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数理解析研究所講究録

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

Published Apr 1, 2006• 0 citations
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Abstract

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.

Related papers

On the Degenerate Multiplicity of the sl 2 Loop Algebra for the 6V Transfer Matrix at Roots of Unity

Tetsuo Deguchi

We review the main result of cond-mat/0503564.The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl 2 loop algebra symmetry if the q parameter is given by a root of unity, q 2N 0 = 1, for an integer N .We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight d± k , which leads to evaluation parameters a j .If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.

arXiv:cond-mat/0602427•Symmetry Integrability and Geometry Methods and Applications•Feb 17, 2006•1 citation

XXZ Bethe states as highest weight vectors of the $sl_2$ loop algebra at roots of unity

Tetsuo Deguchi

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 $U(L(sl_2))$.

arXiv:cond-mat/0212217•Dec 10, 2002•4 citations

Regular XXZ Bethe states at roots of unity as highest weight vectors of thesl2loop algebra

Tetsuo Deguchi

We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the sl 2 loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator S Z , and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities regular Bethe roots . We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a regular Bethe state .

arXiv:cond-mat/0503564•Journal of Physics A Mathematical and Theoretical•Jun 19, 2007•28 citations

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 )

哲生 出口

数理解析研究所講究録•Apr 1, 2006•0 citations

Evaluation Parameters and Bethe Roots for the Six-Vertex Model at Roots of Unity

K. Fabricius, Barry M. McCoy

We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .

arXiv:cond-mat/0108057•Birkhäuser Boston eBooks•Jan 1, 2002•43 citations

Bethe’s equation is incomplete for the XXZ model at roots of unity

K. Fabricius, Barry M. McCoy

We demonstrate for the six vertex and XXZ model parameterized by $\Delta=-(q+q^{-1})/2\neq \pm 1$ that when q^{2N}=1 for integer $N\geq 2$ the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl_2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl_2 loop algebra can be used to complete this specification.

•Jan 1, 2008•94 citations
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.
We review the main result of cond-mat/0503564.The Hamiltonian of the XXZ spin chain and the transfer matrix of the six-vertex model has the sl 2 loop algebra symmetry if the q parameter is given by a root of unity, q 2N 0 = 1, for an integer N .We discuss the dimensions of the degenerate eigenspace generated by a regular Bethe state in some sectors, rigorously as follows: We show that every regular Bethe ansatz eigenvector in the sectors is a highest weight vector and derive the highest weight d± k , which leads to evaluation parameters a j .If the evaluation parameters are distinct, we obtain the dimensions of the highest weight representation generated by the regular Bethe state.
XXZ Bethe states as highest weight vectors of the sl2sl_2sl2​ loop algebra at roots of unity
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 sl2sl_2sl2​ loop algebra. Here qqq is related to the XXZ anisotropic coupling ΔΔΔ by Δ=(q+q−1)/2Δ=(q+q^{-1})/2Δ=(q+q−1)/2, and it is given by a root of unity, q2N=1q^{2N}=1q2N=1, for a positive integer NNN. We show that regular XXZ Bethe states are annihilated by the generators xˉk+{\bar x}_k^{+}xˉk+​'s, for any NNN. We discuss, for some particular cases of N=2, that regular XXZ Bethe states are eigenvectors of the generators of the Cartan subalgebra, hˉk{\bar h}_khˉk​'s. Here the loop algebra U(L(sl2))U(L(sl_2))U(L(sl2​)) is generated by xˉk±{\bar x}_k^{\pm}xˉk±​ and hˉk{\bar h}_khˉk​ for k∈Zk \in {\bf Z}k∈Z, which are the classical analogues of the Drinfeld generators of the quantum loop algebra Uq(L(sl2))U_q(L(sl_2))Uq​(L(sl2​)). A representation of U(L(sl2))U(L(sl_2))U(L(sl2​)) is called highest weight if it is generated by a vector ΩΩΩ which is annihilated by the generators xˉk+{\bar x}_k^{+}xˉk+​'s and such that ΩΩΩ is an eigenvector of the hˉk{\bar h}_khˉk​'s. We also discuss the classical analogue of the Drinfeld polynomial which characterizes the irreducible finite-dimensional highest weight representation of U(L(sl2))U(L(sl_2))U(L(sl2​)).
We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at roots of unity is a highest weight vector of the sl 2 loop algebra, for some restricted sectors with respect to eigenvalues of the total spin operator S Z , and evaluate explicitly the highest weight in terms of the Bethe roots. We also discuss whether a given regular Bethe state in the sectors generates an irreducible representation or not. In fact, we present such a regular Bethe state in the inhomogeneous case that generates a reducible Weyl module. Here, we call a solution of the Bethe ansatz equations which is given by a set of distinct and finite rapidities regular Bethe roots . We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a regular Bethe state .
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 )
We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .
We demonstrate for the six vertex and XXZ model parameterized by Δ=−(q+q−1)/2≠±1\Delta=-(q+q^{-1})/2\neq \pm 1Δ=−(q+q−1)/2=±1 that when q^{2N}=1 for integer N≥2N\geq 2N≥2 the Bethe's ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl_2 loop algebra symmetry group of the model. Therefore in this case the Bethe's ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl_2 loop algebra can be used to complete this specification.