Diabolical touching point in the magnetic energy levels of topological nodal-line metals

Abstract

For three-dimensional metals, Landau levels disperse as a function of the magnetic field and the momentum wave number parallel to the field. In this two-dimensional parameter space, it is shown that two conically dispersing Landau levels can touch at a diabolical point---a Landau-Dirac point. The conditions giving rise to Landau-Dirac points are shown to be magnetic breakdown (field-driven quantum tunneling) and certain crystallographic spacetime symmetry. Both conditions are realizable in topological nodal-line metals, as we exemplify with the material candidates $\mathrm{Ca}{X}_{3}$ $(X=\mathrm{As},\mathrm{P})$. The experimental fingerprints of a Landau-Dirac point include (a) anomalous ``batman''-like peaks in the magnetoresistance, (b) circular Landau-Fermi surfaces revealed by angle-dependent ultrasonic attenuation, and (c) the tunability of the frequency onset of optical absorption to zero.