Keldysh functional renormalization group for electronic properties of graphene

We construct a nonperturbative nonequilibrium theory for graphene electrons interacting via the instantaneous Coulomb interaction by combining the functional renormalization group method with the nonequilibrium Keldysh formalism. The Coulomb interaction is partially bosonized in the forward scattering channel resulting in a coupled Fermi-Bose theory. Quantum kinetic equations for the Dirac fermions and the Hubbard-Stratonovich boson are derived in Keldysh basis, together with the exact flow equation for the effective action and the hierarchy of one-particle irreducible vertex functions, taking into account a possible nonzero expectation value of the bosonic field. Eventually, the system of equations is solved approximately under thermal equilibrium conditions at finite temperature, providing results for the renormalized Fermi velocity and the static dielectric function, which extends the zero-temperature results of Bauer et al., Phys. Rev. B 92, 121409 (2015).

Figure 1

Schwinger-Keldysh time contour in the complex time plane with reference time $t_0$ as starting and end point. ${\mathcal{C}}_{+}$ and ${\mathcal{C}}_{-}$ are the forward and backward branches, respectively.

Figure 2

Cutoff dependent Fermi velocity $v_{\mathrm{\Lambda }}\left(k\right)$ at temperature $T/v_F{\mathrm{\Lambda }}_0={10}^{-3}$. The physical limit corresponds to $\mathrm{\Lambda }=0$. Note that the renormalized Fermi velocity is finite at $\mathrm{\Lambda }=k=0$. Further observe that the figure is almost symmetric around $k=\mathrm{\Lambda }$, suggesting that the momentum $k$ acts as an infrared cutoff for the Fermi velocity viewed as a function of $\mathrm{\Lambda }$ in the same way as $\mathrm{\Lambda }$ acts as a cutoff for the Fermi velocity as a function of $k$.

Figure 3

Fermi velocity versus momentum $k$, for temperatures $T/v_F{\mathrm{\Lambda }}_0=0, 5.0\times {10}^{-4}, 7.5\times {10}^{-4}, 1.0\times {10}^{-3}, 2.5\times {10}^{-3}, 5.0\times {10}^{-3}$, and $7.5\times {10}^{-3}$ (top to bottom data sets). The inset shows the logarithmic temperature dependence of the Fermi velocity at $k=0$. The single data point at $v\left({10}^{-5}\right)/v_F=6$ shows a nonphysical deviation from the logarithmic divergence at zero temperature, indicating that our numerical algorithm breaks down there. This behavior could be expected, since the grids have only been built down to $k/{\mathrm{\Lambda }}_0={10}^{-5}$. At finite temperatures, similar convergence problems occur upon approaching the lower grid cutoff $T/v_F\mathrm{\Lambda }\approx {10}^{-5}$.

Figure 4

Cutoff dependent dielectric function ${\epsilon }_{\mathrm{\Lambda }}\left(q\right)$ at temperature $T/v_{\mathrm{F}}{\mathrm{\Lambda }}_0={10}^{-3}$. Note the sharp feature at $\mathrm{\Lambda }=0$ for momenta $q\lesssim T/v_{\mathrm{F}}$.

Figure 5

Dielectric function as a function of momentum $q$ for temperatures $T/v_{\mathrm{F}}{\mathrm{\Lambda }}_0=0, 5.0\times {10}^{-4}, 7.5\times {10}^{-4}, 1.0\times {10}^{-3}, 2.5\times {10}^{-3}, 5.0\times {10}^{-3}$, and $7.5\times {10}^{-3}$ (bottom to top data sets). For momenta $q/{\mathrm{\Lambda }}_0$ below the reduced temperature $\tilde{T}=T/v_{\mathrm{F}}{\mathrm{\Lambda }}_0$ our data are consistent with the $1/q$ dependence predicted by perturbation theory. The temperature dependence of the prefactor could be fitted by $\epsilon \left(q\right)=1+a\left(\tilde{T}\right){\mathrm{\Lambda }}_0/q$, indicated by dashed lines, where $a\left(\tilde{T}\right)$ is a linear function as shown in the inset.