Single magnetic white dwarfs with Balmer emission lines: a small class with consistent physical characteristics as possible signposts for close-in planetary companions

We report the identification of SDSS J121929.45+471522.8 as the third apparently isolated magnetic (B similar or equal to 18.5 +/- 1.0MG) white dwarf exhibiting Zeeman-split Balmer emission lines. The star shows coherent variability at optical wavelengths with an amplitude of similar or equal to 0.03 mag and a period of 15.26 h, which we interpret as the spin period of the white dwarf. Modelling the spectral energy distribution and Gaia parallax, we derive a white dwarf temperature of 7500 +/- 148 K, a mass of 0.649 +/- 0.022 M-circle dot, and a cooling age of 1.5 +/- 0.1Gyr, as well as an upper limit on the temperature of a sub-stellar or giant planet companion of similar or equal to 250 K. The physical properties of this white dwarf match very closely those of the other two magnetic white dwarfs showing Balmer emission lines: GD356 and SDSS J125230.93-023417.7. We argue that, considering the growing evidence for planets and planetesimals on close orbits around white dwarfs, the unipolar inductor model provides a plausible scenario to explain the characteristics of this small class of stars. The tight clustering of the three stars in cooling age suggests a common mechanism switching the unipolar inductor on and off. Whereas Lorentz drift naturally limits the lifetime of the inductor phase, the relatively late onset of the line emission along the white dwarf cooling sequence remains unexplained.