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Inhomogeneous SSH models and the doubling of orthogonal polynomials

Nicolas Crampé, Quentin Labriet, Lucia Morey, Gilles Parez, Luc Vinet

TL;DR

The paper develops a unifying doubling framework for orthogonal polynomial sequences to diagonalize SSH-type free-fermion Hamiltonians. By associating the standard SSH model with a Chebyshev-doubling construction, it then generalizes to inhomogeneous chains via arbitrary finite orthogonal polynomial sequences. The authors provide two exactly solvable inhomogeneous SSH models based on $Krawtchouk$ and $q$-$Racah$ polynomials, giving closed-form spectra and eigenvectors and exploiting contiguity relations to simplify constructions. This analytic approach enables precise control of spectra and eigenstates in inhomogeneous chains, with potential applications to topological properties and entanglement measures in quantum lattices.

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.

Inhomogeneous SSH models and the doubling of orthogonal polynomials

TL;DR

The paper develops a unifying doubling framework for orthogonal polynomial sequences to diagonalize SSH-type free-fermion Hamiltonians. By associating the standard SSH model with a Chebyshev-doubling construction, it then generalizes to inhomogeneous chains via arbitrary finite orthogonal polynomial sequences. The authors provide two exactly solvable inhomogeneous SSH models based on and - polynomials, giving closed-form spectra and eigenvectors and exploiting contiguity relations to simplify constructions. This analytic approach enables precise control of spectra and eigenstates in inhomogeneous chains, with potential applications to topological properties and entanglement measures in quantum lattices.

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 -Racah polynomials. This work highlights the utility of polynomial-doubling techniques in obtaining exact solutions for physical models.

Paper Structure

This paper contains 13 sections, 3 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Notations.
  4. SSH model and doubling of Chebyshev polynomials
  5. Diagonalisation by Chebyshev polynomials
  6. SSH model with non-vanishing chemical potential
  7. Inhomogeneous SSH model from the doubling procedure
  8. Inhomogeneous SSH model associated to Krawtchouk polynomials
  9. Inhomogeneous SSH model associated to $q$-Racah polynomials
  10. Properties of the $q$-Racah polynomials
  11. First exactly solvable model
  12. Second exactly solvable model
  13. Conclusion

Key Result

Proposition 3.1

The vector $\bold{Q}(x)$, whose components $Q_0(x),Q_1(x),\dots, Q_{2N}(x)$ are defined by eq:Qgen, diagonalizes the $(2N+1)\times (2N+1)$ matrix with the eigenvalue $x$ chosen among if the following constraints are satisfied:

Figures (2)

  • Figure 1: Representation of the usual SSH model with open boundary conditions.
  • Figure 2: Representation of an inhomogeneous SSH model.

Theorems & Definitions (7)

  • Remark 2.1
  • Proposition 3.1
  • Remark 3.2
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof