Drinfeld Twists and Algebraic Bethe Ansatz of the Supersymmetrict-JModel

Abstract

We construct the Drinfeld twists (factorizing F -matrices) for the supersymmetric t-J model. Working in the basis provided by the F -matrix (i.e. the so-called F -basis), we obtain completely symmetric representations of the monodromy matrix and the pseudo-particle creation operators of the model. These enable us to resolve the hierarchy of the nested Bethe vectors for the gl(2|1) invariant t-J model. I IntroductionThe algebraic Bethe ansatz or the quantum inverse scattering method (QISM) provides a powerful tool of solving eigenvalue problems such as diagonalizing integrable two-dimensional quantum spin chains. In this framework, the pseudo-particle creation and annihilation operators are constructed by the off-diagonal entries of the monodromy matrix. The Bethe vectors (eigenvectors) are obtained by acting the creation operators on the pseudo-vacuum state. However, the apparently simple action of creation operators is intricate on the level of the local operators by non-local effects arising from polarization clouds or compensating exchange terms. This makes the exact and explicit computation of correlation functions difficult (if not impossible).Recently, Maillet and Sanchez de Santos [1] showed how monodromy matrices of the inhomogeneous XXX and XXZ spin chains can be simplified by using the factorizing Drinfeld twists. This leads to the natural F -basis for the analysis of these models. In this basis, the pseudo-particle creation and annihilation operators take completely symmetric forms and contain no compensating exchange terms on the level of the local operators (i.e. polarization free). As a result, the Bethe vectors of the models are simplified dramatically and can be written down explicitly.The results of [1] were generalized to certain other systems. In [3], the Drinfeld twists associated with any finite-dimensional irreducible representations of the Yangian Y [gl(2)] were investigated. In [4], the form factors for local spin operators of the spin-1/2 XXZ model were computed and in [5], the spontaneous magnetization of the XXZ chain on the finite lattice was represented. In [2], Albert et al constructed the F -matrix of the gl(m) rational Heisenberg model and obtained a polarization free representation of the creation operators. Using these results, they resolved the hierarchy of the nested

On the Construction of Correlation Functions for the Integrable Supersymmetric Fermion Models

“….f σi k for the Bethe state |Ω N ({λ j }) of the q-deformed supersymmetric t-J model,f σ |Ω N ({λ j }) = |Ω N ({λ σ(j) }) . Then one may prove 38,40 , under the action of the exchange operator, we have the following proposition: Proposition 4. The Bethe state |Ω N ({λ j }) of the q-deformed supersymmetric t-J model satisfies the following exchange symmetrŷ…”

“…Here we outline the proof given in 38,40 . The definition of F 1...N (4.2) and the summation condition (4.4) imply that F 1...N is a lower-triangular matrix.…”

“…The proof for this theorem can be found in 38,40 . This theorem plays an important role in the construction of the symmetric representations of the creation (annihilation) operators.…”

Supersymmetric vertex models with domain wall boundary conditions

“…Let us remark that even though we believe that Drinfeld twists do exist for all algebras, so far only those related to A-type (super) algebras and to XY Z model have been explictly constructed [17,18,19,20,21].…”

“…By using the approach of Drinfeld twists [16,18,19], we derive the determinant representations of the DW partition functions for the two systems. We find that the partition functions degenerate to simple functions in the homogenous limit.…”