The Two-Dimensional Infinite Heisenberg Classical Square Lattice: Exact Theory and Experimental Results

TL;DR

This paper provides an exact theoretical analysis of the 2D classical Heisenberg square lattice, deriving closed-form expressions for thermodynamic quantities, and validates these with experimental susceptibility data for specific compounds.

Contribution

It introduces an exact closed-form solution for the 2D classical Heisenberg lattice's thermodynamics and correlates theoretical predictions with experimental measurements.

Findings

Exact expression for free energy and specific heat at any temperature.

Derived spin-spin correlations, susceptibility, and phase diagram near zero Kelvin.

Good agreement between theoretical and experimental susceptibilities for real compounds.

Abstract

We rigorously examine 2d-infinite square lattices composed of classical spins isotropically coupled between first-nearest neighbors. Each local exchange Hamiltonian is expanded on the basis of its eigenfunctions played by spherical harmonics Yinf{l,m}. The corresponding eigenvalues are modified Bessel functions of the first kind. In the thermodynamic limit a numerical study allows one to select the higher-degree term of the characteristic polynomial associated with the zero-field partition function Zinf{N}(0). A very simple exact closed-form expression is derived, thus permitting to express the free energy F and the specific heat Cinf{V}, for any temperature. We report a thermal study of the basic term appearing in the higher-degree term of Zinf{N}(0). We show that it appears crossovers between two consecutive terms. Coming from high temperatures where the l= 0-term is dominant, near the critical temperature Tinf{c}= 0 K, eigenvalues showing increasing l-values are more and more selected. We derive an exact expression for the spin-spin correlations, the correlation length ksi and the susceptibility khi. Near Tinf{c}= 0 K we obtain a diagram of magnetic phases. We derive the same expressions for xsi, F, Cinf{V} and khi as the corresponding ones derived through a renormalization process. We show that, near 0 K, the lattice is composed of quasi rigid quasi independent Kadanoff blocks of length ksi and magnetic moment M(T), the unit cell moment, so that khi.kinf{B}T=ksi^2.M(T)^2. Finally we compare experimental susceptibilities to the theoretical expression of khi, for two types of 2d-compounds (showing or not organic ligands inside and between sheets of Mn2+ ions). We obtain a remarkable good agreement between the J-values of the exchange energy derived from the fits and the corresponding ones previously measured as well as a value of the Land\'e factor close to the theoretical one.