Effective models for strong correlations and edge magnetism in graphene

We describe a method for deriving effective low-energy theories of electronic interactions at graphene edges. Our method is applicable to general edges of honeycomb lattices (zigzag, chiral, and even disordered) as long as localized low-energy states (edge states) are present. The central characteristic of the effective theories is a dramatically reduced number of degrees of freedom. As a consequence, the solution of the effective theory by exact diagonalization is feasible for reasonably large ribbon sizes. The quality of the involved approximations is critically assessed by comparing the correlation functions obtained from the effective theory with numerically exact quantum Monte-Carlo calculations. We discuss effective theories of two levels: a relatively complicated Fermionic edge state theory and a further reduced Heisenberg spin model. The latter theory paves the way to an efficient description of the magnetic features in long and structurally disordered graphene edges beyond the mean-field approximation.

Figure 1

Spin correlation functions $\langle {\sigma }_0^z{\sigma }_j^z\rangle$ at the edges of a zigzag ribbon ($N_x=7$, $N_y=6$) for three values of $U=0.02,0.1,0.3$. Site 0 in the first index of the correlation function refers to site 0 at the lower edge (as indicated by the circle). The squares represent the exact results quantum Monte-Carlo simulations (error bars are smaller than the symbol size). The curves show the results of the effective theory. The lines between the data points are guides to the eye. The dashed red curves represent the bare correlation function, consisting of edge, bulk, and mixed terms. The solid green and dotted blue lines (mostly on top of each other) show the susceptibility correction without and with the RPA, respectively. The edge sites are labeled as indicated in the inset.

Figure 2

Spin correlation functions $\langle {\sigma }_0^z{\sigma }_j^z\rangle$ at the edges of a zigzag ribbon ($N_x=11$, $N_y=8$) for larger $U=0.2,1.0$. The meaning of the symbols and lines is the same as in Fig. 1.

Figure 3

Definition of the shifted chirality geometry via $\chi ,\mathfrak{S},$ and $N_y$. Dashed lines indicate connections between sites of neighboring unit cells. Part (a) shows a standard chiral nanoribbon. Part (b) shows a nanoribbon with an additional shift $\mathfrak{S}=1$ of the unit cells.

Figure 4

Edge states and bulk states of a chiral ribbon with $\chi =4$ and $N_y=200$. The inset shows the geometry along one edge.

Figure 5

Spin correlation function $\langle {\sigma }_2^z{\sigma }_j^z\rangle$ at the edges of a chiral ribbon for two interaction strengths $U=0.2,1$. The number of unit cells is $N_x=5$ and the number of zigzag lines within each unit cell is $N_y=8$. The positive correlations are intraedge and the negative correlations are interedge. The edge sites labeling is indicated in the inset. The meaning of the symbols and lines is the same as in Fig. 1.

Figure 6

Spin correlation function $\langle {\sigma }_2^2{\sigma }_j^2\rangle$ at the edges of a chiral ribbon with shift $\mathfrak{S}=1$. The number of unit cells is $N_x=5$ and the number of zigzag lines within each unit cell is $N_y=10$. The positive correlations are intraedge and the negative correlations are interedge. The edge sites labeling is indicated in the inset. The meaning of the symbols and lines is the same as in Fig. 1.

Figure 7

Lowest excitation energies of a zigzag ribbon ($N_x=11$, $N_y=20$) as a function of $U$. For orientation the ground-state energy ($E=0$) is included in this plot. The dots are results from the exact diagonalization of the effective edge state Hamiltonian. The lines show the superspin approximation for large $U$ (see text).

Figure 8

$R_{\mathrm{loc}}$ as a function of the ribbon geometry $N_y=2N_x$ and $\mathfrak{S}$, as indicated.

Figure 9

Effective correlation function $\langle {\sigma }_1^z{\sigma }_x^z\rangle$ of the Wannier edge states of chiral ribbons. The parameters are (a) $N_x=5$, $N_y=56$, and $\mathfrak{S}=8$; (b) $N_x=5$, $N_y=120$, and $\mathfrak{S}=8$; and (c) $N_x=5$, $N_y=10$, and $\mathfrak{S}=0$. The insets show the lattice geometry together with the positions of the Wannier states. $U=2$ in all graphs. The reference site is marked by a red circle. The positive correlations correspond to Wannier states at the same edge as the reference site. The dots are calculated from the Fermionic edge state theory. The line connects points calculated from the Heisenberg model, which is an approximation of the Fermionic theory. The line segments connecting the Wannier state numbers are only guides to the eye.