Sum rules for four-spinon dynamic structure factor in XXX model

In the context of the antiferromagnetic spin 12 Heisenberg quantum spin chain (XXX model), we estimate the contribution of the exact four-spinon dynamic structure factor S4 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 S4 is between 1% and 2.5%, depending on the sum rule, whereas the contribution of the exact two-spinon dynamic structure factor S2 is between 70% and 75%. The calculations are numerical and Monte Carlo based. Good statistics are obtained.