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Nuclear Physics B

Null states of the irreducible representations of the Virasoro algebra and hidden symmetries of the finite XXZ Heisenberg chain

Uwe Grimm, V. Rittenberg

Published May 1, 1991• 9 citations
Publisher
DOI
10.1016/0550-3213(91)90361-z

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The $sl_2$ 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 )

Tetsuo Deguchi

The Hamiltonian of the XXZ spin chain has the $sl_{2}$ loop algebra symmetry if the $q$ parameter is given by a root of unity, $q_{0}^{2N}=1$ , for an integer $N$ .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 $sl_{2}$ loop algebra; We prove that every finite-dimensional highest weight representation of the $sl_{2}$ 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.

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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.