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arXiv:q-alg/9606010 · International Journal of Modern Physics B

From Quantum Affine Groups to the Exact Dynamical Correlation Function of the Heisenberg Model

A. Hamid Bougourzi, M. Couture, M. Kacir

Published Jan 20, 1997• 0 citations
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Abstract

The exact form factors of the Heisenberg models XXX and XXZ have been recently computed through the quantum affine symmetry of XXZ model in the thermodynamic limit. We use them to derive an exact formula for the contribution of two spinons to the dynamical correlation function of XXX model at zero temperature.
DOI
10.1142/s0217979297000125

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Inelastic neutron scattering experiments are commonly used to unveil how excitations on Heisenberg spin models play a role in dynamical correlations functions. For a certain class of materials, like CsCoCl3_33​ or CsCoBr3_33​ salts, it turns out that their magnetic properties are fairly well approximated by quasi-one dimensional XXZ models, which enjoy the property of quantum integrability. In these instances, one can in principle use their underlying algebraic structure to describe very precisely how excitations, the so-called spinons, participate in dynamical correlations functions. Even though the available theories (either algebraic Bethe ansatz or quantum group approach) provide all the needed physical quantities such as form factors, complete set of eigenstates and spectrum, it is typically a rather daunting task, however, to obtain sufficiently simple analytical expressions for computing Dynamical Structure Factors (DSFs), valuable, e.g., for parameters estimation based on experimental data. This is particularly the case for the longitudinal DSF of the XXZ model, which has eluded a formal mathematical treatment thus far. Using the quantum group approach, we present here an exact and simple expression of the 2-spinon longitudinal DSF and show our results to be consistent with the expected sum rules and the isotropic and Ising antiferromagnet limiting cases.
We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.