Interactions and Magnetism in Graphene Boundary States

We analyze interaction effects on boundary states of single layer graphene. Near a half filled band, both short- and long-ranged interactions lead to a fully spin-polarized configuration. In addition, the band of boundary states acquires a finite dispersion as a function of the momentum parallel to the edge, induced by the interactions. Away from half filling the wave function develops charge correlations similar to those in a Wigner crystal, and the spin strongly alternates with the occupation of the boundary states. For certain fillings the ground state has a finite linear momentum, leading to the formation of persistent currents.

Figure 1

Left: Atomic structure of an unrolled nanotube with zigzag edges. $j$ labels the zigzag lines, each containing $N$ different $A$ and $B$ sites. $k$ denotes the wave number along the edge. Right: Single-particle spectrum for $N=28,N_B=9$. Gray (red) circles denote boundary states, black circles extended (bulk) states, and blue dashed line shows the Dirac cone. Boundary states are labeled by integer quantum number $m$.

Figure 2

Band dispersion $E_m$ for empty band and half filled band ${\tilde{E}}_m$ (see text) for $N=28,N_B=9$. Dashed lines show continuous limit ($N\to \infty$).

Figure 3

Spin of the ground state (solid red line) and energy gap for spin excitations from the ground state (dashed black line) as a function of the number of boundary electrons. Upper and lower row correspond to $N=28,N_B=9$ and $N=31,N_B=10$, respectively. Energies in units of $e^2/R$ (Coulomb) and $U/N$ (Hubbard).

Figure 4

$N_e$ electron ground state energies as function of total spin $S$ and angular momentum $M$ ($M_{\mathrm{sym}}=N_e{m}_{\mathrm{sym}}$). The favored spin depends on the angular momentum. For certain occupation numbers (e.g., $N_e=8$), the ground state has a finite angular momentum. $N=28,N_B=9$.