Quasiperiodicity-induced bulk localization with self similarity in non-Hermitian systems

Abstract

We analyze the localization behavior in a non-Hermitian system subject to a quasiperiodic onsite potential. We characterize localization transitions using multiple quantitative indicators, including inverse participation ratio (IPR), eigenstate fractal dimension (EFD), extended eigenstate ratio (EER), and spectral survival ratio. Despite the breaking of self-dual symmetry due to non-Hermiticity, our results reveal the existence of a critical potential strength, with its value increasing linearly with the nearest-neighbor antisymmetric hopping term. On the other hand, the inclusion of longer-range hopping not only enriches the topological properties but also gives rise to novel localization phenomena. In particular, it induces the emergence of mobility edges, as evidenced by both IPR and EFD, along with distinct features in the spectrum fractal dimension, which we extract using the box-counting method applied to the complex energy spectrum. Additionally, we uncover self-similar structures in various quantities, such as EER and complex eigenvalue ratio, as the potential strength varies. These findings highlight important aspects of localization and fractal phenomena in non-Hermitian quasiperiodic systems.

Figures (21)

• Figure 1: Schematic setup of a non-Hermitian lattice subject to an onsite potential, as described by Eq. \ref{['Eq:H_nH+qp']}. The arrows represent asymmetric hoppings $(t_{1,2} \pm g_{1,2})$ to the right/left directions, and the wavy curves denote a real onsite potential $V(x) = V_0 \cos ( 2\pi x / \lambda_0 )$ with the wavelength $\lambda_0$.
• Figure 2: (a) PBC (curve) and OBC (dot) spectra of Eq. \ref{['Eq:H_nH+qp']} for $V_0 = 0$ and $N=50$. The PBC spectrum represents a function of momentum $k$, ranging from $0$ (brown) to $2\pi/a_0$ (light brown), while the color of the OBC spectrum indicates the winding number $W(E_r)$ defined in Eq. \ref{['Eq:Winding']}, with the reference point $E_r$ set to the corresponding OBC eigenvalues. (b) Spatial distribution of all the OBC (right) eigenstates. The values of the adopted parameters are given by $(g_1, t_2, g_2) = (-0.7, 0.8, -0.8)$ and $t_{n \ge 3} = g_{n \ge 3} = 0$.
• Figure 3: PBC (curve) and OBC (dot) spectra of Eq. \ref{['Eq:H_nH']}; the latter are plotted with $N=50$. The PBC spectra are obtained from Eq. \ref{['Eq:E_HNL_PBC']}, with $k$ going from $0$ (brown) to $2\pi/a_0$ (light brown). The color of the OBC spectra indicates the winding number $W(E_r)$ given by Eq. \ref{['Eq:Winding']}, with the reference point $E_r$ set to the corresponding eigenvalues. From Panel (a) to (e), the parameters $(g_1 , t_2 , g_2)$ are given by $(-2, 2.6, -3 )$, $(2, -2.6, -3)$, $(0.6, -0.75, -0.35)$, $(-0.6, 0.75, -0.35)$, and $( 1.3, -2.8, -0.4)$, respectively.
• Figure 4: (a)--(c) PBC spectra and IPR of Eq. \ref{['Eq:H_nH+qp']} for $N=1597$ and (a) $V_0 = 2$, (b) $V_0 = 2.25$, and (c) $V_0 = 2.7$. The color of the dots represents the IPR computed from Eq. \ref{['Eq:IPR']}: yellow (blue) color indicates more localized (extended) states. (d)--(f) Spatial profile of the right eigenstates corresponding to (a)--(c), but for $N=89$ and $\lambda_0/a_0=144/89$. The eigenstates are ordered by their eigenvalues, with a larger $n$ labeling a larger Re$(E_n)$. The adopted values of the other parameters are given by $(g_1,t_2,g_2) = (0.05,0.1,0)$.
• Figure 5: EFD ($\Gamma_n$) and averaged EFD ($\langle \Gamma \rangle_H$) computed from Eq. \ref{['Eq:EFD']} and the deduced critical potential strength. (a) $\Gamma_n$ for $g_1 = 0.1$. (b) Similar to (a) but for $g_1 = 0.3$. (c) $\langle \Gamma \rangle_H$ as a function of $V_0$ for various $g_1$ values. (d) Antisymmetric hopping amplitude ($g_1$) dependence of the critical potential strength ($V_c$), deduced from the $V_0$ value at which $\langle \Gamma \rangle_H$ drops below 0.5. The adopted values of the other parameters are given by $(t_2,g_2) = (0,0)$ and $N=1597$.