Vortices, Zero Modes, and Fractionalization in the Bilayer-Graphene Exciton Condensate

A real-space formulation is given for the recently discussed exciton condensate in a symmetrically biased graphene bilayer. We show that in the continuum limit an oddly quantized vortex in this condensate binds exactly one zero-mode per valley index of the bilayer. In the full lattice model, the zero modes are split slightly due to intervalley mixing. We support these results by an exact numerical diagonalization of the lattice Hamiltonian. We also discuss the effect of the zero modes on the charge content of these vortices and deduce some of their interesting properties.

Figure 1

(a) The schematic structure of the system, and the proposal to create an effective axial magnetic field. (b) The spectrum near the nodal point where the dashed (green) line shows the noninteracting, and the solid (red) line the interacting case with the shifted nodes in external potential ($V$) and the exciton gap ($m$) marked.

Figure 2

The “axial charge.” (a) Density in the uniform system, and (b) the extra charge bound to a single vortex as a function of $V$ and $m_0$. The corresponding density profile for $V/t=0.4$ and $m_0/t=0.3$ in (c) the uniform system, and (d) the system with a vortex at the center. In all plots, each layer has $39\times 22$ sites with a lattice size $\approx (35{)}^2$. The lattice sites in (c) and (d) are at the center of the triangles.

Figure 3

(a) The zero-mode energy splitting. The squares show our numerical results for a system size $\approx (35{)}^2$. The solid lines show the fitting with Eq. (9). (b) The exchange phase vs axial charge. The solid (red) line is $\gamma =\pi \delta Q_v$.