Effect of long-range interaction on graphene edge magnetism

It has been proposed that interactions lead to ferromagnetism on a zigzag edge of a graphene sheet. While not yet directly studied experimentally, dramatically improving techniques for making and studying clean zigzag edges may soon make this possible. So far, most theoretical investigations of this claim have been based on mean-field theories or more exact calculations using the Hubbard model. But long-range Coulomb interactions are unscreened in graphene, so it is important to consider their effects. We study rather general nonlocal interactions, including of the Coulomb $1/r$ form, using the technique of projection to a strongly interacting edge Hamiltonian, valid at first order in the interactions. The ground states as well as electron/hole and exciton excitations are studied in this model. Our results indicate that ferromagnetism survives with unscreened Coulomb interactions.

Figure 1

Sketch of a semi-infinite graphene sheet with a zigzag edge.

Figure 2

The ground-state phase diagram of Eq. (15) at half filling on the $U_{2\parallel }/U-U_{2\angle }/U$ plane. The ground states are ferromagnetic below the phase boundary and have reduced degeneracy above the boundary. The boundary is obtained by exact diagonalization in a system with $N=720$ in the $S_z=N/2-1$ sector and is well approximated by the two straight lines corresponding to the trial wave functions $f\left(k\right)\propto sink$ and $f\left(k\right)\propto k-\pi$ (see text). Also shown is the much smaller region where the sufficient condition for ferromagnetic ground states, Eq. (17), is satisfied.

Figure 3

The ground-state energy in the $S_z$ sector $E_{GS}\left(S_z\right)$ versus $\left|S_z\right|$ for $U_{2\parallel }=0$ and various $U_{2\angle }/U$ outside of the ferromagnetic regime. The results are obtained by exact diagonalization in a system with $N=12$.

Figure 4

The minimum of $\tilde{U}\left(K,q\right)$ for $0\le K\le \pi$ and $-2\pi /3\le q\le 2\pi /3,{\tilde{U}}_{min}$, versus $R$, the long-distance cutoff introduced artificially in the Coulomb interaction Eq. (6).

Figure 5

The single-particle/single-hole dispersion for the Coulomb interaction near the Dirac point. ${\epsilon }_{p/h}\left(k\right)/\left[U_0(k-2\pi /3)\right]$ is plotted against $ln\left(k-2\pi /3\right)$ for ${10}^{-5}\le k-2\pi /3\le 0.1$ and different values of long-distance cutoff $R$, with the particle-hole symmetry-breaking perturbation $\mathrm{\Delta }$ set to zero. For comparison we also show the velocity given by Eq. (26) for each $R$ as a horizontal line. The black line has a slope of $2d/\pi$.

Figure 6

The velocity given by Eq. (26) for the Coulomb interaction as a function of the long-distance cutoff $R$. The fitted line has a slope of 0.1836 while $2d/\pi =0.1838$.

Figure 7

The single-particle dispersion for the Coulomb interaction. ${\epsilon }_p\left(k\right)/U_0$ is plotted against $k$ for $2\pi /3\le k\le 4\pi /3$ and different values of long-distance cutoff $R$. The particle-hole symmetry-breaking perturbation $\mathrm{\Delta }$ is either zero (solid symbols) or ${\mathrm{\Delta }}_c\left(R\right)$ (open symbols).

Figure 8

The exciton dispersion for the Coulomb interaction at small momenta. $E\left(Q\right)/\left(U_0{Q}^2\right)$ is plotted against $lnQ$ for $0.01\le Q\le 0.1$ and different values of long-distance cutoff $R$, with the particle-hole symmetry-breaking perturbation $\mathrm{\Delta }$ set to zero. The lowest 100 Chebyshev polynomials are retained in the numerical solution.

Figure 9

The exciton dispersion for the Coulomb interaction. $E\left(Q\right)/U_0$ is plotted against $Q$ for $0\le Q\le 2\pi /3$ and different values of long-distance cutoff $R$. The particle-hole symmetry-breaking perturbation $\mathrm{\Delta }$ is either zero (solid symbols) or ${\mathrm{\Delta }}_c\left(R\right)$ (open symbols). The lowest 100 Chebyshev polynomials are retained in the numerical solution.