Spontaneous inversion symmetry breaking in graphene bilayers

In a mean-field-theory treatment the ground state of a graphene bilayer spontaneously breaks inversion symmetry for arbitrarily weak electron-electron interactions when trigonal-warping terms in the band structure are ignored. We report on a perturbative renormalization-group calculation, which assesses the robustness of this instability, comparing with the closely related case of the charge-density-wave instability incorrectly predicted by mean-field theory in a one-dimensional electron gas. We conclude that spontaneous inversion symmetry breaking in graphene is not suppressed by quantum fluctuations but that, because of trigonal warping, it may occur only in high quality suspended bilayers.

Figure 1

(Color online) (a) The massive-chiral fermion model describes the low-energy sites in a $AB$-stacked graphene bilayer, those atom sites (top layer $B$ sites and bottom layer $A^{\prime }$ sites) which do not have a neighbor in the opposite layer. (b) The conduction and valence bands of a graphene bilayer touch at the Brillouin-zone corner wave vectors, taken as zero momentum in continuum model theories, and separate quadratically with increasing wave vector. (c) In a 1DES left and right-going electrons cross the Fermi energy at a single point; The momentum of right-going (left-going) electrons is plotted relative to $k_F$ $\left(-k_F\right)$ where $k_F$ is the Fermi wave vector.

Figure 2

(Color online) (a) The renormalized interaction ${\Gamma }_D$. (b) and (c) The direct and exchange bare interactions. (d)–(f) They are the one-loop diagrams labeled ZS, ${\text{ZS}}^{\prime }$, and BCS, respectively. The external and internal Green’s function labels refer to layer in the case of graphene and to chirality in 1DES’ case.

Figure 3

(Color online) This illustration plots the inverse interaction strength ${\left({\nu }_0\Gamma {\epsilon }_{\text{eff}}\right)}^{-1}$ versus the scaling parameter $\text{ln}\left(s\right)/{\epsilon }_{\text{eff}}$. ${\epsilon }_{\text{eff}}$ is the effective dielectric constant of the graphene bilayer and $\Gamma ={\Gamma }_{\text{vacuum}}/{\epsilon }_{\text{eff}}$. Interlayer interaction parameters ${\Gamma }_{DSS}$ (green) and ${\Gamma }_{XSD},{\Gamma }_{XDS},{\Gamma }_{XDD}$ (cyan) flow to large values most quickly. According to this estimate the normal state becomes unstable for $V_D\ge 0.6/\text{ln}\left(s\right)$.