First-Principles Computation of G-Factors in Bi and Bi$_2$Se$_3$

In this paper, we propose a first principles computation method for g-factor tensor, which not only gives comparable results with experiments but also establishes a clear physical picture of Zeeman effect in materials. In our method, the Hilbert space of the electronic states is treated as a direct product of "inner" space and "orbital" space, which are spanned by Bloch wave functions and envelope functions respectively. Correspondingly, vector-potential is divided into a periodic part acting only in inner space and a non-periodic part acting only in orbital space. With the above flamework we define the g-factors as coupling coefficients between inner space and magnetic field. By the method we developed, we have further computed the g-factors of bismuth and Bi$_2$Se$_3$ and got satisfactory results, which are in good agreement with the experimental data.