Majorana zero modes in $p+ip$ superconducting rings with half quantum flux in the presence of a $d$-soliton

Half-integer flux quantization has been observed in mesoscopic rings of superconducting $\text{Sr}_2\text{RuO}_4$ \footnote{J. Jang et al. Science \textbf{331}, 186 (2011)}. This finding suggests a chiral $p+ip$ nature of the superconducting order parameter. Under the assumption that the $d$-vector (which parametrizes the triplet pairing) lies in the plane of a 2D superconductor, such rings are expected to support Majorana zero modes at their inner and outer edges. However, such modes have not been directly observed in experiments. More recently, H.-Y. Kee and M. Sigrist argued that the spin-orbit coupling in such systems can stabilize a different spin texture, also consistent with half-quantum vortices\footnote{H.-Y. Kee, M. Sigrist, arXiv:1307.5859}. That spin texture is characterized by the presence of a so-called $d$-soliton \textendash a radial domain wall between the regions where the $d$-vector is oriented in the positive and negative $z$-directions. Our theoretical investigation of superconducting rings with $d$-solitons confirms the existence of two Majorana zero modes, one at each boundary. Furthermore, the presence of a $d$-soliton enhances the hybridization between the localized Majorana modes at the inner and outer boundaries.