Fig. 3: Low-field magneto-infrared spectrum and schematic band structure of EuCd₂As₂.

a False-color plot of relative reflectivity \({R}_{B}/{R}_{0}\) of EuCd₂As₂ at low fields. b Band structure and DOS before and after the critical field \({B}_{{{{{{\rm{c}}}}}}}\). Weyl bands (WB) are located at distant energy from the upper flat band (UFB) and the lower flat band (LFB). Red and blue distinguish the bands with opposite spins. For clarification, the DOS from each set of spin-polarized bands is plotted on the separate energy axis. Below the critical field, the saddle point at zero momentum (yellow dot) only contributes to a weak spectral feature owing to the finite kink in DOS. Increasing external magnetic field leads to the larger exchange splitting and resultant higher Fermi velocity of the Weyl bands. As reaching the critical field \({B}_{{{{{{\rm{c}}}}}}}\), the requirement of \({v}_{z}\ge \sqrt{2}{v}_{{xy}}\) is met in EuCd₂As₂ as proposed in the minimal model. The divergent DOS originates from the newly formed 3D VHS (green dot), leading to the abrupt enhancement of optical features \({{{{{{\rm{T}}}}}}}_{\alpha }\) (blue arrow) and \({{{{{{\rm{T}}}}}}}_{\beta }\) (red arrow), accompanied by satellite peaks from the inter-band Kerr effect.