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arXiv:2609.16146 · Physical review. D/Physical review. D.

Separation of variables for nondiagonalizable models: XXX spin chain with twisted boundary conditions

Juan Miguel Nieto García

Published Oct 9, 2026• 0 citations
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Abstract

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.

DOI
10.1103/xlkt-rqhf

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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.
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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.
Abstract In this paper we apply our new separation of variables approach to completely characterize the transfer matrix spectrum for quantum integrable lattice models associated to fundamental evaluation representations of with quasi-periodic boundary conditions. We consider here the case of generic deformations associated to a parameter q which is not a root of unity. The separation of variables (SoV) basis for the transfer matrix spectral problem is generated by using the action of the transfer matrix itself on a generic co-vector of the Hilbert space, following the general procedure described in our paper (Maillet and Niccoli 2018 J. Math. Phys . 59 091417). Such a SoV construction allows to prove that for general values of the parameters defining the model the transfer matrix is diagonalizable and with simple spectrum for any twist matrix which is also diagonalizable with simple spectrum. Then, using together the knowledge of such a SoV basis and of the fusion relations satisfied by the hierarchy of transfer matrices, we derive a complete characterization of the transfer matrix eigenvalues and eigenvectors as solutions of a system of polynomial equations of order n + 1. Moreover, we show that such a SoV discrete spectrum characterization is equivalently reformulated in terms of a finite difference functional equation, the quantum spectral curve equation, under a proper choice of the set of its solutions. A construction of the associated Q -operator induced by our SoV approach is also presented.
We give explicit formulas for the eigenvectors of the transfer matrix of the Baxter-Bazhanov-Stroganov (BBS) model (N-state spin model) with fixed-spin boundary conditions. We obtain these formulas from the formulas for the eigenvectors of the periodic BBS model by a limit procedure. The latter formulas were derived in the framework of Sklyanin’s method of separation of variables. In the case of fixed-spin boundaries, we solve the corresponding T-Q Baxter equations for the functions of separated variables explicitly. As a particular case, we obtain the eigenvectors of the Hamiltonian of the Ising-like ℤN quantum chain model.
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