In his 1924 PhD thesis, Ising solved the model for the d = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no phase transition. Namely, for any positive β, the correlations ⟨σ i σ j ⟩ decay exponentially in | i − j |: ⟨ ⟨ σ σ i σ σ j ⟩ ⟩ β β ≤ ≤ C exp ⁡ ⁡ ( − − c ( β β ) | i − − j | ) ,