 Ernst Ising

Ernst Ising solved the one-dimensional nearest-neighbor version of the spin model now called the Ising model. For ferromagnetic coupling $J$ and zero external field, its correlations decay with separation as $⟨s_0{s}_r⟩=[\tanh (J/(k_{B}T)){]}^r$. At every positive temperature the factor has magnitude below one, so correlations vanish at arbitrarily large distance and there is no finite-temperature long-range ferromagnetic order in this model. The result makes the role of dimensionality and fluctuations concrete, rather than assuming that an ordering interaction automatically creates macroscopic order.^[ref]