Magnetic properties of two-dimensional (2D) classical square Heisenberg lattices I- zero-field partition function

In this part labelled I we rigorously examine 2D square lattices composed of (2N+1)
²
classical spins isotropically coupled between first-nearest neighbors (i.e., showing Heisenberg cou-plings). Each local operator exp(-β H
_i,j
^ex
) - where H
_i,j
^ex
is the local exchange Hamiltonian associated with each site (i,j) - is expanded on the basis of spherical harmonics, each one being characterized by the couple of integers (ℓ,m), with ℓ≥0 and m∈[-ℓ,+ℓ]. Using spin lattice symmetries, we derive selection rules over these coefficients, thus allowing to express the zero-field partition function Z
_N
(0). In the thermodynamic limit (N→∞), we show that the value m=0 is selected while ℓ=ℓ
₀
≥0, ∀(i,j). A very simple closed-form expression may be then derived for Z
_N
(0). Finally we report a thermal study of the current term of the characteristic polynomial giving Z
_N
(0). We show that it appears crossovers between two consecutive terms characterized by integers ℓ and ℓ+1, respectively. Coming from high temperatures where the term characterized by ℓ=0 is dominant, near absolute zero, increasing ℓ-values are more and more selected when the temperature is cooling down. But, when T reaches zero, all the successive dominant terms become equivalent and are characterized by the value ℓ=0. We show that this property also prevails at T=0 K rigorously. This confirms the fact that, for a 2D lattice showing Heisenberg couplings, the critical temperature is T
_C
=0 K, in agreement with Mermin-Wagner's theorem.