Fate of Dirac points in a vortex superlattice

We consider noninteracting fermions on the honeycomb lattice in the presence of a magnetic vortex superlattice. It is shown that depending on the superlattice periodicity, a gap may open at zero energy. We derive an expression of the gap in the small-flux limit but the main qualitative features are found to be valid for arbitrary fluxes. This study provides an original example of a metal-insulator transition induced by a strongly modulated magnetic field in graphene. At the same time, our results directly apply to Kitaev's honeycomb model in a vortex superlattice.

Figure 1

A piece of the $\mathcal{L}(p=1,q=1)$ magnetic vortex superlattice spanned by primitive vectors ${\mathit{b}}_1$ and ${\mathit{b}}_2$. Vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$ are primitive vectors of the bare honeycomb lattice. Light-center (green) and dark-center (red) plaquettes contain a flux $+\phi$ and $-\phi$, respectively, whereas white plaquettes are flux-free. Blue links with arrows indicate oriented hopping terms “carrying” the flux.

Figure 2

Behavior of $p^2\Delta /{\phi }^2$ as a function of $\ln p$ (for $q=0$) in the small-$\phi$ limit. Exact results are obtained from Eq. (7) and the full line is a linear fit in good agreement with the conjecture discussed in the text.

Figure 3

$\Delta$ as a function of $\phi$ and $p$ (for $q=0$). The maximum is reached for $p=3$ and $\phi =1/2$ where $\Delta \simeq 0.611132$.