Bethe Ansatz Equations for the Broken ZNZ_{N}-Symmetric Model

We obtain the Bethe Ansatz equations for the broken ${\bf Z}_{N}$-symmetric model by constructing a functional relation of the transfer matrix of $L$-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis.Comment: 43 pages, latex, 11 figure

Auxiliary matrices on both sides of the equator

The spectra of previously constructed auxiliary matrices for the six-vertex model at roots of unity are investigated for spin-chains of even and odd length. The two cases show remarkable differences. In particular, it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to Baxter's TQ-equation corresponding to the Bethe ansatz equations above and below the equator. In contrast, one finds for even spin-chains only one linear independent solution and complete strings. The other main result is the proof of a previous conjecture on the degeneracies of the six-vertex model at roots of unity. The proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight-vertex model conjectured by Fabricius and McCoy.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some typos correcte

A Generalized Q-operator for U_q(\hat(sl_2)) Vertex Models

In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.Comment: 22 pages, Latex2e. This replacement is a revised version that includes a simple explicit expression for the Q matrix for the 6-vertex mode

On kernel engineering via Paley–Wiener

A radial basis function approximation takes the form $$s(x)=\sum_{k=1}^na_k\phi(x-b_k),\quad x\in {\mathbb{R}}^d,$$ where the coefficients a 1,…,a n are real numbers, the centres b 1,…,b n are distinct points in ℝ d , and the function φ:ℝ d →ℝ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which φ is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ψ=μ φ is a function of compact support, and when φ is polyharmonic. The novelty of this construction is its use of the Paley–Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ψ, so providing a new form of kernel engineering

Auxiliary matrices for the six-vertex model at roots of 1 and a geometric interpretation of its symmetries

The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$ a three parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged center of the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a three-dimensional complex hypersurface which remains invariant under the action of an infinite-dimensional group of analytic transformations, called the quantum coadjoint action.Comment: 52 pages, TCI LaTex, v2: equation (167) corrected, two references adde

Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz

We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q-operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyze the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.Comment: 32 page

Ground State of the Quantum Symmetric Finite Size XXZ Spin Chain with Anisotropy Parameter Δ=1/2\Delta = {1/2}

We find an analytic solution of the Bethe Ansatz equations (BAE) for the special case of a finite XXZ spin chain with free boundary conditions and with a complex surface field which provides for $U_q(sl(2))$ symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter $\Delta = {1/2}$ corresponding to $q^3 = -1$.Comment: 6 page

XXZ Bethe states as highest weight vectors of the sl2sl_2 loop algebra at roots of unity

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 {\it regular Bethe roots}. We call a nonzero Bethe ansatz eigenvector with regular Bethe roots a {\it regular Bethe state}.Comment: 40pages; revised versio

On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain

We consider the problem of computing form factors of the massless XXZ Heisenberg spin-1/2 chain in a magnetic field in the (thermodynamic) limit where the size M of the chain becomes large. For that purpose, we take the particular example of the matrix element of the third component of spin between the ground state and an excited state with one particle and one hole located at the opposite ends of the Fermi interval (umklapp-type term). We exhibit its power-law decrease in terms of the size of the chain M, and compute the corresponding exponent and amplitude. As a consequence, we show that this form factor is directly related to the amplitude of the leading oscillating term in the long-distance asymptotic expansion of the two-point correlation function of the third component of spin.Comment: 28 page