The self-consistent harmonic approximation is used to study the statistical mechanics, in the low-temperature region, of the XY ferromagnetic model in the quantum and classical cases. In the XY model the spins are not restricted to the XY plane (as in the planar rotator model) due to the presence of ${\mathit{S}}^{\mathit{Z}}$ in the equations of motion. Because a flip of sign of ${\mathit{S}}^{\mathit{Z}}$ costs no change in energy, the thermal fluctuations are dominated by the transverse component. In two dimensions we have a Kosterlitz-Thouless (KT) behavior similar to the planar rotator model, except that, for the classical case, the transition temperature ${\mathit{T}}_{\mathrm{KT}}$ is pushed down slightly from the value expected for the planar model ${\mathit{T}}_{\mathrm{KT}}$/${\mathit{JS}}^2$=0.90 to ${\mathit{T}}_{\mathrm{KT}}$/${\mathit{JS}}^2$=0.79. In the quantum spin S=1/2 case, enhanced quantum fluctuations due to small spin, pushes down the transition point to ${\mathit{T}}_{\mathrm{KT}}$/${\mathit{JS}}^2$=0.30. © 1996 The American Physical Society.

• Received 9 June 1995

DOI:https://doi.org/10.1103/PhysRevB.53.235