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Bethe Algebra of Homogeneous XXX Heisenberg Model has Simple Spectrum

Alexander N. Varchenko, Evgeny Mukhin, V. Tarasov

Published Jul 1, 2021• 4 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 gl2-modules. As a byproduct we show that there exist exactly (nl) − (n l−1) two-dimensional vector subspaces V ⊂ C[u] with a basis f, g ∈ 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.

DOI
10.17615/9g61-mb13

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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 gl2-modules. As a byproduct we show that there exist exactly (nl) − (n l−1) two-dimensional vector subspaces V ⊂ C[u] with a basis f, g ∈ 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.
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 $V \subset \C[u]$ 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 derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin-1 2 Heisenberg chains using the algebraic Bethe ansatz.In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on H, the tensor product of auxiliary spaces.By changing the basis in H, we derive explicit finite-dimensional representations for the matrices.These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo (2006 J. Phys.A: Math.Gen. 39 11335) apart from normalization factors.We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions (Korepin 1982 Commun.Math.Phys.86 391) and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.
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.