Order-disorder transitions to 2 × 2 structures

We study the transitions to 2 × 2 ordered states which can occur in the triangular and honeycomb lattice gases with first- and second-neighbor repulsions only. A simple renormalization group which preserves the symmetry of such states, but not that of the $(\sqrt{3}\times \sqrt{3}) R30°$ phase, is employed. We find that the 2 × 2 phases exist in a very narrow region of density. In addition to phase diagrams, specific heats are calculated.