Peierls instability in the two-dimensional half-filled Hubbard model

We have studied the tendency of the two-dimensional half-filled Hubbard model towards dimerization as a function of the on-site repulsion U. We have found that the phonon with ({π},{π}) wave vector is the most favorable vibration mode when U is less than a critical value Uc, where Uc is a function of the dimerization. When U is greater than Uc, the ({π},0) phonon becomes preferred. In leading order, the energy gain under dimerization appears to be proportional to δ²ln{δ} if the ({π},{π}) phonon is favorable, so that the system displays a Peierls instability in this case. For large U the energy gain to leading order appears to be proportional to δ² for both the ({π},{π}) and ({π},0) phonons, and the latter one is preferred; in that case the system is stable unless the electron-phonon coupling exceeds a critical value. Unlike the effect in the one-dimensional case, the on-site repulsion U seems to have essentially no effect on the tendency towards dimerization for small U, and suppresses it for large U. This difference may be related to the fact that there exists long-range antiferromagnetic order in the two-dimensional half-filled Hubbard model when U is present.