by the formation of domain walls. The transformation is extended to two dimensions by using the Kadanoff-Migdal transformation. A phase diagram is produced, and the displacive region found in one dimension persists. The crossover which occurs above the critical temperature is still identified with the onset of domain-wall formation.