Form factors of integrable higher-spin XXZ chains and the affine quantum-group symmetry

We derive exactly scalar products and form factors for integrable higher-spin XXZ chains through the algebraic Bethe-ansatz method. Here spin values are arbitrary and different spins can be mixed. We show the affine quantum-group symmetry, $U_q(\hat{sl_2})$, for the monodromy matrix of the XXZ spin chain, and then obtain the exact expressions. Furthermore, through the quantum-group symmetry we explicitly derive the diagonalized forms of the $B$ and $C$ operators in the $F$-basis for the spin-1/2 XXZ spin chain, which was conjectured in the algebraic Bethe-ansatz calculation of the XXZ correlation functions. The results should be fundamental in studying form factors and correlation functions systematically for various solvable models associated with the integrable XXZ spin chains.

💡 Research Summary

The paper presents a rigorous derivation of scalar products and form factors for integrable XXZ spin chains with arbitrary (including mixed) spin values, using the algebraic Bethe‑ansatz (ABA) framework. The authors begin by constructing the monodromy matrix (T(u)) from local (L)-operators that realize the spin‑(s_i) representation of the quantum algebra (U_q(sl_2)). By employing the standard (R)-matrix of the six‑vertex model, they verify the fundamental RTT relation (R(u-v)T(u)T(v)=T(v)T(u)R(u-v)).

A central achievement of the work is the explicit proof that the monodromy matrix possesses an affine quantum‑group symmetry, namely (U_q(\widehat{sl_2})). This is accomplished through a Drinfel’d twist (F) which brings (T(u)) into a basis where the generators (A,B,C,D) of the monodromy matrix transform covariantly under the co‑product of (U_q(\widehat{sl_2})). Consequently, the algebraic structure governing the creation ((B)) and annihilation ((C)) operators is fully controlled by the affine quantum group.

The Bethe equations for a chain of length (N) with site‑dependent spins (s_i) are derived in a unified form:
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