Keldysh functional renormalization group for electronic properties of graphene

Abstract

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 non-zero 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).

Chemical-potential flow equations for graphene with Coulomb interactions

“…As explained in Ref. [19], this divergence could be attributed to thermally induced charge carriers. In the presence of a finite chemical potential, that is excess charge carriers, the large momentum components of the dielectric function remain unaffected, whereas the initial 1/q divergence found in the low momentum regime becomes strongly enhanced, leading to an increasingly short ranged renormalized Coulomb interaction.…”

“…A precise estimation of the slope would require a recalculation of the temperature dependence of the renormalized Fermi velocity and dielectric function at the charge neutrality point with a better resolution and accuracy than what was achieved previously in Ref. [19].…”

“…The bosonic field was introduced by a Hubbard-Stratonovich transformation of the Coulomb interaction in the density-density channel. 9,13,[16][17][18][19] In contrast to the standard fRG the main issue of concern in the chemical-potential flow theory is the initial condition of the flow. Since the chemical potential is dif-ferent from an infrared regulator by its analytical structures, the effective action at some -arbitrarily choseninitial chemical potential µ 0 is nontrivial and, in particular, does not coincide with the bare action.…”

“…The truncation scheme in Ref. [19] neglects any dynamical effects, such as plasmons and the quasiparticle wavefunction renormalization, the three-vertex renormalization and higher-order vertices entirely. We use these results as the starting point of the chemical-potential flow.…”

“…which we have obtained by an fRG calculation. 19 The numerical results for the dielectric function are shown in Fig. 2 for the reduced temperature T /v F Λ 0 = 2.5 × 10 −3 , where Λ 0 is the upper band cutoff of the low-energy Hamiltonian (1).…”