Scalar product of Bethe vectors from functional equations

In this work the scalar product of Bethe vectors for the six-vertex model is studied by means of functional equations. The scalar products are shown to obey a system of functional equations originated from the Yang-Baxter algebra and its solution is given as a multiple contour integral.

💡 Research Summary

The paper investigates the scalar product of Bethe vectors in the six‑vertex model by formulating and solving a system of functional equations derived from the Yang‑Baxter algebra. After a concise introduction that situates Bethe vectors and their scalar products within integrable models and highlights the limitations of existing determinant‑based formulas such as Slavnov’s and Izergin‑Korepin’s, the authors turn to the underlying algebraic structure. They begin by recalling the fundamental commutation relations between the R‑matrix and L‑operators that generate the Yang‑Baxter algebra, and they construct an operator that represents the scalar product of two generic off‑shell Bethe vectors characterized by two independent sets of rapidities, ({u_i}) and ({v_i}).

From the algebraic relations they derive a set of functional (difference) equations that the scalar product must satisfy. These equations encode two essential properties: (i) symmetry under the exchange of rapidities within each set, and (ii) specific boundary conditions at special points where the rapidities coincide with the poles of the R‑matrix. By analytically continuing the rapidities to complex variables, the difference equations become functional equations in the complex plane.

The core technical contribution is the solution of these functional equations via a multiple contour integral representation. Introducing integration variables ({z_i}_{i=1}^N), the authors define a kernel function built from the elementary building blocks of the six‑vertex R‑matrix. The kernel takes the form

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