Disorder-tuned crossing of monopole and conventional pairing instabilities in multi-Weyl semimetals

Abstract: We study how non-magnetic impurity scattering affects the emergence and possible coexistence of pairing instabilities in a two-node multi-Weyl semimetal. Within an explicitly specified projected impurity kernel---valley diagonal and momentum independent across each Fermi pocket, the leading behaviour of non-magnetic disorder in the small-pocket window $$q^{\max}_{\rm intra}ξ<em>{\rm dis}\ll1\ll|2\mathbf Q|ξ</em>{\rm dis}$$, treated at leading order in Born and Abrikosov--Gor'kov theory---quenched disorder tunes the leading pairing instability from a topologically nontrivial monopole channel to a conventional ($s$-wave) one, extending our earlier clean-system analysis into the disordered regime. A Born self-energy calculation in the chiral band basis then gives: (i) a band-isotropic conventional channel that is Anderson protected against intra-node scalar disorder ($η<em>s=1$), introduced phenomenologically at the projected-band level; solving the competition for arbitrary $η_s$ yields the crossing criterion $(1-η_s)/(1-η_m)&lt;T</em>{c0}<sup>{(s)}/T_{c0}<sup>{(m)}$, so the mechanism tolerates substantial loss of conventional-channel protection; and (ii) a rank-one monopole sector fixing $f_m$ as the exact eigenfunction with $η<em>m(J)=1/(J+2)$, exact given that kernel. The crossing location in $Γ_N/T</em>{c0}<sup>{(m)}$ is set by $η<em>m(J)$, $η_s$, and $r=T</em>{c0}<sup>{(s)}/T_{c0}<sup>{(m)}$. From the clean projected BdG Hamiltonian the pure monopole nodes carry Berry charge $\pm J$, distinct from the gapped conventional solution; the clean nodal thermodynamics is charge-dependent, $N_{\rm SC}(E)\propto E<sup>{2/J}$ and $C\propto T<sup>{1+2/J}$, and the residual density of states shows a threshold only for $J=1$. The crossing lies in the moderately metallic regime, $μ/Γ<em>N\simeq11$--$14$ for the illustrative $T</em>{c0}<sup>{(m)}/μ=0.133$ used in the figures.