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 6 atoms per unit cell as compared to two atoms for the undistorted lattice.

Figure 1

(a) Shown is the only graphene structure allowed by our theorem. The locations of the three possible bond lengths are indicated by heavy-solid, light-solid, and dashed lines. There are six atoms (or three hexagons) per unit cell. The Kekulé structure is the special case of equal length solid lines. (b) The ALT structure shown breaks some of the reflection symmetry and is not allowed by our theorem.

Figure 2

Some definitions: (a) The top figure is a single row and the bottom figure is a reflected $\theta$ row. They are joined by overlapping the common vertical bonds. (b) A happy bond (heavy line) has the $t$ values of the four adjacent bonds equal in pairs, as shown.

Figure 3

(a) The heavy lines show the more than 50% happy bonds after reflecting in orientations 1 and 2. (b) shows the more than 75% happy bonds after reflecting in orientations 1, 2, and again 1.