Superconductivity in multi-Weyl semimetals: Conditions for the coexistence of topological and conventional phases

In this work, we explore the possible emergence of superconducting phases in a multi-Weyl semimetal. In particular, we show that the presence of a pair of Weyl nodes with chirality $|ν| \ge 1$ leads to an effective description of the intra-nodal pairings in terms of monopole harmonics, in contrast to inter-nodal pairings that preserve the angular dependence of conventional spherical harmonics. Therefore, we explore the conditions for the competition and/or coexistence between both types of superconducting phases, and we identify the presence of the so-called "topological repulsion" mechanism, which was previously reported in the context of simple Weyl semimetals. We identified the critical temperatures corresponding to the monopole and conventional superconducting phases, and calculated the specific heat as a function of temperature, thus showing that this thermodynamical parameter may provide an experimental probe to determine the chirality index $ν$ in the material.