Enhancement of nonlocal exchange near isolated band crossings in graphene

The physics of nonlocal exchange interactions in graphene sheets is studied within a $\pi$-orbital tight-binding model using a Hartree-Fock approximation and Coulomb interactions modified at short distances by lattice effects and at large distances by dielectric screening. We use this study to comment on the strong nonlocality of exchange effects in systems with isolated band crossings at energies close to the Fermi energy. We also discuss the role of lattice scale details of the effective Coulomb interaction in determining whether or not broken-symmetry states appear at strong interaction strengths, and in determining the character of those states when they do appear.

Figure 1

Choice of the irreducible wedge of the primitive cell (left) and the adaptive sampling of the $k$ points in the vicinity of the Dirac point $K$ used for most of our calculations. The density of $k$ points in the dense region shown in the figure corresponds to a sampling density of $512\times 512$ points.

Figure 2

Illustration of sublattice psedospin dependence on momentum for the Dirac-like Hamiltonians. In one dimension (a) the pseudospin changes direction at the Dirac point, in two dimensions (b) it has a vortex, and in three dimensions (c) it has a monopole hedgehog structure. In each case the band state sublattice pseudospin changes direction upon crossing through the Dirac point.

Figure 3

Absolute value of the sublattice off-diagonal density matrix defined in Eq. (21) illustrating the overall power-law decay $r^{-2}$. Left panel: Gray scale map representation where we can notice a regular anisotropy of the off-diagonal density matrix, which follows the triangular Bravais lattice structure of graphene. The density-matrix element falls off with a larger power law along certain discrete directions. Right panel: Density matrix at discrete lattice vectors along the directions ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ indicated in the left panel. For direction ${\mathbf{r}}_1$ we notice a periodic dip in the value of ${\rho }_{AB}\left(\mathbf{r}\right)$ every three lattice constants. The density matrix for these lattice vectors has a $r^{-3}$ decay law. The slow $r^{-2}$ power-law decay reflects the singular dependence of valence-band wave functions on $\mathbf{k}$ at the Dirac points. The values of the fitting coefficients are $c_1=0.0037$, $c_2=0.0015$, and $d_1=0.0019$ when distances are measured in nm.

Figure 4

Upper panel: Tight-binding and Hartree-Fock band structures near the Dirac point. The right panels blow up the small rectangular regions shown in the left panels. We observe that the momentum space Hartree-Fock calculation (HF2) follows the enhancement to smaller momenta than the real-space truncated interaction calculation (HF1). Lower panel: $E\left(k\right)/k$ versus $k$ close to the Dirac point. The momentum space HF2 calculation used $1024\times 1024$ $k$ points in the primitive cell. The velocity enhancement saturates in both calculations. The dashed straight line is the small $k$ fit obtained using Eq. (27) with $p_c=30$.

Figure 5

Phase diagram showing where spin-density-wave (SDW) and charge-density-wave (CDW) broken-symmetry solutions appear in our model as a function of the interaction parameters $U$ and ${\epsilon }_r$. Strong short-distance repulsion (large $U$) favors SDW states, whereas weak short-distance interactions and strong Coulomb interactions (small ${\epsilon }_r$) favor CDW states. Below the the solid line in this figure the Hartree mean-field interaction energy is lowered by forming a CDW state, which has different densities on $A$ and $B$ sublattices. The CDW state boundary lies below this line because the band energy favors uniform densities. The SDW state is a simple antiferromagnet, as expected at large $U$ on bipartite lattices. The arrow in the figure shows the critical value $U=2.23|{\gamma }_0|$ beyond which SDW solutions appear for the pure Hubbard model. The shaded regions in the figure indicate the parameter values thought to be most appropriate for graphene sheets that are suspended and for those that are supported by a dielectric substrate, as discussed further in the main text.

Figure 6

Real-space truncation of the interaction range in graphene as illustrated with two different cutoff values of $L_{\max }\sim 2a$ and $L_{\max }\sim 5a$. As we change the value of the cutoff radius $L_{\max }$ there are oscillations in the relative number of carbon lattice sites $A$ and $B$ enclosed within the cutoff distance.

Figure 7

Longer ranged contributions to the Hartree energy $E_{DI}^{\mathrm{long}}\left(L_{\max }\right)$ as defined in Eq. (A7), which shows a strong cutoff distance $L_{\max }$ dependent oscillation that converges slowly to the limiting value represented with the horizontal line whose behavior is more clearly shown in the inset. We can notice that $E_{DI}^{\mathrm{long}}\left(L_{\max }\right)$ is rather close to the asymptotic limit for certain values of $L_{\max }$. A better estimate of the asymptotic limit can be obtained from the behavior of $E_{DI}^{av,\mathrm{long}}\left(L_{\max }\right)$ defined in the text.