ising

Ising refers to Ernst Ising, a German physicist whose doctoral work at the University of Hamburg, supervised by Wilhelm Lenz, introduced what is now known as the Ising model in 1924–1925. Ising studied a one-dimensional chain of spins and showed that, at finite temperature, there is no spontaneous ferromagnetic order. The model was later generalized and became a central tool in the study of statistical mechanics and critical phenomena.

The Ising model is a simplified mathematical representation of ferromagnetism. It consists of discrete spins s_i = ±1 located on the sites of a lattice, with nearest-neighbor interactions. The energy (Hamiltonian) is H = -J sum over neighboring pairs s_i s_j - h sum_i s_i, where J>0 favors alignment and h is an external magnetic field. Much of the study focuses on the zero-field case (h=0).

Dimensionality matters: in one dimension the system does not exhibit a phase transition at finite temperature, while in two dimensions there is a finite-temperature phase transition. Lars Onsager solved the exact free energy of the square-lattice Ising model in 1944, revealing a critical temperature Tc = 2J/(k_B ln(1+√2)). The spontaneous magnetization below Tc is given by m(T) = [1 - sinh(2J/k_B T)^{-4}]^{1/8} (T<Tc).

Beyond physics, the Ising model serves as a versatile framework in other fields. It is used as a binary Markov random field in statistics and machine learning, in image processing, neuroscience, and social dynamics. Its analytical and computational tractability, along with its rich critical behavior, has led to extensive study and many generalizations to different lattices, couplings, and networks, as well as to advanced simulation methods like Monte Carlo algorithms.