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Journal of Physics A Mathematical and General

The XXZ model with Beraha values

Bo-Yu Hou, Bo-Yuan Hou, Zhong-Qi Ma

Published Jun 21, 1991• 5 citations
Publisher

Abstract

The authors study an XXZ model of spin-1/2 chains with the quantum sl(2) symmetry, where they reduce the Hamiltonian equation to a set of coupled linear algebraic equations. When q is a root of unity, some eigenstates of the Hamiltonian coincide with each other, and a new state is obtained through a limit process from their difference. This eigenspace corresponds to a reducible but indecomposable representation (type 1) of the quantum sl(2) enveloping algebra.

DOI
10.1088/0305-4470/24/12/023

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The authors study an XXZ model of spin-1/2 chains with the quantum sl(2) symmetry, where they reduce the Hamiltonian equation to a set of coupled linear algebraic equations. When q is a root of unity, some eigenstates of the Hamiltonian coincide with each other, and a new state is obtained through a limit process from their difference. This eigenspace corresponds to a reducible but indecomposable representation (type 1) of the quantum sl(2) enveloping algebra.
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The sl2sl_2sl2​ loop algebra symmetry of the XXZ spin chain: an algorithm for the degeneracy of a regular Bethe state(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )
The Hamiltonian of the XXZ spin chain has the sl2sl_{2}sl2​ loop algebra symmetry if the qqq parameter is given by a root of unity, q02N=1q_{0}^{2N}=1q02N​=1 , for an integer NNN .We review an algorithm for determining the dimensions of the degenerate eigenspace generated by a given regular Bethe state in some sectors.The proof of the algorithm has been formulated in Ref.[3] as follows: We show in some sectors that regular Bethe ansatz eigenvectors are highest weight vectors and generate irreducible representations of the sl2sl_{2}sl2​ loop algebra; We prove that every finite-dimensional highest weight representation of the sl2sl_{2}sl2​ loop algebra is irreducible; We then derive the dimensions of the highest weight representation generated by a given regular Bethe state through the Drinfeld polynomial, which is expressed explicitly in terms of the Bethe roots.