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

Using the reflection positivity method, we provide 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 is in contrast with the result obtained using the 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).

Figure 1

Dependence of the four contributions to the order parameter renormalization $\mathrm{\Delta }m$ (22) on the momentum components $\{k_1,k_2\}$ in the LSWT approach: (a) $C_1^{+}\left(\mathbf{k}\right)$, (b) $C_{-1}^{+}\left(\mathbf{k}\right)$, (d) $C_1^{-}\left(\mathbf{k}\right)$, (e) $C_{-1}^{-}\left(\mathbf{k}\right)$. For comparison, the relevant moduli of the structure factors are also shown in (c) $\left|\gamma \right(\mathbf{k}\left)\right|$ and (f) $-\left|\gamma \right(\mathbf{k}\left)\right|$ (see text for further details). The blue (yellow) color indicates for the results found for the honeycomb (square) lattice, respectively.