Universal scalefree non-Hermitian skin effect near the Bloch point

The scalefree non-Hermitian skin effect (NHSE) refers to the phenomenon that the localization length of skin modes scales proportionally with system size in non-Hermitian systems. Authors of recent studies have demonstrated that the scalefree NHSE can be induced through various mechanisms, including the critical NHSE, local non-Hermiticity, and the boundary impurity effect. Nevertheless, these methods require careful modeling and precise parameter tuning. In contrast, in this paper, we suggest that the scalefree NHSE is a universal phenomenon, observable in extensive systems if these systems can be described by non-Bloch band theory and host Bloch points on the energy spectrum in the thermodynamic limit. Crucially, we discover that the geometry of the generalized Brillouin zone determines the scaling rule of the localization length, which can scale either linearly or quadratically with the system size. In this paper, we enriche the phenomenon of the scalefree NHSE.

Figure 1

(a) Generalized Brillouin zone (GBZ; solid curve) and Brillouin zone (BZ; dashed unit circle) are plotted for the model in Eq. (1). The part of the GBZ inside (outside) the BZ is colored red (blue). (b) Open-boundary-condition (OBC) spectrum $E_{\mathrm{OBC}}$ (colored curve) and periodic-boundary-condition (PBC) spectrum $E_{\mathrm{PBC}}$ (dashed loop). The generalized Bloch phase factors and eigenenergies of the open chain with length $L=30$ are plotted as dots in (a) and (b), where the green dots correspond to the specific energy level $E_m$ that we are interested in. (c) Eigenfunctions corresponding to the dots on the red (blue) part of the GBZ are plotted as red (blue) curves, where the green curve corresponds to the energy level $E_m$. In all plots, parameters are set as $t_{-2}=1/4, t_{-1}=1$, and $t_1=1$.

Figure 2

(a) The rescaled profiles of the wave function ${\mathrm{\Psi }}_m\left(x\right)$ at energy level $E_m$ with system sizes $L=40$, 60, 80, and 100 (red, blue, orange, and green, respectively) are displayed. The peaks of the original wave functions are extracted and depicted for clarity. (b) The inverse localization length ${\kappa }_m$ (left ticks, blue) and the difference between $E_m$ and $E_B$ (right ticks, red) are shown to decrease as $\sim L^{-1}$. Parameters are $t_{-2}=t_{-1}=t_1=1$ for both (a) and (b). (c) The dependence of $|{\kappa }_m|$ on $|{\partial }_{\theta }{\left|\beta \left(\theta \right)\right||}_{\theta ={\theta }_B}$ is plotted. The red, green, and blue dots correspond to three cases where $|{\theta }_B|=\pi /2, \pi /3$, and $\pi /6$, respectively, and the system size is $L=40$.

Figure 3

Generalized Brillouin zone (GBZ) and Brillouin zone (BZ), $E_{\mathrm{OBC}}$ and $E_{\mathrm{PBC}}$, and the scalefree power-law behaviors of inverse localization length and eigenenergy near the Bloch point are plotted for (a1)–(a3) $H\left(\beta \right)={\beta }^{-3}+{\beta }^1$, (b1)–(b3) $H\left(\beta \right)={\beta }^{-2}/4+3{\beta }^{-1}/2+2\beta$, and (c1)–(c3) $H\left(\beta \right)={\beta }^{-2}+3{\beta }^{-1}+\beta$. The parts of the GBZ and $E_{\mathrm{OBC}}$ that match in color correspond to each other. (a3), (b3), and (c3) exhibit three distinct scalefree power-law behaviors.

Figure 4

(a) Su-Schrieffer-Heeger (SSH) model with alternating $t_1, t_2$ hoppings and a non-Hermitian $t_3\pm \gamma$ hopping. (b1)–(b3) are plotted with parameters $t1=t2=1, t3=\frac{1}{5}, \gamma =\frac{1}{4}$. Black dots in (b2) represent the eigenenergies of the system with $L=20$. (c1)–(c3) are plotted with parameters $t1=1, t2=\frac{1}{2}, t3=-\frac{1}{10}, \gamma =\frac{1}{4}$. These two different scalefree power-law behaviors shown by the two-band model are consistent with those in the single-band model.

Figure 5

Generalized Brillouin zone (GBZ) and the scalefree power-law behaviors of inverse localization length near the Bloch point are plotted for (a1) and (a2) $H\left(\beta \right)={\beta }^{-2}/2+{\beta }^{-1}+\beta$ and (b1) and (b2)$H\left(\beta \right)={\beta }^{-2}/2+{\beta }^{-1}+(2+\sqrt{2})\beta$/2. The argument difference between the two intersections of GBZ and Brillouin zone (BZ) determines the periodicity of ${\kappa }_m$ with respect to system size $L$.

Figure 6

Scalefree power law behaviors with weak disorders. The model is pictorially shown in Fig. 4, with parameters $t1=t2=1, t3=\frac{1}{5}, \gamma =\frac{1}{4}$. However, the intracell hopping coefficients within each unit cell is randomly generated from a uniform distribution of $[t_1-\delta t,t_1+\delta t]$, where $\delta t=\frac{1}{100}$.