Dynamical Degeneracy Splitting and Directional Invisibility in Non-Hermitian Systems

In this Letter, we introduce the concept of dynamical degeneracy splitting to describe the anisotropic decay behaviors in non-Hermitian systems. We demonstrate that systems with dynamical degeneracy splitting exhibit two distinctive features: (i) the system shows frequency-resolved non-Hermitian skin effect; (ii) Green’s function exhibits anomalous behavior at given frequency, leading to uneven broadening in spectral function and anomalous scattering. As an application, we propose directional invisibility based on wave packet dynamics to investigate the geometry-dependent skin effect in higher dimensions. Our work elucidates a faithful correspondence between non-Hermitian skin effect and Green’s function, offering a guiding principle for exploration of novel physical phenomena emerging from this effect.

Figure 1

(a) Schematic of the relation between dynamical degeneracy splitting (DDS), frequency-resolved NHSE, and the physical consequences via Green’s function. (b)–(f) show the bulk-boundary correspondence between dynamical degeneracy splitting and frequency-resolved NHSE. For the Bloch spectrum in (b), there is no dynamical degeneracy splitting at excitation frequency ${\omega }_1$ in (c), where the color bar corresponds to imaginary energy $\mathrm{Im}E(\mathit{k})$ on the equal-frequency contour (the dashed yellow lines). Correspondingly, NHSE is absent at this frequency in (e). Dynamical degeneracy splitting occurs at ${\omega }_2$ as shown in (d), and consequently, NHSE appears at frequency ${\omega }_2$ in (f).

Figure 2

The illustrations of conventional scattering in (a) and anomalous scattering in (b). Here, the model Hamiltonian reads $H_{\mathrm{NH}}(\mathit{k})=2\sin k_x\cos k_y-2\cos k_x+i(\cos k_x-1)$, and the impurity line ${\mathcal{L}}_{\theta }$ is along the $\theta =3\pi /4$ direction. The dark and light blue dots represents parts of poles calculated in Eq. (4). As $\eta \to 0$, the corresponding poles evolve from dark to light blue dots, as indicated by arrows.

Figure 3

Directional invisibility in Hamiltonian Eq. (5) with the parameters $({\mu }_0,{\mu }_z,t_0,t,t_z,\gamma )=(1.35,-0.05,-0.4,0.4,-0.6,1)$. (a),(b) show the spectral function $A(\omega ,\mathit{k})$ with $\omega =3/2$, of which the intensity corresponds to the opacity as shown in the color bar. Here, ${\mathcal{L}}_{\theta }$ represents the impurity line, and ${\mathit{k}}_i$ denotes the incident wave and ${\mathit{k}}_s$ (${\mathit{k}}_s^{\prime }$) indicates the scattered wave. The incident wave packet with the momentum center at ${\mathit{k}}_i$ hits the oblique impurity line in (c1)–(c5) and vertical impurity line in (d1)–(d5), where the impurity strength $\lambda =1$.