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arXiv:1212.1575 · Journal of Mathematical Physics

On Baxter's Q operator of the higher spin XXZ chain at the Razumov-Stroganov point

Kohei Motegi

Published Jun 1, 2013• 4 citations
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Abstract

Based on the conjecture for the exact eigenvalue of the transfer matrix of the higher half-integer spin XXZ chain at the Razumov-Stroganov point, we evaluate the corresponding Baxter's Q operator in closed form by solving the TQ equation. The combination of the Q operators on the “right side” and the “wrong side” is shown to produce the hierarchy of functional relations.

DOI
10.1063/1.4809931

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Based on the conjecture for the exact eigenvalue of the transfer matrix of the higher half-integer spin XXZ chain at the Razumov-Stroganov point, we evaluate the corresponding Baxter's Q operator in closed form by solving the TQ equation. The combination of the Q operators on the “right side” and the “wrong side” is shown to produce the hierarchy of functional relations.
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.
Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity.
We construct Q-operators for the open spin-12 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang–Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.
We provide two methods of producing the QQQ-operator of XXZ spin chain of higher spin, one for NNNth root-of-unity qqq with odd NNN and another for a general qqq, as the generalization of those known in the six-vertex model. In the root-of-unity case, we discuss the functional relations involving the constructed QQQ-operator for the symmetry study of the theory. The QQQ-operator of XXZ chain of higher spin for a generic qqq is constructed by extending Baxter's argument in spin-1/2 case for the six-vertex QQQ-operator.