Origin of the large anisotropic g-factor of holes in bismuth

The ratio of the Zeeman splitting to the cyclotron energy ($M=\Delta E_{\rm Z}/\hbar \omega_{\rm c}$) for hole-like carriers in bismuth has been quantified with a great precision by many experiments performed during the past five decades. It exceeds 2 when the magnetic field is along the trigonal axis and vanishes in the perpendicular configuration. Theoretically, however, $M$ is expected to be isotropic and equal to unity in a two-band Dirac model. We argue that a solution to this half-a-century-old puzzle can be found by extending the $k\cdot p$ theory to multiple bands. Our model not only gives a quantitative account of magnitude and anisotropy of $M$ for hole-like carriers in bismuth, but also explains its contrasting evolution with antimony doping pressure, both probed by new experiments reported here. The present results have important implications for the magnitude and anisotropy of $M$ in other systems with strong spin-orbit coupling.