Fig. 3: Asymptotic eigenmodes and system states evolving on the Riemann surfaces.

a, b Amplitudes of the normalized eigenmodes X_1,2 of H in Eq. (4) in the parameter space (β ⁄ κ, γ ⁄ κ). The blue and purple sheets represent the amplitudes of the first and second terms of X_1,2. c, d Forward and e, f, backward evolution trajectories in the Riemann surfaces formed by the real and the imaginary part of the energy spectra E of H as the initial state is \({\left[1,0\right]}^{{\rm{T}}}\). The empty circles represent the starting and ending points of the trajectories and the stars represent EPs. The dashed meshes show β ⁄ κ and γ ⁄ κ extending to infinity. The red and purple sheets denote E₁ and E₂, respectively