Critical Non-Hermitian Edge Modes

TL;DR

The paper identifies a critical non-Hermitian edge-mode phenomenon (CNHEM) in which infinitesimal on-site staggered perturbations induce a discontinuous reconfiguration of edge-mode distributions in the thermodynamic limit. Using a 1D non-Hermitian SSH framework with staggered potential $\delta$ and non-reciprocal edge-mode coupling, it derives analytic edge-state energies $\varepsilon_{\pm}$ and non-Bloch wavefunctions, showing $\varepsilon_{\pm}^2 = \delta^2 + c^2(\beta_2/\beta_3)^{N+1}$. The authors construct a perturbation–size phase diagram with a size-dependent boundary $N_c$ given by $\delta = c (\beta_2/\beta_3)^{(N+1)/2}$ and show how edge-mode coupling decays with $N$ while non-reciprocity grows, driving an EP in the thermodynamic limit. For $\delta=0$ the two edge modes coalesce toward the EP as $N\to\infty$, while for any nonzero $\delta$ they remain distinct beyond $N_c$, illustrating a genuinely size-tunable critical phenomenon unique to non-Hermiticity. The findings point to experimental platforms such as active mechanical lattices, phononic/acoustic crystals, and piezophononic media where CNHEM can be probed and controlled via the perturbation–size landscape.

Abstract

We unveil a unique critical phenomenon of topological edge modes in non-Hermitian systems, dubbed the critical non-Hermitian edge modes (CNHEM). Specifically, in the thermodynamic limit, the eigenvectors of edge modes jump discontinuously under infinitesimal on-site staggered perturbations. The CNHEM arises from the competition between the introduced on-site staggered potentials and size-dependent non-reciprocal coupling between edge modes, and are closely connected to the exceptional point (EP). As the system size increases, the coupling between edge modes decreases while the non-reciprocity is enhanced, causing the eigenvectors to gradually collapse toward the EP. However, when the on-site potentials dominate, this weakened coupling assists the eigenvectors to stay away from the EP. Such a critical phenomenon is absent in Hermitian systems, where the coupling between edge modes is reciprocal.

Figures (3)

• Figure 1: (a) Coupled chains $"A"$ and $"B"$ with Hermitian coupling $t_3,t_4$, non-Hermitian coupling $t_1,t_2$ and staggered on-site potentials $\delta, -\delta$. The parameters chosen are $t_1=2,t_2=1.5,t_3=e^{-\pi i/6},t_4=2$. (b) The dependence of $\left| \epsilon_{+} - \delta \right|$ on size when $\delta = 0$ (yellow points), $\delta = 0.005$ (black points) and $\delta = 0.05$ (green points). The inset shows the spectrum under $\delta=0,N=20$ under open boundary condition with red dots denoting the energy of EMs. To exhibit the CNHEM, the distributions of numerically calculated EMs are illustrated with blue $"+"$ and red $"\times"$ in two system sizes $N = 20, 40$ (site numbers 40, 80), when $\delta = 0$ [(c), (d)], $\delta = 0.005$ [(e), (f)] and $\delta = 0.05$ [(g), (h)], while the wavefunction of bulk states are denoted by gray regions. In (b-h), corresponding to these numerical data points, the analytic results are represented by lines of the same color.
• Figure 2: (a) Perturbation-size phase diagram of CNHEM with size-dependent phase boundary. The numerical results (color map) are aligned with the theoretically estimated critical sizes. Under $\delta=0.005$ and the same $t_{1}-t_{4}$ as in Fig. \ref{['fig-model']}, the coupling between two EMs in (b) with the increase of system size leads to the reduction of coupling strength $H_{12}H_{21}$ and the enhancement of non-reciprocity $H_{12}/H_{21}$, as shown in (c). (d) The overlapping magnitude between the EMs, where a complete overlapping signifies the emergence of the EP.
• Figure 3: Comparison of critical length $N_c$ (black dots) and peak location $N_p$ (blue triangles) as the function of $\delta$.