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arXiv:cond-mat/9802271

Exact four-spinon dynamical correlation function in isotropic Heisenberg model

Abdessamad Abada, A. Hamid Bougourzi, S. Seba, B. Si-Lakhal

Published Feb 25, 1998• 1 citation
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Abstract

We discuss some properties of the exact four-spinon dynamical correlation function $S_4$ in the antiferromagnetic spin 1/2 $XXX$-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 $k$ for fixed values of the neutron energy $ω$ and compare it to the one corresponding to the exact two-spinon dynamical correlation function $S_2$. We show that the overall shapes are quite similar, even though the expression of $S_4$ is much more involved than that of $S_2$. We finish with concluding remarks.

DOI
10.48550/arxiv.cond-mat/9802271

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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.
Using Monte Carlo integration methods, we describe the behavior of the exact four-spinon dynamic structure function S4S_{4}S4​ in the antiferromagnetic spin 1/2 Heisenberg quantum spin chain as a function of the neutron energy ωωω and momentum transfer kkk. We also determine the four-spinon continuum, the extent of the region in the (k,ω)(k,ω)(k,ω) plane outside which S4S_{4}S4​ is identically zero. In each case, the behavior of S4S_{4}S4​ is shown to be consistent with the four-spinon continuum and compared to the one of the exact two-spinon dynamic structure function S2S_{2}S2​. Overall shape similarity is noted.
Using Monte Carlo integration methods, we describe the behaviour of the exact four-spinon dynamic structure function S 4 in the antiferromagnetic spin Heisenberg quantum spin chain as a function of the neutron energy ω and momentum transfer k . We also determine the four-spinon continuum, the extent of the region in the ( k , ω) plane outside which S 4 is identically zero. In each case, the behaviour of S 4 is shown to be consistent with the four-spinon continuum and compared to that of the exact two-spinon dynamic structure function S 2 . Overall shape similarity is noted.
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 revisit our initial derivations of the exact 2-spinon S2S_2S2​ and 4-spinon S4S_4S4​ dynamical structure factors (DSF). First we show that the latter derivations had normalization factors that are twice and quadriple the correct ones respectively. This means that S2S_2S2​ contributes not 72% as was previously thought but 36% to the total DSF. We also calculated the contribution of S4S_4S4​ to be between 18% and 20% and not 27% as was calculated by Caux and Hagemans. In fact we show that the latter reference had also the normalization factor twice the correct value and had it done the numeric integrations correctly it would have obtained a contribution between 36% and 40% for S4S_4S4​. Furthermore, we prove that its claim that our initial derivation of S4S_4S4​ was also incorrect in its dependency on the spectral parameters is incorrect because fixing the momentun transfer kkk up to 2π2π2π as the latter reference did to justify its claim only amounts to mutltiplying the overall factor by 2 because as we will prove in this paper S4S_4S4​ is periodic in kkk with period 2π2π2π. Also in this paper we derive SnS_nSn​ for general nnn by following a different approach compared to our initial derivation of S4S_4S4​. Although for S4S_4S4​ both the new derivation and the initial one lead to equivalent formulas that are expressed differently, the new form presented in this paper is much more elegant and compact and also reveals new hidden and nontrivial symmetries which substantially simplify the numeric evaluation of S4S_4S4​ and its sum rules. Moreover based on the results of this paper we propose a simple approximation to the total DSF of the Heisenberg model. Finally we comment on how our work might resolve the discrepancy between the exact theoretic results and experiemntal data as reported by Zaliznyak et al.
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.
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.