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arXiv:hep-th/0703086 · Nuclear Physics B

The Bethe ansatz approach for factorizable centrally extended su(2|2) S-matrices

Marcio Jose Martins, C.S. Melo

Published Jun 7, 2007• 81 citations
arXivHTMLPDF
DOI
10.1016/j.nuclphysb.2007.05.021

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

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We consider both closed and open integrable antiferromagnetic chains constructed with the SU(N)-invariant R-matrix. For the closed chain, we extend the analyses of Sutherland and Kulish — Reshetikhin by considering also complex “string” solutions of the Bethe ansatz equations. Such solutions are essential to describe general multiparticle excited states. We also explicitly determine the SU(N) quantum numbers of the states. In particular, the model has particle-like excitations in the fundamental representations [k] of SU(N), with k = 1, …, N − 1. We directly compute the complete two-particle S-matrices for the cases [1] ⊗ [1] and [1] ⊗ [N − 1 ]. For the open chain with diagonal boundary fields, we show that the transfer matrix has the symmetry SU(l) × SU(N−l) × U(1), as well as a new “duality” symmetry which maps l→N−l. With the help of these symmetries, we compute by means of the Bethe ansatz for particles of types [1] and [N − 1 ] the corresponding boundary S-matrices.

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We consider the spectrum of particles interacting by means of factorized SSS-matrices based on the central extention of the bfsu(2∣2)\\bf{su}(2|2)bfsu(2∣2) symmetry. The underlying bfsu(2∣2)\\bf{su}(2|2)bfsu(2∣2) RRR-matrix is explicitly related to that of the covering Hubbard model through a spectral parameter dependent transformation. This mapping allows us to diagonalize the respective transfer matrix by the algebraic Bethe ansatz. As a consequence of that we derive the quantization condition on the circle for the momenta of particles scattering by the bfsu(2∣2)otimesbfsu(2∣2)\\bf{su}(2|2) \\otimes \\bf{su}(2|2)bfsu(2∣2)otimesbfsu(2∣2) S-matrix. This result may be of relevance to construct the energy spectrum of the AdS5timesS5AdS_5 \\times S^{5}AdS5​timesS5 string sigma model in the thermodynamic limit.
We consider both closed and open integrable antiferromagnetic chains constructed with the SU(N)-invariant R-matrix. For the closed chain, we extend the analyses of Sutherland and Kulish — Reshetikhin by considering also complex “string” solutions of the Bethe ansatz equations. Such solutions are essential to describe general multiparticle excited states. We also explicitly determine the SU(N) quantum numbers of the states. In particular, the model has particle-like excitations in the fundamental representations [k] of SU(N), with k = 1, …, N − 1. We directly compute the complete two-particle S-matrices for the cases [1] ⊗ [1] and [1] ⊗ [N − 1 ]. For the open chain with diagonal boundary fields, we show that the transfer matrix has the symmetry SU(l) × SU(N−l) × U(1), as well as a new “duality” symmetry which maps l→N−l. With the help of these symmetries, we compute by means of the Bethe ansatz for particles of types [1] and [N − 1 ] the corresponding boundary S-matrices.