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arxiv: 2511.15527 · v2 · submitted 2025-11-19 · 🧮 math-ph · cond-mat.stat-mech· cond-mat.str-el· math.MP· quant-ph

Inhomogeneous SSH models and the doubling of orthogonal polynomials

Nicolas Cramp\'e , Quentin Labriet , Lucia Morey , Gilles Parez , Luc Vinet This is my paper

Pith reviewed 2026-05-17 20:41 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechcond-mat.str-elmath.MPquant-ph
keywords Su-Schrieffer-Heeger modelorthogonal polynomialsKrawtchouk polynomialsq-Racah polynomialsexactly solvable modelsinhomogeneous chainsdoubling construction
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The pith

Doubling orthogonal polynomial sequences produces exactly solvable inhomogeneous SSH Hamiltonians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that applying a doubling construction to finite sequences of orthogonal polynomials directly generates the hopping matrices of Su-Schrieffer-Heeger chains. When the original sequence is Chebyshev, the resulting chain is the standard homogeneous SSH model whose spectrum and states are known in closed form. Replacing Chebyshev with Krawtchouk or q-Racah sequences produces chains whose hoppings vary along the lattice yet remain exactly solvable by the same polynomial data. This matters because inhomogeneous SSH models are otherwise solved only numerically, so the method supplies explicit energies, eigenvectors, and therefore all observables.

Core claim

The doubling of a finite orthogonal polynomial sequence maps its three-term recurrence into a 2N-by-2N matrix that can be read as the Hamiltonian of an SSH chain; the eigenvalues of this matrix are then the roots of the original polynomials and the eigenvectors are built from the polynomial values themselves. The standard uniform SSH model arises from Chebyshev polynomials. The same construction applied to Krawtchouk and q-Racah polynomials yields inhomogeneous SSH Hamiltonians whose spectra and eigenstates remain analytically accessible.

What carries the argument

The doubling map that converts the Jacobi matrix of an orthogonal polynomial sequence into the off-diagonal hopping matrix of a bipartite SSH chain.

If this is right

  • The energy eigenvalues of the inhomogeneous model are precisely the zeros of the chosen orthogonal polynomials.
  • Eigenvectors are obtained by evaluating the polynomials at those zeros and applying the doubling transformation.
  • The method supplies closed-form expressions for any finite chain length once the polynomial family is fixed.
  • Chiral symmetry of the doubled matrix forces the spectrum to be symmetric about zero.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same doubling recipe could be tried on other one-dimensional tight-binding models whose hoppings are nearest-neighbor only.
  • Choosing different polynomial families offers a systematic route to engineer solvable position-dependent hoppings.
  • Open-boundary versions of these chains might yield exact expressions for edge states or winding numbers.

Load-bearing premise

The recurrence matrix obtained after doubling can be identified directly with the physical hopping Hamiltonian without extra rescalings or constraints that would destroy the exact solvability.

What would settle it

Numerical diagonalization of the explicit Hamiltonian matrix built from the doubled Krawtchouk recurrence for small N should produce eigenvalues that exactly match the known roots of the Krawtchouk polynomials; any mismatch falsifies the identification.

Figures

Figures reproduced from arXiv: 2511.15527 by Gilles Parez, Lucia Morey, Luc Vinet, Nicolas Cramp\'e, Quentin Labriet.

Figure 1
Figure 1. Figure 1: Representation of the usual SSH model with open boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representation of an inhomogeneous SSH model. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We analyze Su-Schrieffer-Heeger (SSH) models using the doubling method for orthogonal polynomial sequences. This approach yields the analytical spectrum and exact eigenstates of the models. We demonstrate that the standard SSH model is associated with the doubling of Chebyshev polynomials. Extending this technique to the doubling of other finite sequences enables the construction of Hamiltonians for inhomogeneous SSH models which are exactly solvable. We detail the specific cases associated with Krawtchouk and $q$-Racah polynomials. This work highlights the utility of polynomial-doubling techniques in obtaining exact solutions for physical models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using the doubling of orthogonal polynomial sequences to construct and solve inhomogeneous Su-Schrieffer-Heeger (SSH) models exactly. It links the standard SSH model to doubled Chebyshev polynomials and details constructions for inhomogeneous models using doubled Krawtchouk and q-Racah polynomials, yielding analytical spectra and eigenstates.

Significance. This approach could provide a systematic method for finding exactly solvable inhomogeneous SSH chains by associating them with known families of orthogonal polynomials. Such models are relevant for studying position-dependent hoppings in topological systems, and the exact solvability allows for precise analysis of their spectra and states. The connection to standard orthogonal polynomials is a strength if the mapping to physical Hamiltonians is direct.

major comments (2)
  1. [§3] §3 (Krawtchouk doubling): The original Krawtchouk three-term recurrence contains non-zero diagonal coefficients b_n. The doubling construction must be shown explicitly to cancel these and produce a Jacobi matrix with identically zero diagonal (pure off-diagonal hoppings) to identify directly with the SSH Hamiltonian without on-site potentials or rescalings. Please display the explicit doubled recurrence matrix and verify the diagonal vanishes.
  2. [§4] §4 (q-Racah doubling): Analogous verification is required that the doubled recurrence yields a tridiagonal matrix with zero diagonal entries, confirming the resulting operator is a pure nearest-neighbor hopping Hamiltonian for the inhomogeneous SSH chain.
minor comments (2)
  1. [Notation section] The notation distinguishing the original polynomial recurrence coefficients from the doubled SSH hoppings t_i could be clarified to prevent reader confusion.
  2. [Results sections] A short table summarizing the hopping parameters t_i for the Krawtchouk and q-Racah cases would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which help to strengthen the presentation of the doubling construction. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [§3] §3 (Krawtchouk doubling): The original Krawtchouk three-term recurrence contains non-zero diagonal coefficients b_n. The doubling construction must be shown explicitly to cancel these and produce a Jacobi matrix with identically zero diagonal (pure off-diagonal hoppings) to identify directly with the SSH Hamiltonian without on-site potentials or rescalings. Please display the explicit doubled recurrence matrix and verify the diagonal vanishes.

    Authors: We agree that an explicit verification strengthens the exposition. The doubling procedure for finite orthogonal polynomial sequences is constructed precisely so that the diagonal coefficients b_n of the original three-term recurrence are canceled by the symmetric pairing of the doubled sequence, yielding a Jacobi matrix with vanishing diagonal. In the revised manuscript we will insert the explicit 2N × 2N doubled recurrence matrix for the Krawtchouk case in §3 and verify entry-by-entry that every diagonal element is identically zero. This confirms the direct identification with the SSH Hamiltonian consisting solely of nearest-neighbor hoppings, without on-site potentials or additional rescalings. revision: yes

  2. Referee: [§4] §4 (q-Racah doubling): Analogous verification is required that the doubled recurrence yields a tridiagonal matrix with zero diagonal entries, confirming the resulting operator is a pure nearest-neighbor hopping Hamiltonian for the inhomogeneous SSH chain.

    Authors: The same structural cancellation occurs for the q-Racah family. We will add, in the revised §4, the explicit form of the doubled tridiagonal matrix obtained from the q-Racah recurrence and demonstrate that its diagonal vanishes identically. This establishes that the resulting operator is indeed a pure nearest-neighbor hopping Hamiltonian for the inhomogeneous SSH chain, consistent with the general doubling framework introduced earlier in the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent orthogonal polynomial properties

full rationale

The paper constructs inhomogeneous SSH Hamiltonians by doubling sequences of standard orthogonal polynomials (Chebyshev, Krawtchouk, q-Racah) and identifying the resulting Jacobi matrix with the hopping terms. These polynomials, their three-term recurrences, and spectra are established mathematical objects that pre-exist the SSH application and do not depend on any fitted parameters or definitions internal to the paper. The central claim therefore rests on external, independently verifiable properties of orthogonal polynomials rather than on any self-referential loop, self-citation chain, or renaming of a fitted quantity. No load-bearing step reduces by construction to an input defined within the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the assumption that doubled orthogonal polynomial sequences produce valid SSH Hamiltonians; no free parameters, new entities, or ad-hoc axioms beyond standard polynomial theory are indicated in the abstract.

axioms (1)
  • domain assumption The matrix representation obtained from doubling an orthogonal polynomial sequence corresponds exactly to the hopping terms of an inhomogeneous SSH Hamiltonian.
    This mapping is invoked to claim exact solvability for the physical model.

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Two soluble models of an antiferromagnetic chain,

    E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain,”Ann. Phys.16, 407 (1961)

    work page 1961

  2. [2]

    Scaling of entanglement close to a quantum phase transitions,

    A. Osterloh, L. Amico, G. Falci, and R. Fazio, “Scaling of entanglement close to a quantum phase transitions,”Nature416, 608 (2002)

    work page 2002

  3. [3]

    Entanglement in a simple quantum phase transition,

    T. J. Osborne and M. A. Nielsen, “Entanglement in a simple quantum phase transition,”Phys. Rev. A66, 032110 (2002)

    work page 2002

  4. [4]

    Entanglement in quantum critical phenomena,

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, “Entanglement in quantum critical phenomena,” Phys. Rev. Lett.90, 227902 (2003)

    work page 2003

  5. [5]

    Entanglement entropy and quantum field theory,

    P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,”J. Stat. Mech. P06002 (2004)

    work page 2004

  6. [6]

    Calculation of reduced density matrices from correlation functions,

    I. Peschel, “Calculation of reduced density matrices from correlation functions,”J. Phys. A: Math. Gen.36, L205 (2003)

    work page 2003

  7. [7]

    Entanglement in the XY spin chain,

    A. R. Its, B.-Q. Jin, and V. E. Korepin, “Entanglement in the XY spin chain,”J. Phys. A: Math. Gen.38, 2975 (2005). 14

    work page 2005

  8. [8]

    Reduced density matrices and entanglement entropy in free lattice models,

    I. Peschel and V. Eisler, “Reduced density matrices and entanglement entropy in free lattice models,”J. Phys. A: Math. Theor.42, 504003 (2009)

    work page 2009

  9. [9]

    Evolution of entanglement entropy in one-dimensional systems,

    P. Calabrese and J. L. Cardy, “Evolution of entanglement entropy in one-dimensional systems,” J. Stat. Mech.P04010 (2005)

    work page 2005

  10. [10]

    Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field,

    M. Fagotti and P. Calabrese, “Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field,”Phys. Rev. A78, 010306 (2008)

    work page 2008

  11. [11]

    Analytical results for the entanglement dynamics of disjoint blocks in the XY spin chain,

    G. Parez and R. Bonsignori, “Analytical results for the entanglement dynamics of disjoint blocks in the XY spin chain,”J. Phys. A: Math. Theor.55, 505005 (2022)

    work page 2022

  12. [12]

    Entanglement in spin chains with gradients,

    V. Eisler, F. Igl´ oi, and I. Peschel, “Entanglement in spin chains with gradients,”J. Stat. Mech. P02011 (2009)

    work page 2009

  13. [13]

    Conformal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases,

    J. Dubail, J.-M. St´ ephan, J. Viti, and P. Calabrese, “Conformal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases,”SciPost Phys. 2, 002 (2017)

    work page 2017

  14. [14]

    Solitons in polyacetylene,

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,”Phys. Rev. Lett.42, 1698 (1979)

    work page 1979

  15. [15]

    Classification of topological quantum matter with symmetries,

    C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, “Classification of topological quantum matter with symmetries,”Rev. Mod. Phys.88, 035005 (2016)

    work page 2016

  16. [16]

    Observation of the topological soliton state in the Su–Schrieffer–Heeger model,

    E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su–Schrieffer–Heeger model,”Nat. Commun.7, 13986 (2016)

    work page 2016

  17. [17]

    Free-fermion entanglement and orthogonal polynomials,

    N. Cramp´ e, R. I. Nepomechie, and L. Vinet, “Free-fermion entanglement and orthogonal polynomials,”J. Stat. Mech.093101 (2019)

    work page 2019

  18. [18]

    Exactly solvable inhomogeneous XY spin chain,

    P.-A. Bernard, N. Cramp´ e, Q. Labriet, L. Morey, and L. Vinet, “Exactly solvable inhomogeneous XY spin chain,”J. Stat. Mech.103101 (2025)

    work page 2025

  19. [19]

    Entanglement of Free Fermions on Hamming Graphs,

    P.-A. Bernard, N. Cramp´ e, and L. Vinet, “Entanglement of Free Fermions on Hamming Graphs,”Nucl. Phys. B986, 116061 (2023)

    work page 2023

  20. [20]

    Entanglement of Free Fermions on Johnson Graphs,

    P.-A. Bernard, N. Cramp´ e, and L. Vinet, “ Entanglement of Free Fermions on Johnson Graphs,” J. Math. Phys.64, 061903 (2023)

    work page 2023

  21. [21]

    Multipartite information of free fermions on Hamming graphs,

    G. Parez, P.-A. Bernard, N. Cramp´ e, and L. Vinet, “Multipartite information of free fermions on Hamming graphs,”Nucl. Phys. B990, 116157 (2023)

    work page 2023

  22. [22]

    Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes,

    P.-A. Bernard, Z. Mann, G. Parez, and L. Vinet, “Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes,”J. Phys. A: Math. Theor.57, 015002 (2023)

    work page 2023

  23. [23]

    Entanglement of inhomogeneous free fermions on hyperplane lattices,

    P.-A. Bernard, N. Cramp´ e, R. I. Nepomechie, G. Parez, L. Poulain d’Andecy, and L. Vinet, “Entanglement of inhomogeneous free fermions on hyperplane lattices,”Nucl. Phys. B984, 115975 (2022)

    work page 2022

  24. [24]

    How to construct spin chains with perfect state transfer,

    L. Vinet and A. Zhedanov, “How to construct spin chains with perfect state transfer,”Phys. Rev. A85, 012323 (2012)

    work page 2012

  25. [25]

    Entanglement of free-fermion systems, signal processing and algebraic combinatorics,

    P.-A. Bernard, N. Cramp´ e, R. I. Nepomechie, G. Parez, and L. Vinet, “Entanglement of free-fermion systems, signal processing and algebraic combinatorics,” inGravity, Strings and Fields: A Conference in Honour of Gordon Semenoff, p. 273. 2025. 15

    work page 2025

  26. [26]

    Fermionic logarithmic negativity in the Krawtchouk chain,

    G. Blanchet, G. Parez, and L. Vinet, “Fermionic logarithmic negativity in the Krawtchouk chain,”J. Stat. Mech.113101 (2024)

    work page 2024

  27. [27]

    Entanglement Hamiltonian and orthogonal polynomials,

    P.-A. Bernard, R. Bonsignori, V. Eisler, G. Parez, and L. Vinet, “Entanglement Hamiltonian and orthogonal polynomials,”Nucl. Phys. B1020, 117185 (2024)

    work page 2024

  28. [28]

    Entanglement of inhomogeneous free bosons and orthogonal polynomials,

    P.-A. Bernard, R. I. Nepomechie, G. Parez, E. Ragoucy, D. Raveh, and L. Vinet, “Entanglement of inhomogeneous free bosons and orthogonal polynomials,”J. Phys. A: Math. Theor.58, 305001 (2025)

    work page 2025

  29. [29]

    Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings,

    F. Marcell´ an and J. Petronilho, “Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings,”Linear Algebra Appl.260, 169 (1997)

    work page 1997

  30. [30]

    Doubling (dual) Hahn polynomials: classification and applications,

    R. Oste and J. Van der Jeugt, “Doubling (dual) Hahn polynomials: classification and applications,”Symmetry Integr. Geom.: Methods Appl. (SIGMA)12, 003 (2016)

    work page 2016

  31. [31]

    Gasper and M

    G. Gasper and M. Rahman,Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 ed., 2004

    work page 2004

  32. [32]

    Koekoek, P

    R. Koekoek, P. A. Lesky, and R. F. Swarttouw,Hypergeometric Orthogonal Polynomials and Theirq-Analogues. Springer Monographs in Mathematics. Springer-Verlag, 2010

    work page 2010

  33. [33]

    Chihara,An Introduction to Orthogonal Polynomials

    T. Chihara,An Introduction to Orthogonal Polynomials. Gordon and Breach, 1978

    work page 1978

  34. [34]

    Contiguity relations for finite families of orthogonal polynomials in the Askey scheme,

    N. Cramp´ e, L. Morey, L. Vinet, and M. Zaimi, “Contiguity relations for finite families of orthogonal polynomials in the Askey scheme,”The Ramanujan Journal68, 68 (2025). 16

    work page 2025