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arXiv:2006.01417 · Symmetry Integrability and Geometry Methods and Applications

New Separation of Variables for the Classical XXX and XXZ Heisenberg Spin Chains

Universit&, Guido Magnano, #224, Italia degli Studi di Torino, T. Skrypnyk

Published Jun 2, 2020• 0 citations

Abstract

We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix.We show that for the case of such twist matrices one can interchange the role of classical separating functions A(u) and B(u) and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin.We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.

DOI: 10.3842/sigma.2020.047

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We study the Heisenberg XXX spin chain with twisted periodic boundary conditions with a defective twist matrix. Using quantum separation of variables, we show that the generalized eigenvectors of this system can be characterized in terms of a modified TQ equation that we have named the “Jordanian TQ equation.” We also show that the wave function in the separated basis is only separable for true eigenvectors, while the wave function for generalized eigenvectors of higher rank is given instead by a sum over weak compositions.

We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.

We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.