Comment on "Peierls instability in the two-dimensional half-filled Hubbard model"

The analyses of the two-dimensional Peierls-Hubbard model by Tang and Hirsch and by the present author are compared. While there is agreement with the principal conclusion of Tang and Hirsch, viz., that the effects of the Hubbard interaction on the Peierls bond alternation are different in one and two dimensions, I disagree on three related issues. First, the ($\pi , \pi$) phonon which dominates in the uncorrelated limit is different from the one Tang and Hirsch believe wins [($\pi , \pi$) along $x$ axis]. Second, the energy crossing between the ($\pi , \pi$) and ($\pi , 0$) phonons found by Tang and Hirsch for large magnitudes of the Hubbard interaction can occur only in the unrealistic limit of very strong bond alternation where the strength of the weak bond is close to zero. Finally, the disappearance of the bond alternation due to the Hubbard interaction is much more rapid than that calculated by Tang and Hirsch—the bond alternation should become weaker even at small $U$. This suppression of the bond alternation is indeed related to the long-range antiferromagnetism in two dimensions, as surmised by Tang and Hirsch.