Topological multiband $s$-wave superconductivity in coupled multifold fermions

We study three-dimensional time-reversal-invariant topological superconductivity in noncentrosymmetric materials such as RhSi, CoSi, and AlPt which host coupled multifold nodes energetically split by the spin-orbit coupling at the same time-reversal-invariant momentum (TRIM). The topological superconductivity arises from the $s_{+}\oplus s_{-}$ gap function, which is $\mathit{k}$ independent, but with opposite signs for the two nodes split at the same TRIM. We consider various electron-electron interactions in the tight-binding model for RhSi and find that the topological superconducting phase supporting a surface Majorana cone and topological nodal rings is favored in a wide range of interaction parameters.

Figure 1

(a) Band structure of RhSi along the high-symmetry lines in the BZ. The BZ and the (001) SBZ are shown in the inset. (b),(c) The band structure of the continuum model near the nodal points at $\mathrm{\Gamma }$ ($\mathrm{R}$). The ratio of the wave function magnitude between $|\frac{3}{2},\frac{1}{2},\widehat{\mathit{k}}\rangle$ and $|\frac{1}{2},\frac{1}{2},\widehat{\mathit{k}}\rangle$ ($|1,0,\widehat{\mathit{k}}\rangle$ and $|0,0,\widehat{\mathit{k}}\rangle$) is illustrated by different colors. For the bands with $E>0$, ${\mathrm{Ch}}_{\mathrm{in}}$ and ${\mathrm{Ch}}_{\mathrm{out}}$ are the Chern numbers carried by the inner and outer FSSs, respectively, and $\times 2$ in the subscript indicates double degeneracy.

Figure 2

(a),(b) Phase diagram of the TSC in RhSi with respect to ${\beta }_{\mathrm{\Gamma }}$ and ${\mu }_{\mathrm{\Gamma }}$ with ${\alpha }_{\mathrm{\Gamma }}>0$ (${\beta }_{\text{R}}$ and ${\mu }_{\mathrm{R}}$ with ${\alpha }_{\text{R}}>0$). A pair of winding numbers $(w_{3\mathrm{D},\mathrm{\Gamma }\left(\mathrm{R}\right)},w_{1\mathrm{D},\mathrm{\Gamma }\left(\mathrm{R}\right)})$ on each region characterizes the topology of the superconducting phase. $w_{\text{3D},\mathrm{R}}=⌀$ indicates the TSC with NRs in the bulk BdG spectrum. The dashed line represents ${\mu }_{\text{R}}=E_{c1}\left({\beta }_{\text{R}}\right)$ on which the superconducting gap closes/reopens without changing $w_{3\text{D}}$ and $w_{1\text{D}}$. (c) The band structure around $\mathrm{R}$ above the sixfold nodal point colored in red (blue) for positive (negative) ${\delta }_{\text{R},n}$. The four bands are labeled as $n=1,2,3,4$ in the order of closeness to $\mathrm{R}$. (d) The FSS formed by band 2 at $\mu ={\mu }_2$, colored in the same manner as (c).

Figure 3

(a) Phase diagram in $(V/U,J/U)$ at $\mu =1.3m_{\mathrm{\Gamma }}$ with $U>0$. “No SC” and “Trivial” mean no superconductivity and the topologically trivial superconducting phase, respectively, whereas “NR” and “MZM” represent the superconducting phases with the NRs around $\mathrm{R}$, and a MZM at $\overline{\mathrm{\Gamma }}$, respectively. (b) Surface spectral weight around $\overline{\mathrm{\Gamma }}$ at $E=0$. (c) Nodal rings around $\mathrm{R}$ in the BZ.