Competing ordered states in bilayer graphene

We use a perturbative renormalization group approach with short-range continuum model interactions to analyze the competition between isotropic gapped and anisotropic gapless ordered states in bilayer graphene, commenting specifically on the role of exchange and on the importance of spin and valley flavor degeneracy. By comparing the divergences of the corresponding susceptibilities, we conclude that this approach predicts gapped states for flavor numbers $N=1,2,4$. We comment briefly on other approaches which can be used to shed light on the bilayer ground state and on the related ground state of chiral (ABC) trilayers.

Figure 1

Electron-electron scattering processes for a system with a single pseudospin-$1/2$ degree of freedom.

Figure 2

(a) ZS, (b) ZS${}^{\prime }$ loop corrections in the one-loop PRG calculation.

Figure 3

BCS (particle-particle) loop correction for singlet propagation in the one-loop PRG calculation.

Figure 4

BCS (particle-particle) loop correction for triplet propagation in the one-loop PRG calculation.

Figure 5

RG flows of the interaction couplings for (a) $N=1$, (b) $N=2$, and (c) $N=4$. In the $N=1$ case the dashed lines are unphysical since interactions always enter observables in the combination ${\Gamma }_{\mathrm{DS}}={\Gamma }_{\mathrm{D}}-{\Gamma }_{\mathrm{X}}$.

Figure 6

Feynman diagrams for the pseudospin susceptibilities. (a) The noninteracting susceptibility and (b) the interaction correction to the susceptibility. $T,B=x,y,z$ label layer-pseudospin components. The spin-valley flavor labels are not explicitly indicated.

Figure 7

RG flows of interacting susceptibilities for different ordering tendencies. (a) $N=1$, (b) $N=2$, and (c) $N=4$. We use the dimensionless quantity ${\nu }_0{\left[\ln \left(s\right)\right]}^2{\left({\chi }^{\mathrm{T}}\right)}^{-1}$ in the figures.

Figure 8

The particle-hole (PH) and particle-particle (PP) corrections to the layer-interchange interaction ${\Gamma }_{\mathrm{X}}$ at the one-loop level. Each diagram has a pair of triplet (interlayer) propagation lines. The $\uparrow$ and $\downarrow$ labels signify that the incoming (or outgoing) electrons have different spin-valley flavors. $\overline{\sigma }=T\left(B\right)$ when $\sigma =B\left(T\right)$.