Algebraic Bethe Ansatz for deformed Gaudin model

The Gaudin model based on the sl_2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(\lambda) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe Ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl_2-invariant Gaudin model. As opposed to the sl_2-invariant case, the operator t(\lambda) and the Gaudin Hamiltonians are not hermitian. Finally, the inner products and norms of the Bethe states are studied.

💡 Research Summary

The paper investigates a deformation of the Gaudin model obtained by adding a Jordanian term to the standard sl₂‑invariant r‑matrix. This extra term depends on the spectral parameters and breaks the hermiticity of the model while preserving the Yang‑Baxter equation. The authors first construct the L‑operator and the monodromy matrix T(λ) from the deformed r‑matrix and define the generating function of the integrals of motion, t(λ)=Tr T(λ). Because t(λ) commutes with itself for different values of λ, it provides an infinite family of conserved quantities, as in the usual Gaudin model.

A central achievement of the work is the explicit construction of the creation operators that generate Bethe states. These operators, denoted B(μ₁,…,μ_M), are defined recursively: the M‑particle operator is built from the (M‑1)‑particle one multiplied by a basic operator X(μ_M) and corrected by a sum over lower‑order terms weighted by a function F(μ_k,μ_M) that encodes the Jordanian deformation. Although the recursion resembles that of the undeformed sl₂ case, the presence of the Jordanian parameter modifies X and F, leading to additional correction terms.

Using the algebraic relations of the deformed r‑matrix, the authors compute the commutator