Majorana Zero Modes in Superconducting Rings and Arrays

Majorana zero modes are zero energy excitations with unusual (non-Abelian) statistics that offer a promising platform for quantum computation. These modes are theoretically expected to exist in a variety of physical systems, most prominently in chiral $p$-wave superconductors. In the case of a spinless (or spin-polarized) superconductor, a conventional vortex -- a topological defect that carrying one quantum of superconducting magnetic flux -- would host such a zero mode. In the presence of spin degrees of freedom, however, such a mode would require a half quantum vortex, i.e.\ a vortex characterized by a half-integer number of the superconducting flux quanta. To guarantee the single-valued nature of the order parameter, a non-trivial spin texture is required, which would allow the order parameter to pick up an additional factor of $-1$ around the vortex.Half-integer flux quantization has been observed in mesoscopic rings of superconducting $\text{Sr}_2\text{RuO}_4$. 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. 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 alter the hybridization between the localized Majorana modes at the inner and outer boundaries.In addition to chiral p-wave superconductors, some index theorem-like arguments can be used to predict Majorana zero modes in an array of vortices in chiral d-wave superconductors, our numerical studies did not produce any evidence of Majorana zero modes in such systems