The general spin triangle

TL;DR

This paper analyzes the quantum and classical properties of the Heisenberg spin triangle with arbitrary couplings and spins, revealing spectral symmetries, integrability features, and universal polynomials relevant to high-temperature expansions.

Contribution

It introduces a comprehensive analysis of the quantum eigenvalue problem for the spin triangle, identifying spectral symmetries and deriving universal polynomials for high-temperature expansions.

Findings

Classical system is completely integrable.

Quantum eigenvalue problem reduces to tridiagonal matrices of size up to 2s+1.

Spectral symmetries due to permutational symmetry.

Abstract

We consider the Heisenberg spin triangle with general coupling coefficients and general spin quantum number $s$. The corresponding classical system is completely integrable. In the quantum case the eigenvalue problem can be reduced to that of tridiagonal matrices in at most $2s+1$ dimensions. The corresponding energy spectrum exhibits what we will call spectral symmetries due to the underlying permutational symmetry of the considered class of Hamiltonians. As an application we explicitly calculate six classes of universal polynomials that occur in the high temperature expansion of spin triangles and more general spin systems. Aspects of quantum integrability are also discussed in this context.