Fig. 2: The relation between bound state’s geometry and amoeba’s contour.
From: Tailoring bound state geometry in high-dimensional non-hermitian systems
Parameters {t1,1, t−1,−1, t1,0, t−1,0, t0,0} for Hamiltonian in Eq. (1) are set to be {2, 2, i, i, − 2i}. a The red points represent the Bloch spectrum near the Bloch saddle point \({{{{\mathcal{H}}}}}_{0}(0,0)\). The two gray regions indicate the range for energy whose amoeba has two nodes (nnode = 2), which results in a concave wavefunction. And the white region is the range where the amoeba has no node (nnode = 0). The impurity strength is λ = 2.66 + 0.96i for bound state with energy \({E}_{{{{\rm{BS1}}}}}={{{{\mathcal{H}}}}}_{0}(0,0)+0.2\exp (i\frac{\pi }{4})\) and λ = 2.27 + 2.23i for \({E}_{{{{\rm{BS2}}}}}={{{{\mathcal{H}}}}}_{0}(0,0)+0.2\exp (-i\frac{19}{40}\pi )\)). b1, b2 show the corresponding amoeba’s contours for EBS1 and EBS2 respectively outlined by the black curves. The red (blue) dot denotes the point of tangency between the red (blue) dashed line and the amoeba’s contour. The red (blue) dashed line is perpendicular to the red (blue) arrow. c1, c2 depict the amplitude ∣ψ∣of the bound states for EBS1 and EBS2 respectively. The gray dashed line is the equal amplitude curve of ∣ψ(x, y)∣. d1, d2 show a comparison of bound states between the simulated data (colored dots) and the theoretical predictions (colored line). The red (blue) dots and line correspond to the x(y)-axis. The slope of red (blue) line is given by the red(blue) point in (b1, b2). The results are obtained from simulations performed on a 30 × 30 lattice.
