RKKY interaction in gapped or doped graphene

TL;DR

This paper simplifies the calculation of RKKY interactions in gapped or doped graphene using Green's functions and Feynman's trick, providing more efficient methods for theoretical analysis.

Contribution

It introduces a simplified Green's function calculation method for RKKY interactions in gapped or doped graphene, improving computational efficiency.

Findings

Simplified Green's function calculations using Feynman's trick.

Comparison of coordinate-imaginary time and frequency representations.

Advantages of Euclidean space calculations over Minkowski space.

Abstract

In our previous work (E. Kogan, Phys. Rev. B {\bf 84}, 115119 (2011)) we calculated RKKY interaction between two magnetic impurities in pristine graphene using the Green's functions (GF) in the coordinate -- imaginary time representation. Now we show that the calculations of the GF in this representation can be simplified by using the Feynman's trick, which allows to easily calculate RKKY interaction in gapped graphene. We also present calculations of the RKKY interaction in gapped or doped graphene using the coordinate -- imaginary frequency representation. Both representations, corresponding to calculation of the bubble diagram in Euclidean space, have an important advantage over those corresponding to calculation in Minkowskii space, which are very briefly reviewed in the Appendix to the present work. The former, in distinction to the latter, operate only with the convergent integrals from the start to the end of the calculation.