  We consider the Heisenberg antiferromagnet on the square lattice with S=1/2
and very weak easy-plane exchange anisotropy; by means of the quantum Monte
Carlo method, based on the continuous-time loop algorithm, we find that the
thermodynamics of the model is highly sensitive to the presence of tiny
anisotropies and is characterized by a crossover between isotropic and planar
behaviour. We discuss the mechanism underlying the crossover phenomenon and
show that it occurs at a temperature which is characteristic of the model. The
expected Berezinskii-Kosterlitz-Thouless transition is observed below the
crossover: a finite range of temperatures consequently opens for experimental
detection of non-critical 2D XY behaviour. Direct comparison is made with
uniform susceptibility data relative to the S=1/2 layered antiferromagnet
Sr2CuO2Cl2.
