XXZ Bethe states as highest weight vectors of the $sl_2$ loop algebra at roots of unity
| dc.creator | Deguchi, Tetsuo | |
| dc.date | 2002-12-10 | |
| dc.date | 2003-03-22 | |
| dc.date.accessioned | 2026-07-07T02:48:38Z | |
| dc.date.available | 2026-07-07T02:48:38Z | |
| dc.description | We prove some part of the conjecture that regular Bethe ansatz eigenvectors of the XXZ spin chain at roots of unity are highest weight vectors of the $sl_2$ loop algebra. Here $q$ is related to the XXZ anisotropic coupling $Δ$ by $Δ=(q+q^{-1})/2$, and it is given by a root of unity, $q^{2N}=1$, for a positive integer $N$. We show that regular XXZ Bethe states are annihilated by the generators ${\bar x}_k^{+}$'s, for any $N$. We discuss, for some particular cases of N=2, that regular XXZ Bethe states are eigenvectors of the generators of the Cartan subalgebra, ${\bar h}_k$'s. Here the loop algebra $U(L(sl_2))$ is generated by ${\bar x}_k^{\pm}$ and ${\bar h}_k$ for $k \in {\bf Z}$, which are the classical analogues of the Drinfeld generators of the quantum loop algebra $U_q(L(sl_2))$. A representation of $U(L(sl_2))$ is called highest weight if it is generated by a vector $Ω$ which is annihilated by the generators ${\bar x}_k^{+}$'s and such that $Ω$ is an eigenvector of the ${\bar h}_k$'s. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of $U(L(sl_2))$. | |
| dc.description | An almost completed note, 13 pages, no figures | |
| dc.identifier | https://arxiv.org/abs/cond-mat/0212217 | |
| dc.identifier | http://arxiv.org/abs/cond-mat/0212217 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/20484 | |
| dc.subject | Statistical Mechanics | |
| dc.title | XXZ Bethe states as highest weight vectors of the $sl_2$ loop algebra at roots of unity | |
| dc.type | text |