Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum

TL;DR

This paper proves that the Bethe algebra of the homogeneous XXX Heisenberg model has a simple spectrum on certain subspaces, revealing precise combinatorial structures of specific two-dimensional vector subspaces.

Contribution

It establishes the simplicity of the spectrum of the Bethe algebra in the homogeneous XXX Heisenberg model and characterizes associated two-dimensional subspaces with explicit basis properties.

Findings

Bethe algebra has simple spectrum on singular vectors

Exact count of specific two-dimensional subspaces with given properties

Explicit polynomial relations defining these subspaces

Abstract

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $\deg f = l, \deg g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.