  We employ an improved estimator to calculate the constraint effective
potential of the staggered magnetization in the spin $\tfrac{1}{2}$ quantum
Heisenberg model using a loop-cluster algorithm. The first and second moment of
the probability distribution of the staggered magnetization are in excellent
agreement with the predictions of the systematic low-energy magnon effective
field theory. We also compare the Monte Carlo data with the universal shape of
the constraint effective potential of the staggered magnetization and study its
approach to the convex effective potential in the infinite volume limit. In
this way the higher-order low-energy parameter $k_0$ is determined from a fit
to the numerical data.
