Possible Lattice Distortions in the Hubbard Model for Graphene

The Hubbard model on the honeycomb lattice is a well known model for graphene. Equally well known is the Peierls type of instability of the lattice bond lengths. In the context of these two approximations we ask and answer the question of the possible lattice distortions for graphene in zero magnetic field. The answer is that in the thermodynamic limit only periodic, reflection-symmetric distortions are allowed and these have at most six atoms per unit cell as compared to two atoms for the undistorted lattice.