Chiral Anomaly and Strength of the Electron-Electron Interaction in Graphene

The long-standing controversy concerning the effect of electron-electron interaction on the electrical conductivity of an ideal graphene sheet is settled. Performing the calculation directly in the tight binding approach without the usual prior reduction to the massless Dirac (Weyl) theory, it is found that, to leading order in the interaction strength $\alpha =e^2/\hslash v_0$, the dc conductivity $\sigma /{\sigma }_0=1+C\alpha +O({\alpha }^2)$ is significantly enhanced with respect to the independent-electron result ${\sigma }_0$, i.e., with the value $C=0.26$. The ambiguity characterizing the various existing approaches is nontrivial and related to the chiral anomaly in the system. In order to separate the energy scales in a model with massless fermions, contributions from regions of the Brillouin zone away from the Dirac points have to be accounted for. Experimental consequences of the relatively strong interaction effect are briefly discussed.

Figure 1

Honeycomb lattice for graphene. The sublattice $A$ (red points) is spanned by the lattice vectors ${\mathbf{a}}_{1,2}=\frac{a}{2}(\pm 1,\sqrt{3})$ where $a\simeq 3\AA$. The three nearest neighbors on sublattice $B$ (blue points) are displaced by ${\mathbf{d}}_1=\frac{1}{3}({\mathbf{a}}_1-{\mathbf{a}}_2)$, ${\mathbf{d}}_2=\frac{1}{3}({\mathbf{a}}_1+2{\mathbf{a}}_2)$, and ${\mathbf{d}}_3=-\frac{1}{3}(2{\mathbf{a}}_1+{\mathbf{a}}_2)$. Wilson links $W$ describing the minimal coupling to the vector potential $\mathbf{A}(\mathbf{r},t)$ are defined in Eq. (5).

Figure 2

Contours of the tight binding energy ${\epsilon }_{\mathbf{k}}$ in the entire Brillouin zone of the honeycomb lattice. The circles of radius $\mathcal{K}$ around the two Dirac points are the parts described by the effctive low energy Weyl model.