Exact Solution and Large-Scale Scaling Analysis of the Imaginary Creutz–Stark Ladder

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper considers the Creutz ladder with an imaginary Stark field and solves the resulting non-Hermitian Hamiltonian analytically to derive the eigenstates and spectrum. Along with the discrete part of the spectrum encompassing the Stark localized states, and a complex branch, another branch along the real axis is found. The authors show that in the thermodynamic limit, the imaginary parts of this branch decays with a power law and the states becomes extended. As this paper presents a thoughtful methodology for solving an interesting model and provides insight on connection between the wavefunction singularity and open boundary conditions, I recommend publication. I only have a couple of minor comments:
1. In Fig. 2, the authors compare the analytical solution with numerically obtained OBC spectrum. How does it compare with numerically obtained PBC spectrum as the momentum space description correspond to that.

2. In Fig. 2(a), what do the light blue circles represent?

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript, Gui-lu Long et al “Exact Solution and Large-Scale Scaling Analysis of the Imaginary Creutz-Stark Ladder”, investigate the imaginary Creutz-Stark Ladder by analytically solving the complex spectrum. The authors first perform a basis transformation and obtain the non-Hermitian Hamiltonian and the wavefunctions in the momentum-space. Next, the authors focus on varying parameter regimes and analyze singularities in the small-energy regime relative to the inter-site interactions. The results are cross-verified with numerical exact diagonalization. The manuscript is interesting and represents a very important contribution to the field. I am certain that the manuscript will be suitable for publication once my questions and concerns are addressed promptly.

1. In Fig 3, why does \phi_{A} have nodes on m = 0 but not \phi_{B}? Can the authors elaborate on this?

 

2. The power law shown in Figure 6 is quite interesting. Is the number 1.12-1.13 meaningful? I do not think it is unity in the thermodynamic limit.

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

The authors report exact analytical results for part of the spectrum and identify an “asymptotically real” branch whose imaginary component vanishes with increasing system size. The interplay between momentum-space singularities, boundary conditions, and finite-size scaling is potentially interesting and may represent a nontrivial phenomenon in non-Hermitian lattice systems.

However, in its current form, several conceptual aspects and interpretations remain unclear, especially for readers who, including the present referee,  are not specialists in non-Hermitian lattice models. I believe the manuscript would benefit substantially from clarification of the following points.

1. The manuscript states that an exact analytical solution is obtained. Since the model is a single-particle linear Hamiltonian, one might reasonably ask what “exact” means in this context. Does it refer to the closed-form expressions for all eigenvalues and wave functions?

2. Section 6 discusses possible experimental implementations, but the correspondence remains somewhat abstract.
What concrete physical platform does the Creutz–Stark ladder represent in this work? What physical parameter corresponds to the imaginary Stark term? Does the imaginary Stark potential represent generic dissipation in open quantum systems? What microscopic processes (absorption, external coupling, mode-dependent leakage, etc.) would generate a linear dependence on site index?

3. The imaginary term appears to be introduced phenomenologically. While this is common in effective non-Hermitian descriptions, the manuscript would benefit from a clearer statement of its physical origin.

4. The model assumes a specific sign for the imaginary Stark potential. Since the sign distinguishes loss from gain, it would be useful to clarify what physical situation corresponds to the chosen sign. Would reversing the sign qualitatively change the spectral structure?

5. The emergence of algebraic singularities in momentum space plays a central role in the argument. However, the physical meaning of these singularities remains somewhat abstract. Do they correspond to a resonance condition? How should one interpret them in real space?

6. The manuscript introduces the generalized fractal dimension D_2 extracted from the inverse participation ratio. However, the system itself does not possess a geometric fractal structure. 

7. The result D_2→1 appears to indicate extended states in the thermodynamic limit. It would be useful to clarify whether the fractal analysis provides fundamentally new insight in this model, or whether it serves primarily as a consistency check. 

8. One of the central claims is that ∣Im(E)∣→0 as L→∞ for the asymptotically real branch. It is crucial to clarify what this implies. In particular, does this mean that dissipation vanishes in the thermodynamic limit? Or, does it mean that the imaginary part arises solely from finite-size boundary matching?

9. How extent this extreme size calculation L ~ 10^9 was essentially needed; for example, how about L ~ 10^6 enough?

The manuscript presents potentially interesting results on a non-Hermitian lattice model with a structured imaginary potential. However, several conceptual and fundamental aspects require clarification, especially for readers outside the field.
For these reasons, I recommend some optional revisions before the manuscript is published.

Author Response

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