Engineering unique localization transition with coupled Hatano-Nelson chains

The authors investigate the localization properties of coupled one-dimensional quasiperiodic Hatano-Nelson (QHN) chains with asymmetric hopping. They identify two critical points, $V_{c_1}$ and $V_{c_2}$ , which mark transitions between delocalized and localized states. For potential values $V<V_{c_1}$ the system exhibits complete delocalization, while for $V>V_{c_2}$ all states become localized. Between these two points, mobility edges arise, separating delocalized and localized states. The authors further demonstrate that under specific asymmetric hopping conditions,$V_{c_1}$ can approach zero, resulting in localization even for infinitesimally weak potential amplitude. These findings are derived analytically by extending results from prior works and validated through numerical analysis. A potential experimental implementation using coupled waveguide arrays is also referred.
While these results contribute to the active field of non-Hermitian disordered physics, where numerous studies on variants of quasiperiodic models have been conducted, the manuscript suffers from clarity, language, and scientific presentation issues. At this point, I would not recommend publication in SciPost Physics unless the authors provide satisfactory responses or corrections to the issues outlined below.

1. The spectrum of the stacked Hatano-Nelson bilayer is not clearly explained. Beyond plotting the IPR as a function of the real part of the eigenvalue and increasing quasiperiodic potential amplitude, the manuscript should include comments on the structure of the spectrum itself, and whether and how it depends on the applied boundary conditions. The boundary conditions for Figures 2 and 3 are not specified in the main text (e.g., after Eq. 6). Additionally,$V_{n}$ (Eq. 7) is not defined.
2. Fig. 2 / Lines 130-131: In several subplots of Fig.2, for $V<V_{c_1}$ , certain energies are not represented by dark blue, indicating that the eigenstates are not “perfectly” delocalized as claimed in line 131. Also, how do the values of IPR in the delocalized/intermediate/localized phases change with increasing lattice size N?
3. The criterion IPR>0.1 is arbitrary and does not strictly justify labeling states as “delocalized” or “localized”. How does this choice affect the fraction of localized and delocalized eigenstates in the “localized”, “intermediate” and “delocalized” regimes of potential amplitude? The fraction of 0.5 in the intermediate phase would likely differ with a different choice of criterion. To what extent does this choice influence the universality of the results (e.g., the constancy of the fraction in the intermediate phase)?
4. The meaning of lines 157-167 is unclear due to the expression and complicated structure. It is difficult to understand what the authors are trying to convey and how these results differ from those obtained under PBC. Additionally, the purpose of Figures 4. a(i, ii, iii) is ambiguous. What specific eigenvalues of the spectrum do these figures correspond to?
5. The section “Possible Experimental Implementation in Coupled Waveguides” reformulates the problem within the context of coupled-mode theory as an example of a realistic experimental setup in optics. Equation (18) pertains to a single waveguide lattice, not the bilayer lattice model discussed previously. Furthermore, the optical analog of time is the propagation distance z along the waveguides (as shown in Fig. 5) and not the distance between two parallel waveguides, as stated in line 172.

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