The Bethe ansatz approach for factorizable centrally extended S-matrices

TL;DR

The paper investigates integrable models with factorized S-matrices invariant under the centrally extended su(2|2) symmetry, focusing on diagonalizing the associated transfer matrix via the algebraic Bethe ansatz and obtaining circle quantization conditions for asymptotic momenta. It shows that the su(2|2) R-matrix is connected to Shastry’s graded R-matrix of the Hubbard model through a spectral-parameter dependent transformation, enabling a Hubbard-like Bethe ansatz with explicit eigenvalues and Bethe equations. The authors derive nested Bethe equations for two levels, reveal that eigenvectors carry t(λ) dependence while eigenvalues are t-independent, and recast the results in x± variables consistent with the Hubbard framework. In the asymptotic regime, the quantization rules map onto Beisert–Staudacher-type equations for AdS5×S5 worldsheet excitations, with the angular momentum J emerging as the thermodynamic scale.

Abstract

We consider the Bethe ansatz solution of integrable models interacting through factorized $S$-matrices based on the central extention of the $\bf{su}(2|2)$ symmetry. The respective $\bf{su}(2|2)$ $R$-matrix is explicitly related to that of the covering Hubbard model through a spectral parameter dependent transformation. This mapping allows us to diagonalize inhomogeneous transfer matrices whose statistical weights are given in terms of $\bf{su}(2|2)$ $S$-matrices by the algebraic Bethe ansatz. As a consequence of that we derive the quantization condition on the circle for the asymptotic momenta of particles scattering by the $\bf{su}(2|2) \otimes \bf{su}(2|2)$ $S$-matrix. The result for the quantization rule may be of relevance in the study of the energy spectrum of the $AdS_5 \times S^{5}$ string sigma model in the thermodynamic limit. \