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arXiv:1910.07805 · Journal of High Energy Physics

On generalized Q-systems

Zoltán Bajnok, Etienne Granet, Jesper Lykke Jacobsen, Rafael I. Nepomechie

Published Mar 1, 2020• 32 citations
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Abstract

Abstract We formulate Q-systems for the closed XXZ, open XXX and open quantum- group-invariant XXZ quantum spin chains. Polynomial solutions of these Q-systems can be found efficiently, which in turn lead directly to the admissible solutions of the corresponding Bethe ansatz equations.
DOI
10.1007/jhep03(2020)177

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We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.
We consider the -invariant open spin-1/2 XXZ quantum spin chain of finite length N . For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley–Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting -modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case ( ), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic ( ) limit. Numerical solutions of the Bethe equations up to N = 8 are presented. Our results are consistent with the Bethe ansatz solution being complete.