Non-Hermitian comb effect in coupled clean and quasiperiodic chains

TL;DR

The paper analyzes a two-leg lattice where a non-Hermitian quasiperiodic chain is coupled to a clean chain, revealing rich localization physics governed by inter-chain coupling and non-Hermiticity. Using exact diagonalization and standard localization diagnostics ($IPR$, $NPR$, and $\eta$) along with spectral topology ($\rho$ and real/complex/mixed classifications), it maps out delocalized, localized, and intermediate phases and identifies a non-Hermitian comb effect in the intermediate regime. The NHCE manifests as coexistence of localized and extended states without a mobility edge, arising from the energy-dependent effective disorder in the limiting case $t_B=0$ and robust across parameter ranges. The results deepen understanding of NH localization in coupled systems and point to experimental avenues in photonics and atomic circuits.

Abstract

We study localization properties in a system of non-Hermitian quasiperiodic chain coupled to a uniform chain or clean chain by inter-chain hopping. We find that in the limit of weak inter-chain coupling, such a coupled system exhibits transitions from delocalized to intermediate phase with increase in the non-Hermiticity parameter. However, for stronger inter-chain coupling strengths, the delocalized phase undergoes a transition to localized phase and then to an intermediate phase. Interestingly, the intermediate phase in this case exhibits the non-Hermitian comb effect (NHCE), i.e., the coexistence of localized and extended states rather than being well separated from each other by any mobility edge which is conventional in any intermediate phase. We further show that such a NHCE originates from the isolated site limit of the quasiperiodic chain and provide an analytical explanation supporting the numerical signatures.

Figures (7)

• Figure 1: Schematic of a coupled non-Hermitian quasiperiodic an clean chain system. The red filled circles are sites subjected to non-Hermitian quasiperiodic potential, while the blue circles represent the disorder free lattice sites. Nearest-neighbor hopping (solid line) along each chain is denoted by $t_\text{A/B}$, and the inter-chain hopping as $t_r$ (dotted lines).
• Figure 2: (a) Phase diagram in the $t_r\text{-}h$ plane shows the delocalized (D), localized (L) and intermediate (I) phases. The intermediate (I, light and dark gray) phase is identified using $\eta$, while the delocalized (D, light brown) and localized (L, light blue) phases are determined from (b) the maximum IPR (IPR$_\text{max}$) and (c) the minimum IPR (IPR$_\text{min}$). (d) Phase diagram obtained using $\rho$ shows the real (R, blue), complex (C, dark gray), and mixed (M, light gray) regions. Panels (e) and (f) display the variation of $\langle \text{IPR} \rangle$ (red dash-dotted line) and $\langle \text{NPR} \rangle$ (blue dashed line) as functions of the complex phase $h$ for weak ($t_r=2.0$) and strong ($t_r=6.0$) inter-chain couplings, respectively. The other parameters are fixed at $t_{A/B} = 1$, $\lambda = 1$, and $L = 610$.
• Figure 3: (a) Eigenstate index $(n/2L)$ as a function of $h$ shows both delocalized and localized states. (b) Enlarged view of central part of (a). (c) A zoomed picture of IPR and NPR of the states with index $n/2L$ from $0.49$ to $0.51$ for $h = 2.0$ depicts the NHCE. The probability density along the site index for both the clean $(|\psi_{j,A}|^2)$ and disordered chains $(|\psi_{j,B}|^2)$, indicating (d) extended state in blue and (e) localized state in red. (f) Real and imaginary parts of the eigenenergies, along with their corresponding IPR values, are shown for $h = 2.0$. The other parameters are $t_{A/B} = 1$, $\lambda = 1$, $t_r = 2$, and system size $L = 610$.
• Figure 4: (a) IPR of states with respective eigenstate index $(n/2L)$ are plotted as functions of $h$ clearly distinguishes the delocalized, localized, and the intermediate phases. (b) Real eigen-energies are shown as a function of the site index, colored according to their corresponding IPR values. The insets show the probability density versus site index, revealing that the states are mainly confined along the rungs, the upper inset corresponds to clean chain, while the lower inset shows for the disordered chain. (c) For $h = 10$, shows the real eigen-energies as a function of the site index, the inset zooms into the middle of the spectrum to highlight delocalized features. The parameters used are $t_{\mathrm{A/B}} = 1$, $t_{r} = 6$, $\lambda = 1$, and system size $L = 610$.
• Figure 5: Phase diagram in the $t_r \text{--} \lambda$ plane. Intermediate (I) phase appears in the shaded light and dark gray regions, while the blue areas correspond to the localized (L) and delocalized (D) phases determined from $\eta$ for (a) $h=0.5$ and (b) $h=2.0$. IPR of states with respective eigenstate index $(n/2L)$ are plotted as functions of potential strength $\lambda$ for (c) $t_r = 1, h=0.5$, and (d) $t_r = 1, h=2.0$. The other parameters are $t_{A/B} = 1$ and system size $L = 610$.