Duality between dielectricity and diamagnetism in Dirac materials in analogy with quantum electrodynamics*

We theoretically investigate the electrodynamics of narrow-gap electron systems in the presence of strong spin-orbit interaction treated as three-dimensional Dirac materials. In parallel to quantum electrodynamics (QED), we derive analytic expressions of the wave-number- and frequency-dependent electric and magnetic susceptibilities of the narrow-gap systems at zero temperature. When the chemical potential is inside the band gap, the ratio of the electric to magnetic susceptibilities is shown to be a negative constant for arbitrary wave number and frequency in analogy with vacuum polarization in QED [H. Maebashi, M. Ogata, and H. Fukuyama, J. Phys. Soc. Jpn. 86, 083702 (2017)]. On the other hand, when the chemical potential lies in the conduction or valence band, we obtain the plasmon dispersion where the background permittivity is microscopically determined, which is significantly enhanced along with large diamagnetism known as a result of interband effect of magnetic field. Such a duality between dielectricity and diamagnetism can be traced to effective Lorentz covariance which Dirac materials have.