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

TL;DR

This paper rigorously proves the existence of long-range magnetic order in the XY model on the honeycomb lattice for large spins, highlighting the influence of lattice geometry and quantum fluctuations.

Contribution

It provides a rigorous proof of long-range order for the XY model on the honeycomb lattice for spins S ≥ 2, contrasting with previous results on the square lattice.

Findings

Long-range order exists for S ≥ 2 on the honeycomb lattice.

Quantum fluctuations are more significant on the honeycomb lattice due to higher spin wave kinetic energy.

The difference in order stability between lattices is linked to their electronic structure and fluctuations.

Abstract

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).