Time-reversal symmetry breaking type-II Weyl state in YbMnBi2

Spectroscopic detection of Dirac and Weyl fermions in real materials is vital for both, promising applications and fundamental bridge between high-energy and condensed-matter physics. While the presence of Dirac and noncentrosymmetric Weyl fermions is well established in many materials, the magnetic Weyl semimetals still escape direct experimental detection. In order to find a time-reversal symmetry breaking Weyl state we design two materials and present here experimental and theoretical evidence of realization of such a state in one of them, YbMnBi2. We model the time-reversal symmetry breaking observed by magnetization and magneto-optical microscopy measurements by canted antiferromagnetism and find a number of Weyl points. Using angle-resolved photoemission, we directly observe two pairs of Weyl points connected by the Fermi arcs. Our results not only provide a fundamental link between the two areas of physics, but also demonstrate the practical way to design novel materials with exotic properties.


Supplementary
. 0.193,0.015) in YbMnBi2.Upon inclusion of SOC and AFM into the computaIonal scheme all Dirac crossings become gapped in EuMnBi2. In contrast, in YbMnBi2 four 3D-Dirac points are observed. One of them is shown. Supplementary Figure 17. FPLO band structure including canIng. Dispersions along three perpendicular direcIons in the k-space are shown through all inequivalent Weyl points. The three direcIons are 1) perpendicular to the lens 2) along the lens and 3) along the z-direcIon. Note that the resoluIon of the WP posiIon for WP1 is not perfect. Nevertheless, the Berry curvature proves that this is indeed a Weyl point. Type II Weyl character of the points #5 and #6 is clearly seen. Although it looks as if one of the bands forming WP5 and WP6 has a minimum at the WP it is not so. The minimum is slightly shihed off the exact WP posiIon, thus establishing the Type II character.

Supplementary Tables
Supplementary Table 1. Photon energies at which high symmetry points are accessible considering that inner potenIal is 6.5 eV.

Supplementary Table 2.
Coordinates of the Weyl point as a funcIon of canIng angle. In order to understand the complete 3D electronic structure of both materials we have recorded Fermi surface maps at different excitaIon photon energies. The results for EuMnBi2 are shown in Fig. 6. Four lenses Fermi surface remains visible at different photon energies, which is not surprising since it originates from the 2D networks of Bi atoms. On the other hand, it can be due to the probing similar kzs. As is seen from two middle panels, intensity distribuIon at the Fermi level only slightly changes as a funcIon of light polarizaIon. Intensity near the Γ-point does change and represents 3D band which also cross the Fermi level at parIcular kz.

Supplementary
Since EuMnBi2 did not show any evidence for the 3D Dirac or Weyl points in the following we deal with YbMnBi2 only. In spite of very pronounced two-dimensionality of the features originated from the Bi-networks, we were able to detect clearly periodic pagerns of intensity distribuIon along ΓM direcIon (Fig. 7 a,b,f) and determine photon energies which correspond to Γ and Z points in YbMnBi2 (Table 1).
Intensity distribuIons at the energies corresponding to high-symmetry points are compared with calculaIons in panels c and d of Fig. 7. There is a qualitaIve agreement between two datasets confirming our assignment made in Table 1 and defined by the value of inner potenIal. Since the features near k=0 are blurred, we have recorded the similar data from another sample with beger staIsIcs (Fig. 7 e). While qualitaIvely the same due to finite kz-resoluIon of ARPES, the new dataset allows to demonstrate directly the kz-sensiIvity of the features. The exemplary sets of EDCs clearly show the dispersion of the underlying components as well as different number of the features (e.g. marked by red bars).
The decisive evidence for the kz-dispersion comes from the dataset presented in Fig. 7 f. There we measured the k-separaIon of the steepest features at the Fermi level as the distance between the sharp peaks in the corresponding EF-MDCs. This is illustrated with two extreme datasets taken using 20.5 and 38.5 eV photon energies. The distance obtained in this way from mulIple datasets oscillates with photon energy between the values approximately corresponding to Γ and Z points.
We have recorded Fermi surface maps using different photon energies also for YbMnBi2. The results are shown in Fig. 8. We emphasize here that the energies have been selected before the value of the inner potenIal has been found, i.e. before the assignment to G and Z points. The only criterion for the selecIon was the well-defined photoemission signal and sharpness of the features. At other photon energies we were not able to obtain a clear picture of the Fermi surface. Now it can be explained in terms of two regions of kz idenIfied above: at those kz which correspond to either minimal gaps or Weyl crossings the kz dispersion is naturally weaker.
We have also checked that the lihing of the degeneracy is observed at different photon energies to exclude its arIficial origin. Corresponding data sets are shown in Fig. 9. It is seen that the linear features dispersing from higher binding energies split and only one pair reaches the Fermi level for both, posiIve and negaIve angles.

Supplementary Note 2.
As discussed in the main text, magneIc contrast can only be seen on the surface of the YbMnBi2 sample by employing the Voigt effect at perpendicular light incidence in an opIcal polarizaIon microscope, while no Voigt domain contrast shows up on the EuMnBi2 crystal.
Typical for the Voigt effect is its 90° symmetry on rotaIng the sample: the domains show up with maximum contrast if their axes of magneIzaIon are at 90° relaIve to each other and if they are aligned at 45° to polarizaIon axis of the illuminaIng light (see sketches in Fig. 10). The contrast disappears if the sample is rotated by 45° and it shows up again with maximum, but now inverted contrast aher a 90°rotaIon. For our YbMnBi2 sample this typical contrast symmetry is revealed as shown in the leh column of Fig. 10.
For our EuMnBi2 crystal, no Voigt contrast is observed at perpendicular incidence. This indicates an anIferromagneIc alignment of magneIc moments with no in-plane components of the occupied crystallographic axis. Also a canIng of moments along a single axis can be excluded, as in this case the brightness of the images should at least reveal the 90° symmetry of the Voigt effect. This is not the case by comparing the images in the right column of Fig. 10.
Note that the external magneIc fields, available in our microscope setup, are too small to reorient the magneIzaIon of the YbMnBi2 crystal. Therefore the convenIonal difference image technique [1] with reference image in saturated state could not be applied and all shown images are difference images with the analyser been opened (in respect to polarizer) in opposite direcIon for reference image, what inverts the domain contrast. However, this technique provides less contrast to be reasonably seen with microscope, configured for longitudinal Kerr effect (not shown), and, unfortunately, the contrast enhancement upon analyser rotaIon cannot be employed together with selecIve-sensiIvity technique [2], what limits the domain observaIon in our samples to only Voigt effect.

Supplementary Note 3.
In order to explain the spli|ng of the bulk bands in YbMnBi2 seen experimentally we have carried out the calculaIons with canIng. CanIng angle is 10°. It produces net FM magneIzaIon along (1, 1, 0). The results are shown in Fig. 11.
As is seen from the calculaIons, there are two closely separated crossings of the singly degenerate bands on the ΓM ⊥ direcIon, but these are not 3D Weyl points, as evoluIon with kz shows. These crossings make a loop in the verIcal plane. These two symmetry related loops are shown in Fig. 2c of the main text. On the ΓMII direcIon, in contrast, a pair of true 3D Weyl points is observed at kz=0.12 and kz=-0.09. Away from verIcal ZΓM planes all crossings become avoided. The locaIons of 3D Weyl points are symmetrical with respect to earlier detected 3D Dirac point (kz=0.015) for collinear configuraIon of spins, implying that this pair of Weyl points is created by the Imereversal symmetry breaking (canIng). In order to test this, we have carried out the calculaIons for different values of canIng angle. Indeed, the distance between the Weyl point and iniIal 3D Dirac point decreased with decreasing the canIng angle (Table 2).
We have also scanned other porIons of BZ and found another set of Weyl points, lying away from high-symmetry planes and direcIons. First, in Fig. 12 we show the radial cuts from Γ-point towards the Fermi surface contour for kz=0 and kz=π/c.
As is seen, the very small gap is detected in the ZAR-plane signaling the proximity to a Weyl point. Further scanning of momentum space resulted in idenIficaIon of Weyl points with very anisotropic dispersions at kz=0.131 (see Fig.  13). The Weyl point is found at (0.394,0.045, 0.131). The kx and ky of this posiIon corresponds to the point where the hole-like lens is connected to electron-like pocket.
Other symmetry related Weyl points are at (0.394,-0.045,0.131), (0.045, 0.394,0.131) and (-0.045, -0.394,0.131). As follows from the presented data, there are two kz values which define the kx-ky planes where Fermi surface may look conInuous in the experiment: kz=π/c where the gaps are very small and kz~0.1 where the true 3D Weyl points are observed.