The two-dimensional infinite Heisenberg classical square lattice: zero-field partition function and correlation length
Abstract
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 ZinfN(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 ZinfN(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 Tinfc = 0 K. Using an analytical method similar to the one employed for ZinfN(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.