### ARPES measurements

Table of Contents

ARPES measurements were carried out on the 1^{2} and 1^{3}ARPES endstations^{26} at BESSY II synchrotron (Helmholtz-Zentrum Berlin), as well as in the Leibniz-Institut für Festkörper und Werkstoffforschung Dresden (IFW) laboratory using the 5.9 eV laser light source. Samples were cleaved in situ at a pressure lower than 1 × 10^{−10 }mbar and measured at the temperatures of 15 K and 1.5 K at BESSY II and 3-30 K in the IFW laboratory. The experimental data were obtained using the synchrotron light in the photon energy range from 15 to 50 eV with horizontal polarization and laser light with horizontal and circular polarizations. Angular resolution was set to 0.2–0.5° and energy resolution to 2–20 meV. The findings from the experiments were consistent and reproducible across multiple samples.

The simultaneous presence of bulk non-superconducting and surface superconducting states hinders the detection of true coherence peaks with ARPES. Our experiments at the synchrotron, with energy resolution of the order of 5 meV, turned out to be insufficient to detect even the shifts of the leading edges of the corresponding arc peaks having FWHM of the order of 10 meV and peak-to-background ratio of approximately 5. This is because the arc states are always on top of the bulk continuum. Only by measuring with energy resolution of the order of 1–2 meV did we manage to observe sufficiently sharp peaks (Fig. 3c,d and Extended Data Fig. 4) and their sensitivity to temperature. The sharpest features need to be found on the surface.

A superconducting gap on the arcs is most likely anisotropic. We included error bars in Fig. 4e to show the influence of a small shift of the beam spot and thus slightly different emission angle. Taking into account the very high localization in momentum space, this could lead to probing a different part of the arc and thus different *k*_{F}, where the superconducting gap is slightly different.

### Bulk band structure and Fermi arc position

In Extended Data Fig. 1, we show ARPES Fermi surface maps obtained using the photon energies from 15 eV to 43 eV. Relatively strong variation of the pattern suggests a reasonable *k*_{z}-sensitivity of our experiment. We found the optimal value of the inner potential to be equal to 10.5 eV. This agrees with the previous study of Jiang et al.^{17}.

In Extended Data Fig. 2, we present further evidence that our assignment of the surface and bulk features is correct. Extended Data Fig. 2a shows EDCs taken across the Fermi arc for different photon energies (from synchrotron and laser sources), alongside the theoretical EDC for the fully integrated *k*_{z}. The peak corresponding to the Fermi arc remains clearly visible without any noticeable dispersion for different values of *k*_{z}, whereas the peaks located further below the Fermi level disperse. Such absence of the dispersion is peculiar to the surface states.

In Extended Data Fig. 3, we show an analogue of Fig. 1e–g, but here we compare experimental data with the results of band structure calculations carried out using the linear muffin-tin orbital (LMTO) method in the atomic sphere approximation as implemented in PY LMTO computer code^{27}. As is seen from the figure, the agreement is at the same level as earlier, underpinning the previous conclusion as regards the good agreement between experimental and theoretical 3D band structure.

In Extended Data Fig. 4b, we present the sharpest EDCs from among the various samples and cleaves. Most have FWHM below 3 meV and a peak-to-background ratio of over 30.

### Band structure calculations

We performed density functional theory calculations using the full-potential nonorthogonal local-orbital scheme of ref. ^{28} within the general gradient approximation^{29} and extracted a Wannier function model. This allows determination of bulk projected spectral densities (without surface states) and the spectral densities of semi-infinite slabs via Green’s function techniques^{30}. To model surface superconductivity of the semi-infinite slab, the Wannier model is extended into the BdG formalism with a zero-gap function except for a constant Wannier orbital diagonal singlet gap function matrix at the first three PtBi_{2} layers. A modification of the Green’s function method is used to accommodate this surface-specific term.

### Surface superconductivity calculations

To model a system which has a non-zero gap function only at the surface—in the first 30a_{B} which is 3(PtBi_{2}) layers—we modified the standard Green’s function technique for semi-infinite slabs. The system is built by a semi-infinite chain of identical blocks consisting of 3(PtBi_{2}) layers, repeating indefinitely away from the surface. Each block has a Hamiltonian *H*_{k} for each pseudo momentum *k* in the plane perpendicular to the surface and a hopping matrix *V*_{k}, which couples neighbouring blocks. The blocks’ minimum size is determined by the condition that *H* and *V* describe all possible hoppings. To add superconductivity, the BdG formalism is used by extending the matrices in the following way:

$$\begin{array}{rcl}{H}_{k,{\rm{BdG}}} & = & \left(\begin{array}{cc}{H}_{k} & {\varDelta }_{k}\\ {\varDelta }_{k}^{+} & -{H}_{-k}^{* }\end{array}\right),\\ {V}_{k,{\rm{BdG}}} & = & \left(\begin{array}{cc}{V}_{k} & 0\\ 0 & -{V}_{-k}^{* }\end{array}\right),\end{array}$$

where we choose \({\varDelta }_{k}={\delta }_{i{i}^{{\prime} }}\left(\begin{array}{cc}0 & {V}_{0}\\ -{V}_{0} & 0\end{array}\right)\) with *i* being a spinless Wannier function index and the 2 × 2 matrix to act in a single Wannier function’s spin subspace. This choice also leads to \(\varDelta \left[{V}_{k,{\rm{BdG}}}\right]=0\), since *V* is an off-diagonal part of the full Hamiltonian. To model surface-only superconductivity, we let *V*_{0} = 0 for all (infinite) blocks, except the first one, which gets a finite *V*_{0} = 2 meV.

The standard Green’s function solution for this problem consists of determining the propagator *X* which encompasses all diagrams that describe paths that start at a certain block, propagate anywhere towards the infinite side of that block and return to that block. *X* also describes the Green’s function *G*_{00} of the first block and the self-energy to be added to the Hamiltonian to obtain *G*_{00} (a self-consistency condition) \({G}_{00}=X={\left({\omega }^{+}-H-\Sigma \right)}^{-1}\), Σ = *V**X**V*^{+} (in practice, however, self-consistency is obtained by an accelerated algorithm). From this recursion, relations can calculate all other Green’s-function blocks. These can be derived by subdividing propagation diagrams into irreducible parts using known components, in particular *X*.

If the first block differs from all the others (as is the case due to *Δ*_{k}) one needs to modify the method in the following way. Let the first block have Hamiltonian *h* and hoppings to the second block *v* (while all other blocks are described by *H* and *V*). Then the irreducible subdivision of the propagation diagrams for *G*_{00} results in \(g={\left({\omega }^{+}-h\right)}^{-1}\).

$$\begin{array}{l}{G}_{00}=g+gvX{v}^{+}g+(\,gvX{v}^{+})g\\ \,=\,\frac{1}{{\omega }^{+}-h-vX{v}^{+}}\end{array}$$

which contains the surface Hamiltonian and a modified self-energy depending on the *X* of the unmodified semi-infinite slab. From this we can derive the second block’s Green’s function

$${G}_{11}=X+X{v}^{+}{G}_{00}vX$$

and all others

$${G}_{n+1,n+1}=X+X{V}^{+}{G}_{nn}VX,\quad n > 0$$

which can be used to obtain the spectral density up to a certain penetration depth. Note that in our BdG case \(H={H}_{k,{\rm{BdG}}}\left[{V}_{0}=0\right]\), \(V={V}_{k,{\rm{BdG}}}\left[{V}_{0}=0\right]\) and \(h={H}_{k,{\rm{BdG}}}\left[{V}_{0}\ne 0\right]\), *v* = *V*. The BdG spectral density is particle–hole symmetric and to obtain results that resemble ARPES data, one needs to use the particle–particle block *G*^{ee} (the upper left quarter of the *G* matrix) only.

Extended Data Fig. 5b shows the resulting spectra of this method along the path denoted in Extended Data Fig. 5a. Note that a gap is opened at the surface band pockets close to the Fermi energy, while the rest of the spectrum stays gapless (if we let *V*_{0} ≠ 0 for all blocks, we get a completely gapped spectrum). Extended Data Fig. 5c shows a zoomed-in region around the surface state. Note that the bulk bands are gapless (dark blue vertical features) while the surface state shows a gap and corresponding band back-bending. The particle–hole symmetry becomes apparent, although with a larger spectral weight for the occupied part because we use *G*^{ee} only.

### Further discussion

One approach to demonstrate the existence of topologically protected states with a topological insulator is to perform spin-resolved ARPES. In this technique, the spin-locking effect determines the spin structure in the vicinity of the surface Dirac node. However, the situation is quite different for Weyl semimetals. Here, there is no specific spin structure or configuration associated with the Weyl nodes, which can occur at generic points in the Brillouin zone. As inversion is broken and spin-orbital coupling present, each band at a generic *k*-point naturally possesses a spin direction, but this spin texture is smooth. Consequently, spin-resolved ARPES measurements cannot directly reveal Weyl points.

We would like to exclude the interpretation of our data based on density-wave order, which could, in principle, result in the similar features in the spectra. Charge density-waves require a redistribution of the spectral weight in the momentum space, characterized by the particular *k*-vector (vectors). We have always observed almost the same Fermi surface maps and underlying dispersions, independent of temperature. In line with these observations are the results of the STM studies which never detected any kind of a reconstruction. We have never observed any replica of the arcs or of the deeper lying surface states, such as a strong feature at (−0.2, −0.2) in Fig. 3f,g. It is also not clear which *k*-vector would be suitable for characterizing the density-wave order. If the arcs are simply superimposed in momentum, they all are of electron-like topology, so the opening of the hybridization gaps seems very unlikely. Finally, the fundamental difference between the density-wave gaps and superconducting gaps is that the latter are always pinned to the Fermi level. This is the only energy interval where we observe the changes in the spectra of PtBi_{2} with temperature.