Fig. 2: Fractal characteristics of the winding matrix for the fractal braiding of non-Hermitian bands.

a Winding matrix for non-Hermitian fractal-braiding energy bands with N = 3 and β = 2, showcasing the winding behavior of complex energy bands. The top subpanel illustrates the braid diagram, obtained by projecting the starting and ending points of the complex energy bands onto a plane perpendicular to the complex energy plane. The bottom subpanel displays the winding matrix for the fundamental braid, where the highlighted red matrix element indicates the counterclockwise braiding of the 2nd and 3rd bands. b Winding matrix for two-generation fractal braiding, derived from the fundamental braid. The right-hand subpanel corresponds to the band structure and braid diagram, emphasizing the fractal nature of the 9×9 winding matrix, where each 3×3 diagonal submatrix replicates the fundamental winding matrix from (a). The highlighted red elements reflect uniform winding behavior among the second-generation braids. c Winding number matrix for the three-generation fractal braids, demonstrating that the diagonal block matrix of the three-generation 27×27 matrix is identical to the two-generation winding matrix in (b), and the values of elements in each non-diagonal block are equal to the corresponding fundamental matrix elements shown in (a). The highlighted red elements reflect uniform winding behavior among the third-generation braids.