  Using an improved estimator in the loop-cluster algorithm, we investigate the
constraint effective potential of the magnetization in the spin $\tfrac{1}{2}$
quantum XY model. The numerical results are in excellent agreement with the
predictions of the corresponding low-energy effective field theory. After its
low-energy parameters have been determined with better than permille precision,
the effective theory makes accurate predictions for the constraint effective
potential which are in excellent agreement with the Monte Carlo data. This
shows that the effective theory indeed describes the physics in the low-energy
regime quantitatively correctly.
