Chemical-potential flow equations for graphene with Coulomb interactions

We calculate the chemical potential dependence of the renormalized Fermi velocity and static dielectric function for Dirac quasiparticles in graphene nonperturbatively at finite temperature. By reinterpreting the chemical potential as a flow parameter in the spirit of the functional renormalization group (fRG) we obtain a set of flow equations, which describe the change of these functions upon varying the chemical potential. In contrast to the fRG the initial condition of the flow is nontrivial and has to be calculated separately. Our results are consistent with a charge carrier-independent Fermi velocity $v\left(k\right)$ for small densities $n\lesssim k^2/\pi$, supporting the comparison of the zero-density fRG calculation of Bauer et al. [Phys. Rev. B 92, 121409 (2015)], with the experiment of Elias et al. [Nat. Phys. 7, 701 (2011)].

Figure 1

Graphical representation of the chemical potential flow equations for the self-energy ${\widehat{\mathbf{\Sigma }}}_{\mu }$ and the polarization function ${\mathbf{\Pi }}_{\mu }$. Contributions from higher order vertices ${\mathbf{\Gamma }}_{\mu }^{(m,n)}$, with $m>2,n>1$ are already neglected. Straight and wiggly lines represent the flowing fermionic and bosonic propagators, respectively, and the shaded triangle represents the Fermi-Bose three-vertex ${\mathbf{\Gamma }}_{\mu }^{(2,1)}$. The derivative ${\partial /}_{\mu }$ on the right hand side only acts on the flowing fermionic propagators, substituting the latter by the so-called single scale propagator [16]. An analytical representation of these flow equations is given in Appendix pp2.

Figure 2

Dielectric function ${\epsilon }_{\mu }\left(q\right)$ as a function of momentum and chemical potential at the reduced temperature $T/v_F{\mathrm{\Lambda }}_0=2.5\times {10}^{-3}$. The colors blue to green and red to orange separate the two regimes $\mu \le T$ and $\mu >T$, respectively. At the charge neutrality point the long range tail of the bare Coulomb interaction is cut off, due to thermal screening.

Figure 3

Quantitative comparison between the (a) fully self-consistent solution and (b) one loop approximation of the coefficient function $a(\mu ,T)$ for the reduced temperatures $T/v_F{\mathrm{\Lambda }}_0=5\times {10}^{-4},2.5\times {10}^{-3},5\times {10}^{-3}$ (bottom to top data sets). In the small momentum regime, $q\ll T/v_F,\mu /v_F$, the static dielectric function shows a $1/q$ divergence according to ${\epsilon }_{\mu }(q\ll T/v_F)=1+a(\mu ,T){\mathrm{\Lambda }}_0/q$. Observe that the self-consistent solution is about an order of magnitude smaller than the one-loop prediction.

Figure 4

Renormalized Fermi velocity $v_{\mu }\left(k\right)$ as a function of momentum and chemical potential at temperature $T/v_F{\mathrm{\Lambda }}_0=2.5\times {10}^{-3}$ in the fully self-consistent calculation. The renormalized Fermi velocity is finite at $k=0$ at the initial chemical potential ${\mu }_0=0$ due to temperature induced screening of the renormalized Coulomb interaction. Increasing the chemical potential away from the charge neutrality point shows a weak suppression of the Fermi velocity for small momenta. The inset shows a comparison between the $k\to 0$ limits of the Fermi velocity in the self-consistent treatment and in Hartree-Fock approximation as functions of the chemical potential. The dashed vertical line marks $\mu =2T$. For $\mu >2T$ both calculations show a logarithmic suppression of $v_{\mu }(k\to 0)$. This suppression is significantly weakened in the full computation when compared to the result of the Hartree-Fock approximation.