Fig. 4

Evolution of surface states in topological phase transitions. a–d Topological phase transition of the calculated (100)-surface band structure in KHgBi under stress and symmetry breaking. e–g The schematic illustration of surface band structure in each topological phase. Green and brown lines indicate the surface states with different \(\hat M_y\) eigenvalues. The label ±i indicate the \(\hat M_z\) eigenvalues of the surface states depicted in blue and red, respectively. Dotted, dashed, and solid lines represent the first, second, and third pair of the surface states, respectively. The other surface state of each pair locates along \(- \bar Y \to \bar \Gamma\) and is not shown here. a, e The topological non-symmorphic crystalline insulator (TNCI) phase with χ = 2 and C_m = −2 with Δc/c₀ = 0.100. The Mobius-twisted connectivity can be observed along \(\tilde Y \to \tilde T\) and \(\tilde Z \to \tilde \Gamma\), shaping like an hourglass (The shape spoils along \(\tilde Z \to \tilde \Gamma\); however, the same topology remains.). b, f The Dirac semimetal (DSM) phase with C_m = −3 under the stress Δc/c₀ = 0.115. The Mobius-twisted connectivity along \(\tilde Z \to \tilde \Gamma\) is spoiled owing to the occurrence of the bulk-projected Dirac node (DN). A magnified view for the DN is also provided in c. d, g Another TNCI phase with χ = 3 and C_m = −3, transformed from the DSM phase by breaking the rotational symmetry. The quantum spin Hall effect can be observed along \(\tilde Z \to \tilde \Gamma\)