  We rigorously examine 2d-square lattices composed of classical spins
isotropically coupled between first-nearest neighbours. A general expression of
the characteristic polynomial associated with the zero-field partition function
Zinf{N}(0) is established for any lattice size. In the infinite-lattice limit a
numerical study allows to select the dominant term: it is written as a l-series
of eigenvalues, each one being characterized by a unique index l whose origin
is explained. Surprisingly Zinf{N}(0) shows a very simple exact closed-form
expression valid for any temperature. The thermal study of the basic l-term
allows to point out crossovers between l- and (l+1)-terms. Coming from high
temperatures where the l=0-term is dominant and going to 0 K, l-eigenvalues
showing increasing l-values are more and more selected. At T = 0 K l tends to
infinity and all the successive dominant l-eigenvalues become equivalent. As
the z-spin correlation is null for T greater than 0 K but equal to 1 (in
absolute value) for T = 0 K the critical temperature is Tinf{c} = 0 K. Using an
analytical method similar to the one employed for Zinf{N}(0) we also give an
exact expression valid for any temperature for the spin-spin correlations as
well as for the correlation length xsi. In the T=0-limit we obtain a diagram of
magnetic phases which is similar to the one derived through a renormalization
approach. By taking the low-temperature limit of xsi we obtain the same
expressions as the corresponding ones derived through a renormalization
process, for each zone of the magnetic phase diagram, thus bringing for the
first time a strong validation to the full exact solution of the model valid
for any temperature.
