 Easy-plane anisotropy

Easy-plane anisotropy makes spin orientations within one plane energetically preferable to orientations normal to it. For example, a classical single-site term $D(S^z{)}^2$ with $D>0$ penalizes out-of-plane components and leaves the $xy$ plane degenerate. At sufficiently low energies, an in-plane angle can then be the main orientational degree of freedom.

The resulting approximate angular description connects a magnet to the XY model, but the microscopic form matters. For a quantum spin one-half, $(S^z{)}^2$ is proportional to the identity, so that single-site term alone does not generate orientational anisotropy. Anisotropic exchange interactions provide another route. Stating the spin size and Quantum Hamiltonian prevents transferring a classical mechanism to a quantum model where it is ineffective. Residual in-plane anisotropy can further select a finite set of directions rather than a continuous circle.