Exact diagonalization study of the tunable edge magnetism in graphene

The tunable magnetism at graphene edges with lengths of up to 48 unit cells is analyzed by an exact diagonalization technique. For this we use a generalized interacting one-dimensional model which can be tuned continuously from a limit describing graphene zigzag edge states with a ferromagnetic phase, to a limit equivalent to a Hubbard chain, which does not allow ferromagnetism. This analysis sheds light onto the question why the edge states have a ferromagnetic ground state, while a usual one-dimensional metal does not. Essentially, we find that there are two important features of edge states: (a) Umklapp processes are completely forbidden for edge states, which allows a spin-polarized ground state; (b) the strong momentum dependence of the effective interaction vertex for edge states gives rise to a regime of partial spin-polarization and a second-order phase transition between a standard paramagnetic Luttinger liquid and ferromagnetic Luttinger liquid.

Figure 1

(a) Half-infinite honeycomb lattice. The solid ellipses (green) are the complete unit cells in the bulk region and the dashed ellipses indicate the cut unit cells at the $\alpha$ edge. The $n$ and $m$ directions are indicated as well as the sublattice indices A, B. (b) Modulus square of the transverse (in $n$ direction) edge-state wave function $|{\psi }_p(n)|{}^2$ on the B sublattice sites in the reduced Brillouin zone $\frac{2\pi }{3}⩽p⩽\frac{4\pi }{3}$. The extreme momentum dependence of the localization of ${\psi }_p$ in the $n$ direction is crucial for the magnetic properties of edge states.

Figure 2

(Left) The self-energy $~$${\mathcal{N}}_p^2$ of the direct model (dashed line) and the linearized self-energy (solid lines). (Right) The four possible interaction processes. The $g_3$ process is not allowed for graphene edge states.

Figure 3

Lowest eigenenergies in different $S_z$ subspaces for $N=12$ in the Hubbard limit with ${\lambda }_{\mathrm{bs}}=1$, ${\lambda }_{\mathrm{us}}=1$, ${\Gamma }_1=0$, and $U=1$. The inset shows the lowest eigenenergies of the $S_z=0,6$ subspaces (zero and full spin polarization) as the umklapp scattering is suppressed. The lower lines correspond to ${\lambda }_{\mathrm{us}}=1$ and the upper lines to ${\lambda }_{\mathrm{us}}=0$. For the lines in between, ${\lambda }_{\mathrm{us}}$ decreases in steps of 0.2.

Figure 4

Lowest eigenenergies with completely suppressed umklapp scattering ${\lambda }_{\mathrm{us}}=0$ and different momentum dependencies ${\Gamma }_1$. Furthermore, $N=16$, ${\lambda }_{\mathrm{bs}}=1$, and $U=1$. (a) The case of a momentum-independent interaction vertex (${\Gamma }_1=0$), where the ground-state degeneracy jumps from 17 ($=$$2S_{\max }+1$) directly to 1 at the position indicated by the arrow. (b) The case of an interaction vertex with maximal momentum dependence ${\Gamma }_1=6/\pi$. The arrows indicate a change in the ground-state degeneracy.

Figure 5

Dependence of the spin polarization $S$ on $v_F\pi /U$ for different strengths of the momentum dependence ${\Gamma }_1$ from ${\Gamma }_1=0$ (rightmost curve) to ${\Gamma }_1=\frac{6}{\pi }$ (leftmost curve) in steps of $\Delta {\Gamma }_1=\frac{6}{10\pi }$. The length of the edge is $L=48$, ${\lambda }_{\mathrm{bs}}=1$ and ${\lambda }_{\mathrm{us}}=0$. The smooth curve is a power law fit to the plateau centers of the ${\Gamma }_1=\frac{6}{\pi }$ steps with exponent $\beta =0.5$ (see text).

Figure 6

Phase diagram for lengths $L=24$, $L=36$, and $L=48$. For small velocity dependence ${\Gamma }_1$ of the interaction, only the Luttinger liquid (LL) phase and the saturated edge magnetism (SEM) phase exist, whereas above the critical value of ${\Gamma }_1={\Gamma }_1^c$ the weak edge magnetism (WEM) phase appears.

Figure 7

The lowest eigenenergies of different $S_z$ subspaces with (solid blue lines) and without (dashed green lines) backscattering calculated for an edge of length $L=48$. The broken SU(2) symmetry in the case of ${\lambda }_{\mathrm{bs}}\ne 1$ lifts the degeneracy of the ground states in the different $S_z$ subspaces.

Figure 8

Lowest eigenenergies in the $S_z=0,\pm 2,\pm 4,\pm 6,\pm 8$ subspaces (from bottom to top) for the direct model calculated for an edge of length $L=48$. The inset shows the dependence of the spin polarization $S$ as a function of $\Delta /U$ determined from the degeneracy of the ground state.