Chiral condensate with topological degeneracy in graphene and its manifestation in edge states

The role of chiral symmetry in many-body states of graphene in strong magnetic fields is theoretically studied with the honeycomb lattice model. For a spin-split Landau level where the leading electron-electron interaction is the nearest-neighbor repulsion, a chiral condensate is shown to be, within the subspace of the $n=0$ Landau level, an exact many-body ground state having a finite gap, for which a calculation of the Chern number reveals that the ground state is a Hall insulator with a topological degeneracy of two. The topological nature of the ground state is shown to manifest itself as a Kekuléan bond order along armchair edges, while the pattern melts in the bulk due to quantum fluctuations. These can be regarded as a realization of the bulk-edge correspondence that is peculiar to a chiral-symmetric system. We have also obtained the ground state when point defects are introduced in the honeycomb lattice to reveal how the presence or absence of the chiral symmetry affects the defect states.

Figure 1

Magnetic-field dependence of the many-body gap $\Delta$ in the $n=0$ LL. The result is displayed for 30 electrons with the consecutive values of the linear dimension of the system $L$ connected by a line with the results for $L\equiv 3$ (mod 3) forming a lower envelope ($\Delta \propto {\varphi }^2$). The inset shows the gap at $L=3l$ against $1/L$ for the magnetic flux $\varphi =1/12$ (circle), $\varphi =1/27$ (triangle), and $\varphi =1/48$ (square).

Figure 2

Mean-field results for the energy spectrum (a), its blowup around the $n=0$ LL (b), and the bond strength $|{\Delta }_{ij}|$ plotted in a real space (c). The parameters are $L=15$, $\varphi =1/15$, and $V/t=0.25$. A dashed hexagon in (c) is an enlarged unit cell with arrows primitive vectors.

Figure 3

Bond strength near an armchair (left) or zigzag (right) edge of the doubly degenerate chiral condensate. The magnetic flux is $\varphi =1/192$ for which the magnetic length is $l_B\simeq 8.9a$. The bond order decays in a bulk away from the edges.

Figure 4

Bond strength of the chiral condensate near a divacancy composed of two adjacent sites that are missing. The magnetic flux is $\varphi =1/1200$ for which the magnetic length is $l_B\simeq 22.3a$. The bond order reflects twofold axial symmetry of the divacancy.