In the context of the antiferromagnetic spin 1/2 Heisenberg quantum spin chain (XXX model), we estimate the contribution of the exact four-spinon dynamic structure factor S 4 by calculating a number of sum rules the total dynamic structure factor S is known to satisfy exactly. These sum rules are: the static susceptibility, the integrated intensity, the total integrated intensity, the first frequency moment and the nearest-neighbor correlation function. We find that the contribution of S 4 is between 1% and 2.5%, depending on the sum rule, whereas the contribution of the exact two-spinon dynamic structure factor S 2 is between 70% and 75%. This is consistent with the expected scattering weight of states from outside the spin-wave continuum. The calculations are numerical and Monte Carlo based. Good statistics are obtained.

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We derive the exact expression of the four-spinon contribution S4 to the dynamical correlation function S of the spin 1 2 isotropic (XXX) Heisenberg model in the antiferromagnetic regime using the quantum group symmetry of the model. We first give the exact expression for the n-spinon contribution in the form of contour integrals and display known results regarding the two-spinon contribution S2. Then we specialize the n-spinon formula to the case n = 4 and compute three sum rules for S4 that the total S is known to satisfy exactly. These are: the total integrated intensity, the first frequency moment and the nearest-neighbor correlation function. We find that S4 corrects only by a small amount the contribution from S2.