Inhomogeneous SSH models and the doubling of orthogonal polynomials

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.

💡 Research Summary

The manuscript presents a unified analytical framework for solving both the standard Su‑Schrieffer‑Heeger (SSH) chain and a broad class of inhomogeneous SSH‑type models by exploiting the “doubling” construction of orthogonal polynomial families. The authors first revisit the homogeneous SSH model, whose alternating hopping amplitudes t₊ and t₋ are conventionally expressed through a dimerisation parameter δ. By introducing a pair of Chebyshev polynomials of the second kind, Uₙ(x), they define a new polynomial sequence Qₙ(x) that combines even‑ and odd‑indexed Chebyshev functions with a δ‑dependent linear map π(x). This doubled sequence satisfies a three‑term recurrence whose coefficients are precisely the alternating hoppings of the SSH Hamiltonian. Consequently, the eigenvectors of the single‑particle Hamiltonian are given by the components of Qₙ evaluated at the eigenvalue x, and the eigenvalues are obtained from the condition Q₂N+1(x)=0, i.e. xU_N(πx)=0. The spectrum thus consists of a zero‑energy topological mode and a set of symmetric non‑zero energies x=±√