Current orderings of interacting electrons in bilayer graphene

By taking into account the possibility of all the intralayer as well as the interlayer current orderings, we derive an eight-band model for interacting electrons in bilayer graphene. With the numerical solution to the model, we show that only the current orderings between the same sublattice sites can exist within the range of the physical interacting strength. This result confirms our previous model of spin-polarized-current phase for the ground-state of interacting electrons in bilayer graphene that resolves a number of experimental puzzles.

Figure 1

Bond currents between atoms $a$ (black) and $b$ (white) in top layer of BLG. The purple lines denote the current from atom $b$ to atom $a$; the red and green lines are, respectively, the left-turn and right-turn currents from $a$ to $b$. The red and green circles with arrows indicate the circulations of the current in the hexagon bonds. The diamond area enclosed by the dashed lines is the unit cell.

Figure 2

Unit cell of the BLG containing three $a$ atoms (black)—$a_1, a_2$, and $a_3$—and three $b$ atoms (white with red edge)—$b_1, b_2$, and $b_3$—on the top layer and the corresponding $a^{\prime }$ (red) and $b^{\prime }$ (blue) atoms on the bottom layer. The arrows denote the interlayer currents between the $a$ and the $b^{\prime }$ atoms.

Figure 3

Lattice points in the reciprocal space. The solid blue points are for the original honeycomb lattice with the area enclosed by the black solid line as the first BZ. The red-edge circles are the additional points for the superlattice with unit cell containing three $a$ atoms and three $b$ atoms. The two vectors represent the basis of the reciprocal lattice. The diamond area with the dashed line edge is the first BZ for the superlattice. The inner hexagon is the equivalent high-symmetry first BZ.

Figure 4

Currents through the central atom (black) from/to the same-type atoms on a layer. The currents are incoming (outgoing) from (to) the white (red) atoms. There are no currents between the black and the blue sites.

Figure 5

The same as Fig. 1 but with some currents of longer bonds connected to the original atom depicted.

Figure 6

Current order ${\mathrm{\Delta }}^{a{b}^{\prime }}$ as function of the relative interaction strength $v^{a{b}^{\prime }}/v^{aa}$ with $v_{11}^{ab}\left(0\right)/v_{12}^{aa}\left(0\right)=5$. (Inset) Orders ${\mathrm{\Delta }}^a$ and ${\mathrm{\Delta }}^b$.

Figure 7

Current orders ${\mathrm{\Delta }}^a$ and ${\mathrm{\Delta }}^b$ as functions of the external potential $u$.

Figure 8

(Left) Functions $f_a\left(p\right)$ (main panel) and $f_b\left(p\right)$ (inset) at $v^{a{b}^{\prime }}/v^{aa}=1$. (Right) Function $f_{a{b}^{\prime }}\left(p\right)$ at $v^{a{b}^{\prime }}/v^{aa}=1.5$.