© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2511.15527

Inhomogeneous SSH models and the doubling of orthogonal polynomials

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

math-phcond-mat.stat-mechcond-mat.str-elquant-phSubmitted Nov 19, 2025• Announced Nov 20, 2025• Revised Feb 20, 2026 (v2)

v1 · Nov 19, 2025 v2 · Feb 20, 2026

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.

Journal reference: SciPost Phys. 20, 124 (2026)
DOI: 10.21468/SciPostPhys.20.5.124
Comments: 19 pages. V2: Minor modifications, improved discussion on zero modes

Related papers

Perfect state transfer in inhomogeneous XX model of q-Racah type

Nicolas Crampé, Lafrance, Simon, Charles Robillard +1 more

New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q-Racah and para q-Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of perfect state transfer across these spin chains.

arXiv:2512.12419•Dec 13, 2025•0 citations

Perfect state transfer in inhomogeneous XX model of q -Racah type

Nicolas Crampé, S. Lafrance, Charles Robillard +1 more

Abstract New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer (PST) are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q -Racah and para q -Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of PST across these spin chains.

Journal of Physics A Mathematical and Theoretical•May 22, 2026•0 citations

Exactly Solvable Inhomogeneous Fermion Systems

Ryu Sasaki B

Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on 1D lattices are constructed explicitly based on the discrete orthogonal polynomials of the Askey scheme, e.g. Krawtchouk, Hahn, Racah, Meixner, and q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground-state two-point correlation functions are derived explicitly. The multipoint correlation functions are obtained by Wick’s theorem. 15 corresponding exactly solvable XX spin systems are also displayed. They all have nearest-neighbor interactions. The exact solvability of the Schrödinger equation means that the corresponding Fokker–Planck equation is also exactly solvable. This leads to 15 exactly solvable birth and death fermions and 15 birth and death spin models. These provide plenty of material for calculating interesting quantities, e.g. entanglement entropy, etc.

arXiv:2410.07614•Progress of Theoretical and Experimental Physics•Nov 14, 2024•4 citations

Topological phases and spontaneous symmetry breaking: the revenge of the original Su-Schrieffer-Heeger model

Polina Matveeva, Dmitri B. Gutman, Sam T. Carr

We study the interplay of spontaneous symmetry breaking and topological properties in interacting one-dimensional models. We solve these models using bozonization and identify topologically non-trivial phases by counting the additional degeneracy (affiliated with the edge modes) of a finite-size system relative to the infinite one. We find even if the mean-field solution is topological, this may not be true when it arises from spontaneous symmetry breaking, including in the Su-Schrieffer-Heeger (SSH) model. This implies that the original SSH model, as presented by Su, Schrieffer, and Heeger, is topologically trivial, as opposed to its mean-field version. A spinful version, on the other hand, does exhibit a topologically non-trivial phase. In that state, both mean-field solutions are topologically non-trivial and correspond to non-interacting SSH chains in the opposite phases with the winding number $ν=1$. We show that this phase is protected by a chiral symmetry, similar to the non-interacting phases.

arXiv:2508.01985•Aug 4, 2025•0 citations

Large Polarons in the Discrete SSH Model

R. Ruckh, E. Sigmund

Abstract Stationary polaron solutions are derived for the discrete version of the Su‐Schrieffer‐Heeger (SSH) model in two different ways. First Holstein's large polaron theory is applied. Second, the polaron solution is derived from Heisenberg's equations of motion. It turns out that the results obtained from the continuum version of the SSH Hamiltonian are different from those obtained from Holstein's molecular crystal model (MCM).

physica status solidi (b)•May 1, 1988•0 citations

Exactly solvable inhomogeneous XY spin chain

Pierre-Antoine Bernard, Nicolas Crampé, Quentin Labriet +2 more

Abstract Analytical expressions for the eigenvalues of certain inhomogeneous XY spin chains are computed. These models are rewritten in terms of free-fermion models using a well-known Jordan–Wigner transformation. Finding the spectrum of such models amounts to diagonalizing a matrix whose size is equal to the number of sites in the chain. This is achieved by recognizing and exploiting contiguity relations satisfied by specific orthogonal polynomials.

arXiv:2507.06331•Journal of Statistical Mechanics Theory and Experiment•Oct 1, 2025•2 citations

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.

New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q-Racah and para q-Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of perfect state transfer across these spin chains.

Abstract New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer (PST) are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q -Racah and para q -Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of PST across these spin chains.

Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on 1D lattices are constructed explicitly based on the discrete orthogonal polynomials of the Askey scheme, e.g. Krawtchouk, Hahn, Racah, Meixner, and q-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground-state two-point correlation functions are derived explicitly. The multipoint correlation functions are obtained by Wick’s theorem. 15 corresponding exactly solvable XX spin systems are also displayed. They all have nearest-neighbor interactions. The exact solvability of the Schrödinger equation means that the corresponding Fokker–Planck equation is also exactly solvable. This leads to 15 exactly solvable birth and death fermions and 15 birth and death spin models. These provide plenty of material for calculating interesting quantities, e.g. entanglement entropy, etc.

We study the interplay of spontaneous symmetry breaking and topological properties in interacting one-dimensional models. We solve these models using bozonization and identify topologically non-trivial phases by counting the additional degeneracy (affiliated with the edge modes) of a finite-size system relative to the infinite one. We find even if the mean-field solution is topological, this may not be true when it arises from spontaneous symmetry breaking, including in the Su-Schrieffer-Heeger (SSH) model. This implies that the original SSH model, as presented by Su, Schrieffer, and Heeger, is topologically trivial, as opposed to its mean-field version. A spinful version, on the other hand, does exhibit a topologically non-trivial phase. In that state, both mean-field solutions are topologically non-trivial and correspond to non-interacting SSH chains in the opposite phases with the winding number

ν=1

. We show that this phase is protected by a chiral symmetry, similar to the non-interacting phases.