Baxters Q-operators and operatorial Backlund flow for quantum (super)-spin chains

We propose the operatorial form of Baxter’s TQ-relations in a general form of the operatorial B"acklund flow describing the nesting process for the inhomogeneous rational gl(K|M) quantum (super)spin chains with twisted periodic boundary conditions. The full set of Q-operators and T-operators on all levels of nesting is explicitly defined. The results are based on a generalization of the identities among the group characters and their group co-derivatives with respect to the twist matrix, found by one of the authors and P.Vieira [V.Kazakov and P.Vieira, JHEP 0810 (2008) 050 [arXiv:0711.2470]]. Our formalism, based on this new “master” identity, allows a systematic and rather straightforward derivation of the whole set of nested Bethe ansatz equations for the spectrum of quantum integrable spin chains, starting from the R-matrix.

💡 Research Summary

The paper presents a fully operatorial formulation of Baxter’s T‑Q relations for inhomogeneous rational gl(K|M) quantum (super)‑spin chains with twisted periodic boundary conditions. The authors introduce a “master identity” that generalizes earlier character identities and their group‑derivatives with respect to the twist matrix. This identity underlies a systematic construction of all Q‑operators and T‑operators at every nesting level, thereby providing a unified description of the entire nested Bethe‑Ansatz hierarchy directly from the R‑matrix.

The work begins by recalling the standard R‑matrix for the rational gl(K|M) model, incorporating site‑dependent inhomogeneities ξ_i and a diagonal twist matrix T. The monodromy matrix is built from these ingredients, and its transfer matrices T_a(u) (with a labeling the auxiliary space representation) are the usual commuting family of operators. The novelty lies in treating the twist matrix as a variable and applying a group‑derivative operator D (the co‑derivative) to characters χ_R(T) of gl(K|M) representations. The master identity, \