Minimal conductivity in graphene: interaction corrections and ul- traviolet anomaly

Conductivity of a disorder-free intrinsic graphene is studied to the first order in the long-range Coulomb interaction and is found to be σ = σ0(1+0.01g), where g is the dimensionless (“fine structure”) coupling constant. The calculations are performed using three different methods: i) electron polarization function, ii) Kubo formula for the conductivity, iii) quantum transport equation. Surprisingly, these methods yield different results unless a proper ultraviolet cut-off procedure is implemented, which requires that the interaction potential in the effective Dirac Hamiltonian is cut-off at small distances (large momenta). Introduction. – Low-frequency optical conductivity of undoped (intrinsic) graphene free of disorder is known to have a universal value of σ0 = e /4h̄ [1–13]. Experimental measurements [14,15], which yielded a value somewhat bigger than the theoretical predictions, motivated the studies of the possible role played by electron-electron interactions. The findings of Ref. [16] that the combined effect of self energy (velocity renormalization) and vertex corrections leads to a suppression of the optical conductivity at low frequencies have been questioned in Refs. [17,18] on the basis of scaling arguments. The latter indicate that the large logarithmic (momentum cut-off dependent) terms in the self-energy and vertex corrections cancel each other. We note that Ref. [16] and Refs. [17, 18] agree on this cancellation in the lowest order in electron-electron interaction but differ on whether the higher order terms feature similar cancellation. It appears that the analysis of Ref. [16], though valid in the first order, fails for higher orders, and that the conclusion of the suppression of the conductivity at low frequencies is not valid. The theory presented in Refs. [17, 18] implies that the low-frequency dependence is properly described by the lowest order correction. Indeed, to the first order in interaction the conductivity is expected to yield, σ/σ0 = 1+Cg, where C is some constant, g = e/κv is the interaction strength; κ is the dielectric constant of a substrate and v is the electron velocity in graphene. Renormalization group approach for 2D Dirac fermions predicts that the interaction strength g is a running coupling constant that depends on frequency g → g̃(ω) [20, 21]. At low frequencies g̃(ω) flows to zero, so that higher order corrections to the electron velocity become progressively negligible and it is sufficient to consider only the first order renormalization of velocity (electric charge is not renormalized): g̃(ω) = g/[1 + g 4 ln (Kv/ω)], where K is the momentum cut-off. Combining these expressions gives, σ/σ0 = 1 + Cg 1 + g 4 ln (Kv/ω) , (1) with the low-frequency behavior of the conductivity being determined by the constant C alone. Calculation of this constant, therefore, becomes an important task. While Ref. [17] did not calculate C, Ref. [18] provided the following value C = 25− 6π 12 ≈ 0.51 . (2) This result predicts quite a considerable variation of σ with the frequency for typical values of the bare graphene interaction constant g (which can exceed 1). In the present Letter we test the above prediction (2) by performing a perturbative calculation of the minimal conductivity to the first order in electron-electron interaction using three different methods, based on, a) electron polarization operator, b) Kubo formula for the conductivity, c) kinetic equation. We point out that crucial anomaly, which does not appear in a non-interacting case, occurs for the interaction correction. Three above mentioned methods would give essentially different values for the constant