SciPost Phys. 16, 129 (2024) ·
published 22 May 2024
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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.
SciPost Phys. 16, 113 (2024) ·
published 26 April 2024
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Bethe Ansatz equations for spin-s Heisenberg spin chain with s≥1 are significantly more difficult to analyze than the spin-$\tfrac{1}{2}$ case, due to the presence of repeated roots. As a result, it is challenging to derive extra conditions for the Bethe roots to be physical and study the related completeness problem. In this paper, we propose the rational Q-system for the XXX$_s$ spin chain. Solutions of the proposed Q-system give all and only physical solutions of the Bethe Ansatz equations required by completeness. This is checked numerically and proved rigorously. The rational Q-system is equivalent to the requirement that the solution and the corresponding dual solution of the TQ-relation are both polynomials, which we prove rigorously. Based on this analysis, we propose the extra conditions for solutions of the XXX$_s$ Bethe Ansatz equations to be physical.
