Ab initio calculation of low-energy collective charge-density excitations in MgB2

We present {it ab-initio} time-dependent density-functional theory calculation results for low-energy collective electron excitations in $textrm{MgB}_2$. The existence of a long-lived collective excitation corresponding to coherent charge density fluctuations between the boron $sigma$- and $pi$- bands ($sigmapi$ mode) is demonstrated. This mode has a sine-like oscillating dispersion for energies below 0.5 eV. At even lower energy we find another collective mode ($sigmasigma$ mode). We show the strong impact of local-field effects on dielectric functions in MgB$_2$. These effects account for the long q-range behavior of the modes. We discuss the physics that these collective excitations bring to the energy region typical for lattice vibrations.