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.
The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.
We revisit our initial derivations of the exact 2-spinon
S2 and 4-spinon
S4 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
S2 contributes not 72% as was previously thought but 36% to the total DSF. We also calculated the contribution of
S4 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
S4. Furthermore, we prove that its claim that our initial derivation of
S4 was also incorrect in its dependency on the spectral parameters is incorrect because fixing the momentun transfer
k up to
2π 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
S4 is periodic in
k with period
2π. Also in this paper we derive
Sn for general
n by following a different approach compared to our initial derivation of
S4. Although for
S4 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
S4 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 discuss some properties of the exact four-spinon dynamical correlation function
S4 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
S2. We show that the overall shapes are quite similar, even though the expression of
S4 is much more involved than that of
S2. We finish with concluding remarks.
We consider the transverse dynamical structure factor of the anisotropic Heisenberg spin-1/2 chain ( XXZ model) in the gapped antiferromagnetic regime (Δ>1). Specializing to the case of zero field, we use two independent approaches based on integrability (one valid for finite size, the other for the infinite lattice) to obtain the exact two-spinon part of this correlator. We discuss in particular its asymmetry with respect to the π/2-momentum line, its overall anisotropy dependence, and its contribution to sum rules.
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.