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arXiv:0704.3542 · Journal of Physics A Mathematical and Theoretical

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

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

Published Sep 11, 2007• 32 citations
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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 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.

DOI
10.1088/1751-8113/40/39/009

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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. 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.
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 give an integral representation for solutions to the quantized Knizhnik- Zamolodchikov equation (qKZ) associated with the Lie algebra glN+1gl_{N+1}glN+1​. Asymptotic solutions to qKZ are constructed. The leading term of an asymptotic solution is the Bethe vector -- an eigenvector of the transfer-matrix of a quantum spin chain model. We show that the norm of the Bethe vector is equal to the product of the Hessian of a suitable function and an explicitly written rational function. This formula is a generalization of the Gaudin-Korepin formula for a norm of the Bethe vector. We show that, generically, the Bethe vectors form a base for the gl2gl_2gl2​ case.
Abstract The nineteen-vertex model of Fateev and Zamolodchikov on a periodic lattice with an anti-diagonal twist is investigated. Its inhomogeneous transfer matrix is shown to have a simple eigenvalue, with the corresponding eigenstate displaying intriguing combinatorial features. Similar results were previously found for the same model with a diagonal twist. The eigenstate for the anti-diagonal twist is explicitly constructed using the quantum separation of variables technique. A number of sum rules and special components are computed and expressed in terms of Kuperberg’s determinants for partition functions of the inhomogeneous six-vertex model. The computations of some components of the special eigenstate for the diagonal twist are also presented. In the homogeneous limit, the special eigenstates become eigenvectors of the Hamiltonians of the integrable spin-one XXZ chain with twisted boundary conditions. Their sum rules and special components for both twists are expressed in terms of generating functions arising in the weighted enumeration of various symmetry classes of alternating sign matrices (ASMs). These include half-turn symmetric ASMs, quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and horizontally perverse ASMs and double U-turn ASMs. As side results, new determinant and pfaffian formulas for the weighted enumeration of various symmetry classes of alternating sign matrices are obtained.
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.