Tailoring Bound State Geometry in High-Dimensional Non-Hermitian Systems

TL;DR

The paper addresses impurity-induced bound states in high-dimensional non-Hermitian systems and shows that geometry-dependent skin effect (GDSE) and Bloch saddle points (BSPs) remove the usual barrier to bound-state formation, enabling zero-threshold bound states. It develops a Green's-function framework with a key mapping $lambda_inv(E_BS)=G0(E_BS;0,0)$ and reveals that BSPs eliminate the finite impurity threshold, while the bound-state geometry is encoded by amoeba contours $f(E_BS,kx,ky)=E_BS-H0(kx,ky)$, governing anisotropic decay. The bound-state wavefunction decays as a direction-dependent exponential with localization lengths satisfying $mu_x l_x + mu_y l_y = 1$, linking geometry to the amoeba contour and predicting a convex-to-concave transition controlled by impurity parameters. The results suggest experimental observables such as LDOS patterns and provide a route to manipulate bound states with arbitrarily weak impurities in 2D non-Hermitian systems.

Abstract

It is generally believed that the non-Hermitian effect (NHSE), due to its non-reciprocal nature, creates barriers for the appearance of impurity bound states. In this paper, we find that in two and higher dimensions, the presence of geometry-dependent skin effect eliminates this barrier such that even an infinitesimal impurity potential can confine bound states in this type of non-Hermitian systems. By examining bound states around Bloch saddle points, we find that non-Hermiticity can disrupt the isotropy of bound states, resulting in concave dumbbell-shaped bound states. Our work reveals a geometry transition of bound state between concavity and convexity in high-dimensional non-Hermitian systems, offering theoretical insights for the experimental manipulation of bound states.

Figures (4)

• Figure 1: Energy response of impurity strength. (a) shows the PBC spectrum of the Hamiltonian $e^{i\pi/6}\cos{(k_x+k_y)} + e^{i\pi/3}\cos{k_x}+2\cos 2k_y$, with four black points denoting the energies at its BSPs. The black cross represents the bound state energy induced by the impurity. (b) illustrates the 1D Bloch saddle lines (BSLs) in the BZ, with brown lines representing $\partial_{k_y}\mathcal{H}_0(k_x,k_y)=0$ and gray lines for $\partial_{k_x+k_y}\mathcal{H}_0(k_x,k_y)=0$. The corresponding spectral lines $\mathcal{H}_0(k_x,k_y)$ are shown in the same color in panel (a). The four intersection points, i.e., high-symmetry $\textbf{k}$ points in the BZ, are the BSPs and correspond to the four vertices in the spectrum shown in (a). (c) and (d) show the function $|\lambda(\delta E)|$, corresponding to the blue and orange trajectories in (a), respectively. Here, $\delta E$ is defined as $E - \mathcal{H}_0(0,0)$ in (c) and $E - \mathcal{H}_0(\pi/3,0)$ in (d). The insets in (c) and (d) show zoomed-in results as $|\lambda| \rightarrow 0$.
• Figure 2: The relation between bound state's geometry and amoeba's contour. Parameters $\{t_{1,1}, t_{-1,-1}, t_{1,0}, t_{-1,0}, t_{0,0} \}$ 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 ($n_{\text{node}}=2$), which results in a concave wavefunction. And the white region is the range where the amoeba has no node ($n_{\text{node}}=0$). The impurity strength is $\lambda = 2.27+2.23i$ for bound state with energy $E_{\text{BS1}}=\mathcal{H}_0(0,0)+0.2\exp(i \frac{\pi}{4})$ and $\lambda=2.66+0.96i$ for $E_{\text{BS2}}=\mathcal{H}_0(0,0)+0.2\exp(-i \frac{19}{40}\pi)$. (b1) and (b2) show the corresponding amoeba's contours for $E_{\text{BS1}}$ and $E_{\text{BS2}}$ 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) and (c2) depict the amplitude $|\psi|$of the bound states for $E_{\text{BS1}}$ and $E_{\text{BS2}}$ respectively. The gray dashed line is the equal amplitude curve of $|\psi(x,y)|$. (d1) and (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) and (b2). The results are obtained from simulations performed on a $30\times 30$ lattice.
• Figure A1: amoeba's contour in Supplementary Equation.\ref{['weak impurity contour']} for different energy phase. $f(\mu_x,\mu_y)=0$ in Supplementary Equation.\ref{['weak impurity contour']} for different $\alpha$. Fix $\theta = \frac{\pi}{3}$ (a)$\alpha = \frac{\pi}{2}$ ; (b)$\alpha = \frac{2\pi}{3}$; (c)$\alpha = \frac{5\pi}{6}$ ; (d)$\alpha = \pi$ ; (e) $\alpha = \frac{7\pi}{6}$ ; (f)$\alpha = \frac{4\pi}{3}$ ; (g)$\alpha = \frac{3\pi}{2}$ ; (h)$\alpha = \frac{5\pi}{3}$ ;(i)$\alpha = \frac{11\pi}{6}$ ;
• Figure A2: The relation between bound state's shape and Newton polytope. Parameters $\{t_{1,1}, t_{1,0}, t_{-1,0}, t_{0, -1}, t_{0,1},t_{-1,-1} \}$ for Hamiltonian are chosen to be $\{0 ,1, 1, i, i,0\}$ in (a) and (b), and $\{2, i, i, 0, 0, 2\}$ in (c) and (d). Panels (a) and (c) depict amplitude of bound state wave function and the gray dashed lines represent the shape of bound state, while panels (b) and (d) show the corresponding Newton polytope. The red arrow in (b) and (d) represent the hopping direction and the red numbers denote the hopping strength in lattice model. The results are obtained from simulations performed on a $30\times30$ lattice, with data extracted from the central $11\times11$ sites with the impurity strength set to $\lambda = 50$.