Fig. 4: Nontrivial helical corner states.

Energy levels of a finite lattice composed of 6 × 6 × 6 conventional unit cells for a, Tl₄PbTe₃ and b, Tl₄SnTe₃. There are 16 degenerate corner states as indicated with red color, which is magnified in the inset of a, b. Insets show the probability of one corner state, where we sum all contributions of probability in one unit cell and each unit cell is represented with a cubic. c–e 1^st order Wilson bands \(v_x\), \(v_y\), and \(v_z\) of the occupied bulk bands. The generalized’Fermi levels’ are indicated with red dash lines, where we take the 1^st order Wilson bands among the two red lines as occupied 1^st order Wilson bands. f–h 2^nd order Wilson bands \(v_z^{ + x}\), \(v_x^{ + y}\), and \(v_y^{ + z}\) of the selected occupied 1^st order Wilson bands. The generalized ‘Fermi levels’ are indicated with red dash lines, where we take the 2^nd order Wilson bands among the two red lines as occupied 2^nd order Wilson bands.