Critical and crossover behavior of the two-dimensionalφ4model on a lattice

A real-space renormalization-group transformation has been carried out on the classical ${\phi }^4$ model on a lattice in two dimensions. The critical line has been found for $0<\mathrm{}\theta <\infty$ where $\theta$ is the ratio of the site well depth to the nearest-neighbor harmonic coupling energy. For $\theta \ll 1$ (near the displacive limit) the critical temperature qualitatively agrees with a rigorously derived function for $T_c\left(\theta \right)$ in that limit. In contrast, a previously published Kadanoff-Migdal transformation on the same model predicts that $T_c\left(\theta \right)$ is linear in $\theta$ for $\theta \ll 1$. A crossover to noncritical Gaussian-like behavior is found at temperature $T_0\left(\theta \right)>\mathrm{}{T}_c\left(\theta \right)$ for sufficiently small $\theta$.