The Two-Dimensional Infinite Heisenberg Classical Square Lattice: Zero-Field Partition Function and Correlation Length

Abstract

The nonlinear σ-model has known a new interest for it allows to describe the properties of two-dimensional quantum antiferromagnets which, when properly doped, become superconductors up to a critical temperature notably high compared to other types of superconducting materials. This model has been conjectured to be equivalent at low temperatures to the two-dimensional Heisenberg model. In this article we rigorously examine 2d-square lattices composed of classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg couplings). A general expression of the characteristic polynomial associated with the zero-field partition function 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 the zero-field partition function 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 zero Kelvin, l-eigen¬values showing increasing l-values are more and more selected. At absolute zero l becomes infinite and all the successive dominant l-eigenvalues become equivalent. As the z-spin correlation is null for positive temperatures but equal to unity (in absolute value) at absolute zero the critical temperature is absolute zero. Using an analytical method similar to the one employed for the zero-field partition function we also give an exact expression valid for any temperature for the spin-spin correlations as well as for the correlation length. In the zero-temperature 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 the correlation length we obtain the same expressions as the corresponding ones derived through a renor¬malization 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.