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POLYNOMIAL SOLUTIONS OF qKZ EQUATION AND GROUND STATE OF XXZ SPIN CHAIN AT ∆ = −1/2

A. V. Razumov, Yu. G. Stroganov, Paul Zinn-Justin

Published Aug 4, 2013• 30 citations
OpenAlex

Abstract

Abstract. Integral formulae for polynomial solutions of the quantum Knizhnik– Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to e ±2πi/3 and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit it is a ground state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter ∆ equal to −1/2 and odd number of sites. The obtained integral representations for the components of this eigenvector allow to prove some conjectures on its properties formulated earlier. A new conjecture relating the ground state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved. 1.

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Polynomial solutions ofqKZ equation and ground state of XXZ spin chain at Δ = −1/2

A V Razumov, Yu G Stroganov, Paul Zinn-Justin

Integral formulae for polynomial solutions of the quantum Knizhnik–Zamolodchikov equations associated with the R -matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit, it is a ground-state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Δ equal to −1/2 and an odd number of sites. The obtained integral representations for the components of this eigenvector allow us to prove some conjectures on its properties formulated earlier. A new statement relating the ground-state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved.

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Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

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Loop symmetry of integrable vertex models at roots of unity

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It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$. These new symmetries have been linked to an $U(A^{(1)}_1)$ invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups $U_q(\\hat g)$ into account. Here $\\hat g$ is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of $U(\\hat{g})$. For the simplest case $\\hat g=A_1^{(1)}$ the discussion is also extended to even primitive roots of unity.

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Thesl2loop algebra symmetry of the twisted transfer matrix of the six-vertex model at roots of unity

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We discuss a family of operators which commute or anti-commute with the twisted transfer matrix of the six-vertex model at q being roots of unity, q 2 N = 1. The operators commute with the Hamiltonian of the XXZ spin chain under the twisted boundary conditions, and they are also valid for the inhomogeneous case. For the case of the anti-periodic boundary conditions, we show explicitly that the operators generate the sl 2 loop algebra in the sector of the total spin operator S Z ≡ N /2(mod N ). The infinite-dimensional symmetry leads to exponentially-large spectral degeneracies, as shown for the periodic boundary conditions (Deguchi T, Fabricius K and McCoy B M 2001 J. Stat. Phys. 102 701). Furthermore, we derive explicitly the sl 2 loop algebra symmetry for the periodic XXZ spin chain with an odd number of sites in the sector S Z ≡ N /2(mod N ) when q is a primitive N th root of unity with N odd. Interestingly, in the case of N = 3, various conjectures of combinatorial formulae for the XXZ spin chain with odd sites have been given by Stroganov and other authors. We also note a connection to the spectral degeneracies of the eight-vertex model.

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Christian Korff

The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s = 1/2 Heisenberg chain at roots of unity are investigated. Two conjectures are formulated both of which are proved for N = 3 and are verified numerically for several examples with N > 3. The first conjecture identifies an Abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeros of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N = 3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra . This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.

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Spin chains and combinatorics

A. V. Razumov, Yu. G. Stroganov

In this paper we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in our previous article (Stroganov Yu G 2001 J. Phys. A: Math. Gen. 34 L179-85 ). As it turned out, for a special value of the asymmetry parameter Δ = -1/2 the Hamiltonian of the system has an eigenvalue, which is exactly proportional to the number of sites E = -3 N /2. Using Mathematica we have found explicitly the corresponding eigenvectors for N ⩽17. The obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector. We make a lot of conjectures concerning the correlations of the model. Many remarkable relations between the wavefunction components are noted. It turns out, for example, that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices.

arXiv:cond-mat/0012141•Journal of Physics A Mathematical and General•Mar 30, 2001•217 citations
Abstract. Integral formulae for polynomial solutions of the quantum Knizhnik– Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to e ±2πi/3 and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit it is a ground state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter ∆ equal to −1/2 and odd number of sites. The obtained integral representations for the components of this eigenvector allow to prove some conjectures on its properties formulated earlier. A new conjecture relating the ground state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved. 1.
Integral formulae for polynomial solutions of the quantum Knizhnik–Zamolodchikov equations associated with the R -matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit, it is a ground-state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Δ equal to −1/2 and an odd number of sites. The obtained integral representations for the components of this eigenvector allow us to prove some conjectures on its properties formulated earlier. A new statement relating the ground-state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved.
Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.
It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter (q+q−1)/2(q+q^{-1})/2(q+q−1)/2 is evaluated at roots of unity qN=1q^{N}=1qN=1. These new symmetries have been linked to an U(A1(1))U(A^{(1)}_1)U(A1(1)​) invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups Uq(hatg)U_q(\\hat g)Uq​(hatg) into account. Here hatg\\hat ghatg is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of U(hatg)U(\\hat{g})U(hatg). For the simplest case hatg=A1(1)\\hat g=A_1^{(1)}hatg=A1(1)​ the discussion is also extended to even primitive roots of unity.
We discuss a family of operators which commute or anti-commute with the twisted transfer matrix of the six-vertex model at q being roots of unity, q 2 N = 1. The operators commute with the Hamiltonian of the XXZ spin chain under the twisted boundary conditions, and they are also valid for the inhomogeneous case. For the case of the anti-periodic boundary conditions, we show explicitly that the operators generate the sl 2 loop algebra in the sector of the total spin operator S Z ≡ N /2(mod N ). The infinite-dimensional symmetry leads to exponentially-large spectral degeneracies, as shown for the periodic boundary conditions (Deguchi T, Fabricius K and McCoy B M 2001 J. Stat. Phys. 102 701). Furthermore, we derive explicitly the sl 2 loop algebra symmetry for the periodic XXZ spin chain with an odd number of sites in the sector S Z ≡ N /2(mod N ) when q is a primitive N th root of unity with N odd. Interestingly, in the case of N = 3, various conjectures of combinatorial formulae for the XXZ spin chain with odd sites have been given by Stroganov and other authors. We also note a connection to the spectral degeneracies of the eight-vertex model.
The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s = 1/2 Heisenberg chain at roots of unity are investigated. Two conjectures are formulated both of which are proved for N = 3 and are verified numerically for several examples with N > 3. The first conjecture identifies an Abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeros of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N = 3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra . This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.
In this paper we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in our previous article (Stroganov Yu G 2001 J. Phys. A: Math. Gen. 34 L179-85 ). As it turned out, for a special value of the asymmetry parameter Δ = -1/2 the Hamiltonian of the system has an eigenvalue, which is exactly proportional to the number of sites E = -3 N /2. Using Mathematica we have found explicitly the corresponding eigenvectors for N ⩽17. The obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector. We make a lot of conjectures concerning the correlations of the model. Many remarkable relations between the wavefunction components are noted. It turns out, for example, that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices.