Exactly solvable inhomogeneous XY spin chain

TL;DR

This paper derives analytical eigenvalue expressions for inhomogeneous XY spin chains by transforming them into free-fermion models and utilizing orthogonal polynomial relations for diagonalization.

Contribution

It provides a novel method to compute spectra of inhomogeneous XY chains using orthogonal polynomials and Jordan-Wigner transformation.

Findings

Eigenvalues expressed analytically for certain inhomogeneous XY chains

Diagonalization achieved via orthogonal polynomial contiguity relations

Model transformation simplifies spectral analysis

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.