DFT calculations and BdG model
We performed DFT calculations using the full-potential local orbital code FPLO38 within the generalized gradient approximation39, using the tetrahedron method with 123 points for the Brillouin integration. Subsequently, a maximally projected symmetry-conserving Wannier model40 containing Wannier orbitals for each Bi 6p and Pt 6s, 5d basis orbital was constructed. The model is mapped onto a semi-infinite slab with a surface block consisting of three Bi 6 Pt 3 layers on which only a nonzero gap function is added. In detail, the gap function reads
$$\Delta ={\delta }_{\text{orbital-qns}}{\rm{i}}{\sigma }_{y}D(k){V}_{0}$$
with \(D(k)=1,020{k}_{x}{k}_{y}(3{k}_{x}^{4}-10{k}_{x}^{2}{k}_{y}^{2}+3{k}_{y}^{4})\) being a scaled Taylor expansion of sin(6ϕ). The scaling factor was chosen to fulfil D(k) = 1 at k = (0.4, 0.0325) Å−1. The Bloch spectral density for a penetration depth of three surface blocks is obtained by solving the BdG equations using Green function recursion2,41 in this semi-infinite geometry.
In Extended Data Fig. 1, we show results for the SC gap on the (001) surface as a function of the distance from the node for different values of V 0 (compare with Fig. 3b). Also, for this termination, we observe the same features: a V-shaped gap that increases with V 0 .
Symmetry-allowed SC states
As noted in the main text, the symmetry of possible SC order parameters can be classified in terms of the irreps A 1 , A 2 and E of the point group C 3v of PtBi 2 . For trigonal PtBi 2 , the mirror planes contain the Γ–K lines and not the Γ–M lines, and thus they do not include the centre points of the Fermi arcs, at which the apparent nodes are located. However, time-reversal symmetry acts like twofold rotation symmetry about the z-axis for momenta k = (k x , k y ). Consequently, any time-reversal-symmetric function of k that is even under all mirror reflections of the lattice is also even, and that which is odd is also odd, under mirror reflections with respect to vertical planes through the arc centres.
Owing to the lack of inversion symmetry, spin-singlet and spin-triplet pairing generically mix, and it is necessary to consider 2 × 2 pairing matrices Δ(k) appearing in the BdG Hamiltonian
$${\mathcal{H}}({\bf{k}})=\left(\begin{array}{rc}{H}_{N}({\bf{k}}) & \varDelta ({\bf{k}})\\ {\varDelta }^{\dagger }({\bf{k}}) & -{H}_{N}^{{\rm{T}}}(-{\bf{k}})\end{array}\right).$$ (1)
To construct possible pairing matrices Δ(k), we have to consider the symmetry properties of k-dependent form factors and of matrices acting on spin space.
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