Reply to "Comment on "Electric conductivity in graphene: Kubo model versus a nonlocal quantum field theory model"" (ArXiv:2506.10792v2)

TL;DR

This paper defends the validity of previous models for graphene's electric conductivity against critiques, clarifying misinterpretations and reaffirming the correctness of established theoretical results.

Contribution

The authors clarify misconceptions about their previous work, reaffirming the validity of their conductivity model and addressing concerns about gauge invariance and physical consistency.

Findings

The conductivity model correctly predicts zero current without external field.

The electric permittivity does not have a double pole in frequency.

Inclusion of losses is standard in transport property studies.

Abstract

In the Comment by Bordag et al. [Phys. Rev. B 113, 207401 (2026) and ArXiv:2506.10792], concerns are raised regarding the validity of the results presented in [Phys. Rev. B 111, 115428 (2025)], where the theoretical descriptions of the electric conductivity of graphene obtained from the Kubo formula and from quantum field theory via the polarization tensor are compared. In this Reply, we show that these concerns arise from misinterpretations of Phys. Rev. B 111, 115428 (2025), in which the results are either inaccurately represented or applied outside the domain of validity of the model. We address the comments concerning the derivation of the Luttinger formula for the electric conductivity from the Kubo formula and clarify why the results of Phys. Rev. B 111, 115428 (2025) cannot be arbitrarily extended to make claims on the gauge invariance. We further demonstrate that our findings are fully consistent with the established and widely accepted literature cited in the Comment. We confirm that the model for electric conductivity discussed in Phys. Rev. B 111, 115428 (2025) correctly predicts a vanishing electric current in the absence of an external electric field, as physically required, and in contrast with the model advocated by the Authors of the Comment. We also show that the electric permittivity does not exhibit a double pole in $\omega$, contrary to the claim made in the Comment. Finally, we emphasize that the inclusion of losses is a standard and well-established approach in the study of transport properties of materials, including graphene, and we take the opportunity to correct a few minor typographical errors in Phys. Rev. B 111, 115428 (2025). We show and maintain that all results derived in Phys. Rev. B 111, 115428 (2025) are fully valid and correct.