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arXiv:0706.0688

Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum

Evgeny Mukhin, V. Tarasov, A. Varchenko

Published Jun 5, 2007• 7 citations
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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 $°f = l, °g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.

DOI
10.48550/arxiv.0706.0688

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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 gl2gl_2gl2​-modules. As a byproduct we show that there exist exactly (nl)−(nl−1)\binom {n}{l}-\binom{n}{l-1}(ln​)−(l−1n​) two-dimensional vector subspaces V⊂\C[u]V \subset \C[u]V⊂\C[u] with a basis f,g∈Vf,g\in Vf,g∈V such that °f=l,°g=n−l+1°f = l, °g = n-l+1°f=l,°g=n−l+1 and f(u)g(u−1)−f(u−1)g(u)=(u+1)nf(u)g(u-1) - f(u-1)g(u) = (u+1)^nf(u)g(u−1)−f(u−1)g(u)=(u+1)n.
We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.
We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l -cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S , the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys–Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur–Weyl duality.