Monopole versus spherical harmonic superconductors: Topological repulsion, coexistence and stability

TL;DR

This paper explores the coexistence and competition between monopole harmonic superconductors and conventional spherical harmonic pairings, revealing a topological repulsion effect rooted in their topological properties.

Contribution

It demonstrates, using weak coupling BCS theory, that monopole and spherical harmonic superconductors can coexist or repel based on their topological charges, introducing the concept of topological repulsion.

Findings

Monopole and spherical harmonic SCs can coexist in certain parameter regions.

Topological repulsion occurs when monopole charge matches spherical harmonic angular momentum.

Majorana surface modes may serve as experimental signatures.

Abstract

The monopole harmonic superconductor (SC), proposed in doped Weyl semimetals as a pairing between the Fermi surfaces enclosing the Weyl points, is rather unusual, as it features the monopole charge inherited from the parent metallic phase. However, this state can compete with more conventional spherical harmonic pairings, such as an $s$-wave. We here demonstrate, within the framework of the weak coupling mean-field BCS theory, that the monopole and a conventional spherical harmonic SC quite generically coexist, while the repulsion can take place when the absolute value of the monopole charge matches the angular momentum quantum number of the spherical harmonic. As we show, this feature is a direct consequence of the topological nature of the monopole SC, and we dub it \emph{topological repulsion}. We illustrate the above principle with the example of the conventional $s-$ and $(p_x\pm ip_y)-$wave pairings competing with the monopole SC $Y_{-1,1,0}(\theta,\phi)$, which coexist in a finite region of the parameter space, and repel, respectively. Furthermore, the s-wave pairing is more stable both when the chemical potentials at the nodes are unequal, and in the presence of point-like charged impurities. Since the phase transition is discontinuous, close to the phase boundary, we predict that the Majorana surface modes at the interfaces between domains featuring the monopole and the trivial phases, such as an $s-$wave, will be the experimental signature of the monopole SC.