Extended Data Fig. 7: Calculated hinge states for a four-layer α-Bi4Br4.
a, The edge-projected band structure for a (001) four-layer ribbon on the top surface of α-Bi4Br4. Purple bands represent gapless hinge states, cyan bands are from the bulk and (001) surfaces, and orange bands are from the (100) and (−100) side surfaces of the ribbon. Due to the inversion asymmetry inherent in even-layer systems, the bands are singly degenerate at each kb. The ribbon is infinitely long in the b-direction and 50-chain wide in the a-direction. For a four-layer system, the left bottom and the left top states (depicted in panel b) undergo hybridization, resulting in a small gap of 4.1396 meV. Note that the energy gap may be slightly overestimated in this calculation due to its derivation from three-dimensional bulk band structure calculations. In reality, our tunneling spectroscopy measurements (see Fig. 4) reveal the presence of a gapless state on the left edge. Additionally, there exists a quantitative disparity in the positioning of the Fermi energy when comparing the calculation with the tunneling spectroscopy measurements. Nevertheless, the calculation provides a qualitative depiction of well-defined hinge states within the energy gaps of both the bulk and surface states. b, Schematic representations in real space illustrating the bulk, surface, and hinge states of a four-layer α-Bi4Br4.
