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<p>We present a closed inhomogeneous quantum <i>XX</i> spin chain on the periodic integer lattice <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}_m\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> which is diagonalized by means of Slater determinants built from Rogers’ <i>q</i>-ultraspherical polynomials with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(q^m=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mi>m</mi> </msup> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The hermiticity of our periodic quantum spin Hamiltonian encodes a discrete orthogonality relation for these <i>q</i>-ultraspherical polynomials that is of a type studied previously by Spiridonov and Zhedanov.</p>

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Exactly solvable inhomogeneous periodic quantum spin chain from q-ultraspherical polynomials at roots of unity

  • 期刊论文

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  • 发表日期：

    2025年11月10日

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  • Andreo Cazares,
  • Jan Felipe van Diejen

摘要

We present a closed inhomogeneous quantum XX spin chain on the periodic integer lattice \(\mathbb {Z}_m\) Z m which is diagonalized by means of Slater determinants built from Rogers’ q-ultraspherical polynomials with \(q^m=1\) q m = 1 . The hermiticity of our periodic quantum spin Hamiltonian encodes a discrete orthogonality relation for these q-ultraspherical polynomials that is of a type studied previously by Spiridonov and Zhedanov.