Magnetic Correlations in Short and Narrow Graphene Armchair Nanoribbons

Electronic states at the ends of a narrow armchair nanoribbon give rise to a pair of nonlocally entangled spins. We propose two experiments to probe these magnetic states, based on magnetometry and tunneling spectroscopy, in which correlation effects lead to a striking, nonlinear response to external magnetic fields. On the basis of low-energy theories that we derive here, it is remarkably simple to assess these nonlinear signatures for magnetic edge states. The effective theories are especially suitable in parameter regimes where other methods such as quantum Monte Carlo simulations are exceedingly difficult due to exponentially small energy scales. The armchair ribbon setup discussed here provides a promisingly well-controlled (both experimentally and theoretically) environment for studying the principles behind edge magnetism in graphene-based nanostructures.

Figure 1

Lattice geometry of an armchair nanoribbon with $W=3$ hexagons in zigzag direction and $L=10$ hexagons in armchair direction. On top of the lattice the weight of the edge-localized low-energy states $|{\psi }_{\mathrm{L}}(i){|}^2+|{\psi }_{\mathrm{R}}(i){|}^2$ is shown. Both the dot size and color scales with the weight. The blue circle indicates a typical site at which the spectral function is to be measured (see text).

Figure 2

Singlet-triplet splitting $J$ as a function of ribbon length $L$. The blue dotted line is $J_{\mathrm{f}}$ calculated from $H_{\mathrm{f}}$ [Eq. (2)]. The red solid line corresponds to $J_{\mathrm{H}}$ [Eq. (3)]. The black circles with error bars are the results of the QMC simulations $J_{\mathrm{exact}}$. (a) is with $U=0.5t$ and (b) with $U=t$.

Figure 3

Total magnetic moment $M$ (in ${10}^3{\mu }_{\mathrm{B}}$) of an ensemble of nanoribbons as a function of the magnetic field $B$ for different temperatures. The ensemble contains 3000 ribbons for each length $L=8,10,\dots ,20$. Each individual ribbon contributes two Bohr magnetons ${\mu }_{\mathrm{B}}$ to the total moment if it is in the triplet state.

Figure 4

Spectral function $A_{\mathrm{L}\downarrow }(\omega )$ for $t^{*}=0.006\mathrm{eV}$ and $U^{*}=0.6\mathrm{eV}$, corresponding to a ribbon with $W=3$ and $L=10$. The critical field $B_c=\pm J/2\approx 2.1\mathrm{T}$ is indicated by dashed lines. In the main plot the Gaussian smearing $\eta =200$ Kelvin (see text). The insets show the fine structure of the spectral peaks with narrower Gaussian smearing of $\eta =10$ Kelvin (b) and $\eta =0.1$ Kelvin (a).