The exploration of spin symmetry (SS) in nuclear physics has been instrumental in identifying atomic nucleus structures. In this study, we solve the Dirac equation from the relativistic mean field (RMF) in complex momentum representation. We investigated SS and its breaking in single-particle resonant states within deformed nuclei, with a focus on the illustrative nucleus \(^{168}\) Er. This was the initial discovery of a resonant spin doublet in a deformed nucleus, with the expectation of the SS approaching the continuum threshold. With increasing single-particle energy, the splitting of the resonant spin doublets widened significantly. This escalating splitting implies diminishing adherence to the SS, indicating a departure from the expected behavior as the energy levels increase. We also analyzed the width of the resonant states, showing that lower orbital angular momentum resonances possess shorter decay times and that SS is preserved within broad resonant doublets, as opposed to narrow resonant doublets. Comparing the radial density of the upper components for the bound-state and resonant-state doublets, it becomes evident that while SS is well-preserved in the bound states, it deteriorates in the resonant states. The impact of nuclear deformation ( \(\beta _2\) ) on SS was examined, demonstrating that an increase in \(\beta _2\) resulted in higher energy and width splitting in the resonant spin doublets, which is attributed to increased component mixing. Furthermore, the sensitivity of spin doublets to various potential parameters such as surface diffuseness (a), radius (R), and depth ( \(\Sigma _0\) ) is discussed, emphasizing the role of these parameters in SS. This study provides valuable insights into the behavior of spin doublets in deformed nuclei and their interplay with the nuclear structure, thereby advancing our understanding of SS in the resonance state.
