Fig. 1. Topological band inversion induced by bilayer structure in CaKFe₄As₄.

a The crystal structure of CaKFe₄As₄ with the inset of the As-Fe-As tetrahedron. The bond lengths (d₁, d₂) and angles are different between Fe-As₁ and Fe-As₂¹⁵. b–d DFT+DMFT calculation results for the band structures of b CaFe₂As₂, c CaKFe₄As₄, and d CaKFe₄As₄ with SOC, respectively. In the b, c, the red symbols of “+” and “−” represent the band parity, and the red (black) dashed line box marks the position of the topological nontrivial (trivial) band inversion. The glide-mirror symmetry breaking effect in 1144 system is visible by comparing the band structures of CaFe₂As₂ with CaKFe₄As₄. In CaKFe₄As₄, a large hybridization gap (~0.5 eV) is formed between the folded p_z bands, and a band inversion with a SOC gap of 20 to 30 meV is found near E_F at d. e Fermi surfaces measured by ARPES at a photon energy of 35 eV, which is around the middle point of Γ–Z, note that the ‘a’ in scale is 2.73 Å referring to the Fe–Fe distance of CaKFe₄As₄ in the real space; the red contours on the right are extracted from DFT+DMFT calculation at Z and A, which shows good consistence with experimental Fermi surface. f–h ARPES spectral intensity plots along the blue dashed line in e with p-polarized photons at energies of 42 eV (Γ) and 27 eV (Z and A). i ARPES spectral intensity plot along the Γ–Z direction measured under photon energies from 21 to 45 eV. j The momentum distribution curve (MDC) second derivative of the ARPES intensity plot along the A–Z direction obtained from g, h, which enhances the vertical part of the band but suppresses the horizontal part of the band³², with comparison to the calculated results from d (plotted as colored lines).