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arXiv:cond-mat/0012501 · Journal of Statistical Physics

Completing Bethe's Equations at Roots of Unity

K. Fabricius, Barry M. McCoy

Published Aug 1, 2001• 66 citations
arXivHTMLPDF
DOI
10.1023/a:1010372504158

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The solution of Bethe Ansatz equations for XXZ spin chain with the parameter q being a root of unity is infamously subtle. In this work, we develop the rational Q-system for this case, which offers a systematic way to find all physical solutions of the Bethe Ansatz equations at root of unity. The construction contains two parts. In the first part, we impose additional constraints to the rational Q-system. These constraints eliminate the so-called Fabricius-McCoy (FM) string solutions, yielding all primitive solutions. In the second part, we give a simple procedure to construct the descendant tower of any given primitive state. The primitive solutions together with their descendant towers constitute the complete Hilbert space. We test our proposal by extensive numerical checks and apply it to compute the torus partition function of the 6-vertex model at root of unity.
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.