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The spectrum of particles interacting through centrally extended su(2|2) S-matrices

Marcio Jose Martins, C.S. Melo

Published Mar 8, 2007• 2 citations
OpenAlex

Abstract

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 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.
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] X [1] and [1] X [N-1]. For the open chain with diagonal boundary fields, we show that the transfer matrix has the symmetry SU(l) X SU(N-l) X U(1), as well as a new ``duality'' symmetry which interchanges l and 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.
We investigate the integrable structure of spin chain models with centrally extended and symmetry. These chains have their origin in the planar anti-de Sitter/conformal field theory correspondence, but they also contain the one-dimensional Hubbard model as a special case. We begin with an overview of the representation theory of centrally extended . These results are applied in the construction and investigation of an interesting S -matrix with symmetry. In particular, they enable a remarkably simple proof of the Yang–Baxter relation. We also show the equivalence of the S -matrix to Shastry's R -matrix and thus uncover a hidden supersymmetry in the integrable structure of the Hubbard model. We then construct eigenvalues of the corresponding transfer matrix in order to formulate an analytic Bethe ansatz. Finally, the form of transfer matrix eigenvalues for models with symmetry is sketched.
The off-diagonal Bethe ansatz method is generalized to the high spin integrable systems associated with the su(2) algebra by employing the spin-s isotropic Heisenberg chain model with generic integrable boundaries as an example. With the fusion techniques, certain closed operator identities for constructing the functional T − Q relations and the Bethe ansatz equations are derived. It is found that a variety of inhomogeneous T − Q relations obeying the operator product identities can be constructed. Numerical results for two-site s = 1 case indicate that an arbitrary choice of the derived T − Q relations is enough to give the complete spectrum of the transfer matrix.
The algebraic structure of the 1D Hubbard model is studied by means of the fermionic R-operator approach. This approach treats the fermion models directly in the framework of the quantum inverse scattering method. Compared with the graded approach, this approach has several advantages. First, the global properties of the Hamiltonian are naturally reflected in the algebraic properties of the fermionic R-operator. We want to note that this operator is a local operator acting on fermion Fock spaces. In particular, SO(4) symmetry and the invariance under the partial particle hole transformation are discussed. Second, we can construct a genuinely fermionic quantum transfer transfer matrix (QTM) in terms of the fermionic R-operator. Using the algebraic Bethe Ansatz for the Hubbard model, we diagonalize the fermionic QTM and discuss its properties.
Models generalizing the su(2) XX spin-chain were recently introduced. These XXC models also have an underlying su(2) structure. Their construction method is shown to generalize to the chains based on the fundamental representations of the Am Lie algebras. Integrability of the new models is shown in the context of the quantum inverse scattering method. Their R-matrix is found and shown to yield a representation of the Hecke algebra. The diagonalization of the transfer matrices is carried out using the algebraic Bethe Ansatz. I comment on eventual generalizations and possible links to reaction-diffusion processes.