Long-range order in the XY model on the honeycomb lattice

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).