  Collective nuclear excitations, like giant resonances, are sensitive to
nuclear deformation, as evidenced by alterations in their excitation energies
and transition strength distributions. A common theoretical framework to study
these collective modes, the random-phase approximation (RPA), has to deal with
large dimensions spanned by all possible particle-hole configurations
satisfying certain symmetries. This work aims to establish a new theoretical
framework to study the impact of deformation on spin-isospin excitations, that
can provide fast and reliable solutions of the RPA equations. The nuclear
ground state is determined with the axially-deformed relativistic
Hartree-Bogoliubov (RHB) model based on relativistic point-coupling energy
density functionals (EDFs). To study the excitations in the charge-exchange
channel, an axially-deformed proton-neutron relativistic quasiparticle RPA
(pnRQRPA) is developed in the linear response approach. After benchmarking the
axially-deformed pnRQRPA in the spherical limit, a study of spin-isospin
excitations including Fermi, Gamow-Teller (GT), and Spin-Dipole (SD) is
performed for selected $pf$-shell nuclei. For GT transitions, it is
demonstrated that deformation leads to considerable fragmentation of the
strength function. A mechanism inducing the fragmentation is studied by
decomposing the total strength to different projections of total angular
momentum $K$ and constraining the nuclear shape to either spherical, prolate or
oblate. A similar fragmentation is also observed for SD transitions, although
somewhat moderated by the complex structure of these transitions, while the
Fermi strength is almost shape-independent. The axially-deformed pnRQRPA
introduced in this work opens perspectives for future studies of deformation
effects on astrophysically relevant weak interaction processes, in particular
beta decay and electron capture.
