Inhomogeneous SSH models and the doubling of orthogonal polynomials

1. The mathematical framework is rigorous.
2. The analytical results are interesting.

1. The presentation is currently too dense and opaque for the broad readership.
2. The connection between the mathematical machinery (recurrence relations, doubling) and the physical observables (edge states, topological phases) is obscured by a lack of pedagogical structure.

1. Lack of Pedagogical Background on Orthogonal Polynomials: The manuscript relies heavily on the properties of discrete orthogonal polynomials (Askey scheme) without providing sufficient background. Terms like "dual of the backward shift relation" or specific "contiguity relations" are introduced with minimal context. I request that the authors must include a dedicated Section or a comprehensive Appendix summarizing the relevant properties of the orthogonal polynomials used. Specifically, the derivations hinge on the properties of discrete orthogonal polynomials; these must be defined clearly for a general physics audience.

2. Clarification of the Derivation Logic: The flow of the derivation is currently buried in technical details. Readers need a high-level roadmap to navigate the math. I suggest the authors insert a summary of the logical flow early in Section 3, clearly distinguishing the steps:

(a) Ansatz Construction: Using the doubling method to define Q_{2n}(x) and Q_{2n+1}(x) (Eqs. 3.5a, 3.5b).

(b) Mapping to Hamiltonian: Proposition 3.1 demonstrates that this form satisfies the (inhomogeneous) SSH eigen-equations subject to specific parameter constraints (Eqs. 3.9a-3.9c).

(c) Solving for Spectrum: Identifying the roots of the characteristic polynomial via the condition Q_{2N+1}=x R_N(\pi_x )=0, where the roots yield the eigenenergies and \mathbf{Q}(x_k) yield the eigenvectors.

1. Summary Table of Models: The paper discusses multiple cases (Chebyshev, Krawtchouk, two q-Racah variants) with different parameters, making it difficult to compare them. I suggest provide a summary table listing: (a) The Polynomial R_n(e.g., Chebyshev, Krawtchouk). (b)The parameters \tau_2 and \tau_0. (c) The resulting hopping amplitudes t_{n}^{\pm}. (d) The analytical form of the eigenvalues x_k^\pm.

2. Explicit Analysis of Edge States: A interesting feature of the SSH model is the existence of topologically protected edge states which decay exponentially into the bulk. While the authors analytically obtained the eigenvectors, they do not explicitly analyze its edge states. I request the authors explicitly demonstrate how the analytical form of the eigenvectors (e.g., Eq. 2.15 or 4.13) exhibits exponential localization (or lack thereof) for the edge states. Also I would like to ask the authors do similar analysis for the inhomogeneous models (Krawtchouk, q-Racah) and check do they host localized edge states? If so, how does the inhomogeneity affect the localization length? This physical insight is essential for publication in a physics journal. ​

3. Boundary Conditions and System Size Parity: The manuscript focuses almost exclusively on Open Boundary Conditions (OBC) for odd system sizes (2N+1). I suggest there is a discussion of periodic boundary conditions. For example, can this method handle periodic boundary conditions? If not, this limitation should be explicitly stated and discussed.

4. Parity (2N vs 2N+1): In Section 2, the authors state that for 2N sites, the roots are not known explicitly. However, in Section 5.3, they successfully solve a q-Racah model for 2N sites. Why is the 2N case solvable for this specific q-Racah example but not generally for the others? The authors should clarify the mathematical or physical reason for this distinction.

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