  Hybrid semiconductor-superconductor quantum dot devices are tunable physical
realizations of quantum impurity models for a magnetic impurity in a
superconducting host. The binding energy of the localized sub-gap Shiba states
is set by the gate voltages and external magnetic field. In this work we
discuss the effects of the Zeeman spin splitting which is generically present
both in the quantum dot and in the (thin-film) superconductor. The unequal
$g$-factors in semiconductor and superconductor materials result in respective
Zeeman splittings of different magnitude. We consider both classical and
quantum impurities. In the first case we analytically study the spectral
function and the sub-gap states. The energy of bound states depends on the
spin-splitting of the Bogoliubov quasiparticle bands as a simple rigid shift.
For the case of collinear magnetization of impurity and host, the Shiba
resonance of a given spin polarization remains unperturbed when it overlaps
with the branch of the quasiparticle excitations of the opposite spin
polarization. In the quantum case, we employ numerical renormalization group
calculations to study the effect of the Zeeman field for different values of
the $g$-factors of the impurity and of the superconductor. We find that in
general the critical magnetic field for the singlet-doublet transition changes
non-monotonically as a function of the superconducting gap, demonstrating the
existence of two different transition mechanisms: Zeeman splitting of Shiba
states or gap closure due to Zeeman splitting of Bogoliubov states. We also
study how in the presence of spin-orbit coupling, modeled as an additional
non-collinear component of the magnetic field at the impurity site, the Shiba
resonance overlapping with the quasiparticle continuum of the opposite spin
gradually broadens and then merges with the continuum.
