In the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available. The Hamiltonian of the one-dimensional Ising model on a lattice of L sites with periodic boundary conditions is H ( σ σ ) = − − J ∑ ∑ i = 1 , … … , L − − 1 σ σ i σ σ i + 1 − − h ∑ ∑ i σ σ i , where J and h can be any number, since in this simplified case J is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites. Then the free energy is f ( β β , h ) = − − lim L → → ∞ ∞ 1 β β L ln ⁡ ⁡ Z ( β β ) = − − 1 β β ln ⁡ ⁡ ( e β β J cosh ⁡ ⁡ β β h + e 2 β β J ( sinh ⁡ ⁡ β β h ) 2 + e − − 2 β β J ) , and the spin-spin correlation (i.e. the covariance) is ⟨ ⟨ σ σ i σ σ j ⟩ ⟩ − − ⟨ ⟨ σ σ i ⟩ ⟩ ⟨ ⟨ σ σ j ⟩ ⟩ = C ( β β ) e − − c ( β β ) | i − − j | , where C (β) and c (β) are positive functions for T > 0. For T → 0, though, the inverse correlation length c (β) vanishes.