Pair separation in parallel electric field in magnetar magnetosphere and narrow spectra of fast radio bursts

When the magnetosphere of a magnetar is perturbed by crustal deformation, an electric field $E_{\parallel }$ parallel to the magnetic field line would appear via Alvf'en waves in the charge starvation region. The electron-positron pair bunches will be generated via two-stream instability in the magnetosphere, and these pairs will undergo charge separation in the $E_{\parallel }$ and in the meantime emit coherent curvature radiation. Following the approach of Yang & Zhang (2018), we find that the superposed curvature radiation becomes narrower due to charge separation, with the width of spectrum depending on the separation between the electron and positron clumps. This mechanism can interpret the narrow spectra of FRBs, in particular, the spectrum of Galactic FRB 200428 recently detected in association with a hard X-ray burst from the Galactic magnetar SGR J1935+2154.