Low-energy subband wave-functions and effective $g$-factor of one-dimensional hole gas

One-dimensional (1D) hole gas confined in a cylindrical Ge nanowire has potential applications in quantum information technologies. Here, we analytically study the low-energy properties of this 1D hole gas. The subbands of the hole gas are two-fold degenerate. The low-energy subband wave-functions are obtained exactly, and the degenerate pairs are related to each other via a combination of the time-reversal and the spin-rotation transformations. In evaluating the effective $g$-factor of these low-energy subbands, the orbital effects of the magnetic field are shown to contribute as strongly as the Zeeman term. Also, near the center of the $k_{z}$ space, there is a sharp dip or a sharp peak in the effective $g$-factor. At the site $k_{z}=0$, the longitudinal $g$-factor $g_{l}$ is much less than the transverse $g$-factor $g_{t}$ for the lowest subband, while away from the site $k_{z}=0$, $g_{l}$ can be comparable to $g_{t}$.