Form-factors and complete basis of observables via separation of variables for higher rank spin chains

TL;DR

This paper develops a comprehensive framework using Functional Separation of Variables and Character Projection to compute form-factors, matrix elements, and operators in higher rank sl(N) spin chains, enabling advanced correlation function calculations.

Contribution

It introduces a novel method combining FSoV and Character Projection to derive determinant formulas for form-factors and operators in higher rank spin chains, including the B operator.

Findings

Derived determinant formulas for form-factors of principal operators.

Proved the principal operators generate the full Yangian algebra.

Represented operators in SoV bases for correlation function computations.

Abstract

Integrable sl(N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin`s SoV B operator. As a result, we are able to derive the B operator for sl(N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.