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