Rational \(Q\)-systems at root of unity. I: Closed chains. (English) Zbl 07912424
Summary: 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.
MSC:
| 82B23 | Exactly solvable models; Bethe ansatz |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82D40 | Statistical mechanics of magnetic materials |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
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