Two-band description of the strong `spin'-orbit coupled one-dimensional hole gas in a cylindrical Ge nanowire

The low-energy effective Hamiltonian of the strong spin'-orbit coupled one-dimensional hole gas in a cylindrical Ge nanowire in the presence of a strong magnetic field is studied both numerically and analytically. Basing on the Luttinger-Kohn Hamiltonian in the spherical approximation, we show this strong spin’-orbit coupled one-dimensional hole gas can be accurately described by an effective two-band Hamiltonian $H^{\mathrm{e}\mathrm{f}}={\mathrm{\hslash }}^2{k}_z^2\mathrm{/}(2m_h^{\ast })+\alpha {\sigma }^x{k}_z+g_h^{\ast }{\mu }_{B}B{\sigma }^z\mathrm{/}2$, as long as the magnetic field is purely longitudinal or purely transverse. The explicit magnetic field dependent expressions of the `spin’-orbit coupling $\alpha \equiv \alpha (B)$ and the effective $g$-factor $g_h^{\ast }\equiv g_h^{\ast }(B)$ are given. When the magnetic field is applied in an arbitrary direction, the two-band Hamiltonian description is still a good approximation.