Fig. 3: Topological understanding of the unconventional BFA and the paired EP3s.
a Two closed loops encircling each of the paired EP3s (purple loop L1, and blue loop L2), and another loop encircling both (green loop L3). The eigenvalues on the loops are shown in the upper panel, and the structures of the loops in the 2D parameter space are shown in the lower panel. b, c The eigenvalue evolution processes along L1 (b) and L2 (c), respectively. The eigenvalues j1, j2, and j3 are colored in yellow, blue, and green, respectively. The order exchange processes are inverse along L1 and L2. For L1, j1 and j2 coalesce and swap with j3 (j1,2 ↔ j3), while for L2, j2 and j3 coalesce and swap with j1 (j1 ↔ j2,3). d The eigenvalue evolution processes along L3. The eigenvalues do not experience order exchange along L3. The solid lines and rhombus symbols in panels (b, c, and d) correspond to theoretical and experimental results, respectively. e, f Demonstration of the nontrivial topology of the paired EP3s with the evolution process of eigenframe along L3, corresponding to the real and imaginary parts of eigenstates, respectively. The eigenstates φ1, φ2, and φ3 are also colored in yellow, blue, and green, respectively. The lower panels show the corresponding accumulated angles to the initial states. We see that the imaginary parts of eigenstates serve as an intermediate process. Eigenstates φ1 and φ3 accumulate nontrivial quantized angle π, and thus their final states are on the antipodal points to the initial states. The accumulated angle for φ2 is zero. The experimental raw data for plotting the symbols in (b–d) is provided in the Supplementary Data.
