Excitonic instabilities and insulating states in bilayer graphene

The competing ground states of bilayer graphene are studied by applying renormalization group techniques to a bilayer honeycomb lattice with nearest neighbor hopping. In the absence of interactions, the Fermi surface of this model at half-filling consists of two nodal points with momenta $\mathbf{K}$, ${\mathbf{K}}^{\prime }$, where the conduction band and valence band touch each other, yielding a semimetal. Since near these two points the energy dispersion is quadratic with perfect particle-hole symmetry, excitonic instabilities are inevitable if interband interactions are present. Using a perturbative renormalization group analysis up to the one-loop level, we find different competing ordered ground states, including ferromagnetism, superconductivity, spin and charge density wave states with ordering vector $\mathbf{Q}=\mathbf{K}-{\mathbf{K}}^{\prime }$, and excitonic insulator states. In addition, two states with valley symmetry breaking are found in the excitonic insulating and ferromagnetic phases. This analysis strongly suggests that the ground state of bilayer graphene should be gapped, and with the exception of superconductivity, all other possible ground states are insulating.

Figure 1

Bilayer graphene with $AB$ stacking: $a_1$, $b_1$ are the two sublattice sites in the upper layer; $a_2$, $b_2$ are the two sublattice sites in the lower layer. ${\gamma }_0$ is the tight-binding hopping constant between $a_1$; $b_1$, ${\gamma }_1$ is the hopping between $a_1$ and $a_2$; ${\gamma }_3$ is the hopping between $b_1$ and $b_2$. ${\mathbf{a}}_1=\frac{a}{2}(3,\sqrt{3})$ and ${\mathbf{a}}_2=\frac{a}{2}(3,-\sqrt{3})$ are the primitive lattice vectors.

Figure 2

(a) $\mathbf{K}=\frac{2\pi }{3a}(1,\frac{1}{\sqrt{3}})$ and ${\mathbf{K}}^{\prime }=\frac{2\pi }{3a}(1,-\frac{1}{\sqrt{3}})$ are the two points, constituting the Fermi surface of the noninteracting system. $\Lambda$ is the momentum cutoff of the theory, $d\Lambda$ is a thin shell containing high energy modes to be integrated out. (b) ${\epsilon }_c\left(K\right)$ and ${\epsilon }_f\left(K\right)$ are the dispersion energy of the conduction and the valence band, respectively. The other two bands are gapped by ${\gamma }_1$.

Figure 3

Feynman diagrams: $1,2,3,4$ represent the low-energy modes with momentum $K_{1,2,3,4}$, band index $I$, valley index $\alpha$, and spin $\sigma$. The momentum inside the loop, $K$, must lie within the shell $d\Lambda$, and $Q=K_3-K_1$, $Q^{\prime }=K_4-K_1$, and $P=K_1+K_2$. Note that the interaction lines are suppressed.

Figure 4

Flow of the susceptibilities: here, we set $h_0=u_0=v_0$, $h_1=u_1=v_1=0.8h_0$, $h_2=u_2=v_2=0.1h_0$, and $g_0=g_1=g_2=0$ ($h_0>0$). The FM instabilities occupy a large region in parameter space, and fine-tuning is not necessary.

Figure 5

Phase diagram for a representative choice of bare couplings $h_0=u_0=v_0=1$, $h_1=v_1=u_1=0.8h_0$, and $h_2=u_2=v_2=g_2=0.1h_0$. The phase diagram is determined by monitoring which channels diverge first during the RG flow.

Figure 6

Flow of $g_0$ as the bare value of $g_1$ varies. We set $g_0=g_2=0.1h_0$, and the remaining bare couplings are the same as Fig. 4.