Monopole Superconductivity in Magnetically Doped Cd$_3$As$_2$

When superconducting pairing occurs between Fermi surfaces with different Chern numbers, the Cooper pairs possess nontrivial pair Berry phase, which enforces pairing gap nodes. The resulting pairing order is further distinguished from the familiar $s$-, $p$-, and $d$-wave pairing orders by a nonzero pair monopole charge and is described by monopole harmonics. To date, this exotic monopole pairing order is yet to be achieved experimentally. We therefore study the magnetically doped Dirac semimetal Cd$_3$As$_2$ as a candidate material for realizing monopole superconductivity with pair monopole charges $q_p=1$ or $2$, depending on the chemical potential. For each case of pair monopole charge, we explore representatives of uniform pairing orders in all allowed irreducible representations of the $C_{4h}$ symmetry of magnetically doped Cd$_3$As$_2$. We demonstrate the distinctions in the monopole analogs of different higher partial wave superconducting orders that result from the combination of topological Fermi surfaces of higher Chern number and crystalline symmetry. In all cases, the patterns of the superconducting phase winding around a Fermi surface are constrained by the topologically invariant total winding number, which depends only on the pair monopole charge and requires nodes in the $q_p\neq 0$ superconducting order. Our work can guide further experimental investigation of monopole orders in the topological semimetal Cd$_3$As$_2$ and can be generalized to other materials with Fermi surfaces of higher Chern number, which enforce monopole pairing order in higher partial waves, such as the monopole analog of $f$-wave pairing.