Abstract. The ratio of the Zeeman splitting to the cyclotron energy (M = ∆E Z / ω c ), which characterizes the relative strength of the spin-orbit interaction in crystals, is examined for the narrow gap IV-VI semiconductors PbTe, SnTe, and their alloy Pb 1−x Sn x Te on the basis of the multiband k · p theory. The inverse mass α, the g-factor g, and M are calculated numerically by employing the relativistic empirical tight-binding band calculation. On the other hand, a simple but exact formula of M is obtained for the six-band model based on the group theoretical analysis. It is shown that M < 1 for PbTe and M > 1 for SnTe, which are interpreted in terms of the relevance of the interband couplings due to the crystalline spin-orbit interaction. It is clarified both analytically and numerically that M = 1 just at the band inversion point, where the transition from trivial to nontrivial topological crystalline insulator occurs. By using this property, one can detect the transition point only with the bulk measurements. It is also proposed that M is useful to evaluate quantitatively a degree of the Dirac electrons in solids.The spin-orbit interaction (SOI) affects the eigenstate of electrons in solids in a variety of ways. It strongly depends on the crystal structure and the momentum of the carrier. One of the most fundamental such effects is the modification of the band structure. For example, in semiconductors of the diamond and zincblende structures, the band modification can be characterized by the spin-orbit splitting energy. But this is not the whole information of the crystalline SOI. Another important information can be obtained under a magnetic field, where we cannot attain to only with the band calculations. The one-body Hamiltonian under the magnetic field can be separated into two part in general: the symmetric and the antisymmetric part with respect to the commutation of the kinematical momentum operator [1,2,3,4,5]. The eigenenergy of the symmetric part is given in terms of the cyclotron energy as ω c (n + 1/2) with an anisotropic cyclotron mass. The eigenenergy of the antisymmetric part is given by the Zeeman energy with an anisotropic g-factor. This Zeeman energy does not originate from the bare electron spins, but originates from the orbital motion of electrons. The antisymmetric part is relevant only in the case with the sizable crystalline SOI. Therefore, the effect of the crystalline SOI is clearly reflected by the antisymmetric part, whose relative strength is characterized by the ratio of the Zeeman splitting to

The quantum correction to electrical conductivity is studied on the basis of two-dimensional Wolff Hamiltonian, which is an effective model for a spin-orbit coupled (SOC) lattice system. It is shown that weak anti-localization (WAL) arises in SOC lattices, although its mechanism and properties are different from the conventional WAL in normal metals with SOC impurities. The interband SOC effect induces the contribution from the interband singlet Cooperon, which plays a crucial role for WAL in the SOC lattice. It is also shown that there is a crossover from WAL to weak localization in SOC lattices when the Fermi energy or band gap changes. The implications of the present results to Bi-Sb alloys and PbTe under pressure are discussed.

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