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

Abstract

Integrable $$ \mathfrak{sl} $$ 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 $$ \mathfrak{sl} $$ sl (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

Current mean values in the XYZ model

“…Numerical solutions to the Bethe equations for the XYZ model were considered previously in [59], in the framework of the so called Off-Diagonal Bethe Ansatz. However, to the best of our knowledge, in that paper the explicit values of the Bethe roots were only given for a special choice of η, which satisfies the constraint (56). Our result works for that special choice as well, but here we consider the general case.…”

“…Correlation functions in the higher spin generalizations of the Heisenberg chain have been considered for example in [44][45][46][47]. Correlation functions in higher rank spin chains solvable by the nested Bethe Ansatz were considered for example in [48][49][50][51][52][53] and also in [54][55][56]. The results of [19] for the mean values of current operators in the SU(3)-symmetric model also belong to this list, together with certain factorized mean values found in [57] via lattice generalizations of the T T -deformation known from quantum field theories.…”

A study of form factors in relativistic mixed-flux AdS3

“…In particular the bound state axiom and JHEP03(2024)082 the kinematical singularities would have to be carefully analysed. It would also be interesting to approach the problem using Lukyanov's method [107][108][109][110][111][112][113][114][115][122][123][124] and see whether we obtain a different result. We plan to come back to these issues in future work.…”

Fast QSC solver: tool for systematic study of $$ \mathcal{N} $$ = 4 Super-Yang-Mills spectrum

“…The Q-functions are not all independent and are related by the QQ-relations described below. In the well studied cases of the spin-chains the Q-functions (or Baxter polynomials for the compact symmetry group) are the building blocks for the wave-functions of the states in a separation of variables basis (for recent developments see [80][81][82][83][84][85][86][87][88][89][90]). In particular, they contain all the quantum numbers of the state in the large-u asymptotic, which includes, in particular, the exact scaling dimension ∆.…”