Klaus Fabricius

The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity

We demonstrate that the six vertex model (XXZ spin chain) with Δ=(q+q-1)/2 and q2N=1 has an invariance under the loop algebra of sl2 which produces a special set of degenerate eigenvalues. For Δ=0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.

Bethe's Equation Is Incomplete for the XXZ Model at Roots of Unity

We demonstrate for the six vertex and XXZ model parameterized by Δ= −(q+q-1)/2≠±1 that when q2N=1 for integer N≥2 the Bethes ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl2 loop algebra symmetry group of the model. Therefore in this case the Bethes ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function. We discuss how the evaluation parameters of the finite dimensional representations of the sl2 loop algebra can be used to complete this specification.

New developments in the eight vertex model

We demonstrate that the Q matrix introduced in Baxters 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for Q(v) which both contains the Bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues.

Evaluation Parameters and Bethe Roots for the Six-Vertex Model at Roots of Unity

We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .

Completing Bethe's equations at roots of unity

In a previous paper we demonstrated that Bethes equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the sl2 loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethes ansatz.

New Developments in the Eight Vertex Model II. Chains of Odd Length

AbstractWe study the transfer matrix of the 8 vertex model with an odd number of lattice sites N. For systems at the root of unity pointsη=mK/L with m odd the transfer matrix is known to satisfy the famous ‘‘TQ’’ equation where Q(υ) is a specifically known matrix. We demonstrate that the location of the zeroes of this Q(υ) matrix is qualitatively different from the case of evenN and in particular they satisfy a previously unknown equation which is more general than what is often called ‘‘Bethe’s equation.’’ For the case of even m where no Q(υ) matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the TQ equation. The ground state for the particular case of η=2K/3 and N odd is investigated in detail.

SPIN DIFFUSION AND THE SPIN-1/2 XXZ CHAIN AT T= FROM EXACT DIAGONALIZATION

We study the long time behavior of the zz and xx time dependent autocorrelation function of the spin 1/2 XXZ chain at

A new Q-matrix in the Eight-Vertex Model

T=\infty

An elliptic current operator for the eight-vertex model

by exact diagonalizations on a chain of 16 sites. We find that the numerical results for the zz correlation are very well fit by the formula

TEMPERATURE-DEPENDENT SPATIAL OSCILLATIONS IN THE CORRELATIONS OF THE XXZ SPIN CHAIN

t^{-d}[A+Be^{-\gamma (t-t_0)}\cos \Omega(t-t_0)].