arXiv:2405.20177

On the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$-symmetry

Allan John Gerrard

Published May 30, 2024• 1 citation

Abstract

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with $\mathfrak{g}$-symmetry for any simple Lie algebra $\mathfrak{g}$. Starting the nesting process by removing a single simple root from $\mathfrak{g}$, we use the residual $U(1)$ charge and the block Gauss decomposition of the $R$-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.

DOI: 10.48550/arxiv.2405.20177

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arXiv:2405.20177

On the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$ -symmetryOn the nested algebraic Bethe ansatz for spin chains with simple $\mathfrak{g}$-symmetry

Allan John Gerrard

Published May 30, 2024• 1 citation

arXiv HTML PDF

Abstract

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with

\mathfrak{g}

-symmetry for any simple Lie algebra

\mathfrak{g}

. Starting the nesting process by removing a single simple root from

\mathfrak{g}

, we use the residual

U(1)

charge and the block Gauss decomposition of the

R

-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.

DOI: 10.48550/arxiv.2405.20177

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On the nested algebraic Bethe ansatz for spin chains with simple

\mathfrak{g}

-symmetry

We propose a new framework for the nested algebraic Bethe ansatz for a closed, rational spin chain with

\mathfrak{g}

-symmetry for any simple Lie algebra

\mathfrak{g}

. Starting the nesting process by removing a single simple root from

\mathfrak{g}

, we use the residual

U(1)

charge and the block Gauss decomposition of the

R

-matrix to derive many standard results in the Bethe ansatz, such as the nesting of Yangian algebras, and the AB commutation relation.

Completeness of Wronskian Bethe equations for rational

\mathfrak{gl}_{m|n}

spin chain

\theta_\ell

\ell=1,\ldots,L

satisfy

\theta_\ell+\hbar\neq\theta_{\ell'}

for

\ell<\ell'

. Counting of solutions is done in two independent ways: by computing a character of the Wronskian algebra and by explicitly solving the Bethe equations in certain scaling regimes supplemented with a proof that the algebraic number of solutions is the same for any value of

\theta_\ell

. In particular, we consider the regime

\theta_{\ell+1}/\theta_{\ell}\gg 1

for the twist-less chain where we succeed to provide explicit solutions and their systematic labelling with standard Young tableaux.