  Using the reflection positivity method we provide the rigorous proof of the
existence of long range magnetic order for the XY model on the honeycomb
lattice for large spins $S\ge 2$. This stays in contrast with the result
obtained using the {\it same} method but on the square lattice -- which gives a
stable long-range order for spins $S\ge 1$. We suggest that the difference
between these two cases stems from the enhanced quantum spin fluctuations on
the honeycomb lattice. Using linear spin-wave theory we show that the enhanced
fluctuations are due to the overall much higher kinetic energy of the spin
waves on the honeycomb lattice (with Dirac points) than on the square lattice
(with good nesting properties).
