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arXiv:q-alg/9604019 · Physical review. B, Condensed matter

Exact two-spinon dynamical correlation function of the one-dimensional Heisenberg model

A. Hamid Bougourzi, M Couture, M. Kacir

Published Nov 1, 1996• 85 citations
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Abstract

We derive the exact contribution of two spinons to the dynamical correlation function of the spin-\textonehalf{} Heisenberg model. For this, we use the isotropic limits of the exact form factors that have been recently computed through the quantum affine symmetry of the anisotropic Heisenberg model $\mathrm{XXZ}$.

DOI
10.1103/physrevb.54.r12669

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We derive the exact contribution of two spinons to the dynamical correlation function of the spin-\textonehalf{} Heisenberg model. For this, we use the isotropic limits of the exact form factors that have been recently computed through the quantum affine symmetry of the anisotropic Heisenberg model XXZ\mathrm{XXZ}XXZ.
We discuss some properties of the exact four-spinon dynamical correlation function S4S_4S4​ in the antiferromagnetic spin 1/2 XXXXXXXXX-model, the expression of which we derived recently. We show that the region in which it is not identically zero is different from and larger than the spin-wave continuum. We describe its behavior as a function of the neutron momentum transfer kkk for fixed values of the neutron energy ωωω and compare it to the one corresponding to the exact two-spinon dynamical correlation function S2S_2S2​. We show that the overall shapes are quite similar, even though the expression of S4S_4S4​ is much more involved than that of S2S_2S2​. We finish with concluding remarks.
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 compute the exact four-spinon contribution to the zero-temperature dynamical structure factor of the spin-1/2 Heisenberg isotropic antiferromagnet in zero magnetic field, directly in the thermodynamic limit. We make use of the expressions for matrix elements of local spin operators obtained by Jimbo and Miwa using the quantum affine symmetry of the model, and of their adaptation to the isotropic case by Abada, Bougourzi and Si-Lakhal (correcting some overall factors). The four-spinon contribution to the first frequency moment sum rule at fixed momentum is calculated. This shows, as expected, that most of the remaining correlation weight above the known two-spinon part is carried by four-spinon states. Our results therefore provide an extremely accurate description of the exact structure factor.