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arXiv:0912.2163

Eight-vertex model and Painlevé VI equation. II. Eigenvector results

Vladimir V. Mangazeev, Vladimir V. Bazhanov

Published Dec 11, 2009• 16 citations
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Abstract

We study a special anisotropic XYZ-model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with Picard's elliptic solutions of the Painlevé VI equation. Connections with other problems related to the eight-vertex model are briefly discussed.

DOI
10.48550/arxiv.0912.2163

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We study a special anisotropic XYZ-model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with Picard's elliptic solutions of the Painlevé VI equation. Connections with other problems related to the eight-vertex model are briefly discussed.
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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 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 We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T – Q functional equations, where the Q -functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.
The construction of creation operators of exact strings in eigenvectors of the eight-vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the ‘naïve’ string operator vanishes. In this paper, we present a proof of this conjecture. Furthermore, we show that for chains of odd length the string operator is either proportional to the symmetry operator S or vanishes depending on the precise form of the crossing parameter.
The construction of creation operators of exact strings in eigenvectors of the eight-vertex model at elliptic roots of unity of the crossing parameter which allow the generation of the complete set of degenerate eigenstates is based on the conjecture that the 'naïve' string operator vanishes. In this paper, we present a proof of this conjecture. Furthermore, we show that for chains of odd length the string operator is either proportional to the symmetry operator S or vanishes depending on the precise form of the crossing parameter.