Spin and Pseudo-Spin Collective Modes in Doped Graphene

Collective modes in graphene has been an active area of research during the last decade. As any two-dimensional electron system, graphene harbors a gapless plasmon mode which disperses as q^1/2. Gapped collective modes in graphene are, however, less explored. The goal of this work is two-fold: a comprehensive study of collective modes in graphene (a) without spin-orbit coupling (SOC) and (b) in the presence of Rashba and valley-Zeeman SOC. For a short-range electron-electron interaction, we find that in case (a) there exists a collective mode below the continuum, corresponding to oscillations of pseudospin. In case (b), we show that, in graphene with Rashba SOC, there exists a collective spin mode consisting of three branches: a double-degenerate branch, corresponding to oscillations of in-plane magnetization, and a non-degenerate branch corresponding to oscillations of out of plane magnetization. For graphene with valley-Zeeman SOC, there exists only a single branch of the collective mode corresponding to out of plane oscillation of the magnetization. We also study the case of an unscreened Coulomb interaction, solved by reducing the sum of ladder diagrams for a vertex function to a Dirac equation with non-local interaction.