Why Does Graphene Behave as a Weakly Interacting System?

We address the puzzling weak-coupling perturbative behavior of graphene interaction effects as manifested experimentally, in spite of the effective fine structure constant being large, by calculating the effect of Coulomb interactions on the quasiparticle properties to next-to-leading order in the random phase approximation (RPA). The focus of our work is graphene suspended in vacuum, where electron-electron interactions are strong and the system is manifestly in a nonperturbative regime. We report results for the quasiparticle residue and the Fermi velocity renormalization at low carrier density. The smallness of the next-to-leading order corrections that we obtain demonstrates that the RPA theory converges rapidly and thus, in contrast to the usual perturbative expansion in the bare coupling constant, constitutes a quantitatively predictive theory of graphene many-body physics for any coupling strength.

Figure 1

(a) Schwinger-Dyson equation for the RPA interaction. (b),(c) Diagrams that renormalize the propagator (b) and the vertex (c) to leading order in the RPA. (d),(e) The logarithmic divergences in perturbation theory are subtracted by propagator and vertex counterterms, respectively, as defined in Eq. (4). (f) Diagrams that renormalize the wave function and the Fermi velocity to next-to-leading order in the RPA.

Figure 2

Anomalous dimension of the Fermi velocity as a function of the fine structure constant $\alpha$. The dashed blue line is the leading-order RPA result from Ref. [20]. The solid blue line is the result including the next order in the RPA as obtained in this Letter. The correction is small and changes the quantitative features of the RG flow. In contrast, perturbation theory at second order in $\alpha$ (red dashed dotted) gives a strikingly different result from the first-order (red dashed) flow, and predicts an unphysical fixed point at ${\alpha }^{*}=0.78$ [16].

Figure 3

Renormalization of the Fermi velocity in suspended graphene. The Fermi velocity is obtained by integrating the flow equations (8) starting at a density $n_0=100\times 10^{10}{\mathrm{cm}}^{-2}$ with $v_F^0=1.24\times 10^8\mathrm{cm}/\mathrm{s}$. While first-order perturbation theory (red dashed) is qualitatively consistent, the second-order result (red dashed dotted) is in striking disagreement with the experimental data. In contrast, the LO (blue dashed) and NLO (blue solid) RPA results are both in close agreement. The experimental data points are taken from Ref. [8].

Figure 4

Running of the quasiparticle residue for suspended graphene as the initial scale ${\mu }_0\sim \sqrt{n_0}$ is reduced to $\mu (s)=e^{-s}{\mu }_0$. The residue is not renormalized to zero but approaches a constant value in the low-density limit. The initial conditions for the flows are the same as in Fig. 3.