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arXiv:cond-mat/0506613

States with v1=lambda, v2=-lambda and reciprocal equations in the six-vertex model

M. J. Rodríguez-Plaza

Published Jun 23, 2005• 1 citation
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Abstract

The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v1,...,vn, the zeros of the function Q(v). A peculiar class of eigenvalues are those in which two of the constants v1, v2 are equal to lambda, -lambda, with Delta=-cosh lambda and Delta related to the Boltzmann weights of the six-vertex model by the usual combination Delta=(a^2+b^2-c^2)/2 a b. The eigenvectors associated to these eigenvalues are Bethe states (although they seem not). We count the number of such states (eigenvectors) for n=2,3,4,5 when N, the columns in a row of a square lattice, is arbitrary. The number obtained is independent of the value of Delta, but depends on N. We give the explicit expression of the eigenvalues in terms of a,b,c (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary N in the blocks n considered. For real a,b,c such eigenvalues are real.

DOI
10.48550/arxiv.cond-mat/0506613

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The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of n constants v1,...,vn, the zeros of the function Q(v). A peculiar class of eigenvalues are those in which two of the constants v1, v2 are equal to lambda, -lambda, with Delta=-cosh lambda and Delta related to the Boltzmann weights of the six-vertex model by the usual combination Delta=(a^2+b^2-c^2)/2 a b. The eigenvectors associated to these eigenvalues are Bethe states (although they seem not). We count the number of such states (eigenvectors) for n=2,3,4,5 when N, the columns in a row of a square lattice, is arbitrary. The number obtained is independent of the value of Delta, but depends on N. We give the explicit expression of the eigenvalues in terms of a,b,c (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary N in the blocks n considered. For real a,b,c such eigenvalues are real.
States with v1=λ,v2=−λv_1 = \lambda, v_2 = -\lambdav1​=λ,v2​=−λ and reciprocal equations in the six-vertex model(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )
The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of nnn constants v1,v_{1},v1​, …,\ldots,…, vnv_{n}vn​ , the zeros of the function Q(v)Q(v)Q(v) .A peculiar class of eigenvalues are those in which two of the constants v1,v_{1},v1​, v2v_{2}v2​ are equal to λ,\lambda,λ, −λ-\lambda−λ , with Δ=−cosh⁡\Delta=-\coshΔ=−cosh A and Δ\DeltaΔ related to the Boltzmann weights of the six-vertex model by the usual combination Δ=(a2+b2−c2)/2ab\Delta=(a^{2}+b^{2}-c^{2})/2abΔ=(a2+b2−c2)/2ab .The eigenvectors associated to these eigenvalues are Bethe states (although they seem not).We count the number of such states (eigenvectors) for n=2,3,4,5n=2,3,4,5n=2,3,4,5 when NNN , the columns in a row of a square lattice, is arbitrary.The number obtained is independent of the value of Δ\DeltaΔ , but depends on NNN .We give the explicit expres- sion of the eigenvalues in terms of a,a,a, b,b,b, ccc (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary NNN in the blocks nnn considered.For real a,a,a, b,cb,cb,c such eigenvalues are real.
It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz and both a simple component and its square norm are expressed in terms of the Izergin–Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.
The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)
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 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.
Abstract This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.