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The renormalization group equations describing a finite 2DXY system withfourfold anisotropy are solved in two steps, in order to study the magnetictransition to paramagnetism in an ultrathin film. First, the equations arelinearized near the critical coupling $K=J/k_BT=2/\pi$. An exact solution ofthe linearized equations confirms that a finite-size Kosterlitz-Thouless (KT)transition occurs in the presence of weak fourfold anisotropy, and that anIsing transition occurs for strong anisotropy. The behaviour of a given systemis determined by the product of an anisotropy parameter and the logarithm ofthe system size. The linearized RG equations are not quantitatively reliableacross the full temperature range of the finite-size transition, but theydefine the parameter space where a second approach, where the fourfoldanisotropy is treated as a perturbation, is valid. This perturbative treatmentprovides a quantitative determination of the renormalized exchange coupling,vortex density, and anisotropy throughout the transition. In particular, thecoupling has a universal point of inflection where vortex-antivortex pairsunbind (as opposed to a "universal jump"), and goes to zero asymptotically inthe paramagnetic state, as is expected for a finite system. These results areused to calculate the magnetic susceptibility as the system moves from onedominated by spin waves to one dominated by a free vortex gas. The presence ofanisotropy makes it necessary to include both the susceptibility $\chi_{||}$due to fluctuations of the magnitude of the magnetization, and $\chi_\perp$ dueto angular fluctuations of the magnetization about a fourfold easy axis. Acomparison to recent measurements of the magnetic susceptibility of ultrathinFe/W(001) films is very encouraging.

physical review. b./physical review. b

Renormalization group analysis of the finite 2DXY model with fourfold  anisotropy: Application to the magnetic susceptibility of a ferromagnetic  ultrathin film