Only a specific version of the two-dimensional Ising Model can be solved exactly, and the three-dimensional version cannot be solved exactly in any form. It should also be noted that the Ising Model is a special case of the more general Heisenberg Model, which is a magnetic model that is still in use to this day. This is an example of short-range local interactions giving rise to extended long-range behavior, which can be a counter-intuitive and unexpected outcome. The Ising Model is interesting due to the two- and three-dimensional versions exhibiting a phase transition at a critical temperature, above which the model no longer exhibits permanent magnetism. It is named after Ernst Ising, who solved the one-dimensional version exactly as part of his 1924 thesis. a graph) forming a d-dimensional lattice.The Ising Model is a model of a permanent magnet. 9.6 Cayley tree topologies and large neural networksĬonsider a set Λ of lattice sites, each with a set of adjacent sites (e.g.8.8 4 − ε dimensions – renormalization group.7.1 Istrail's NP-completeness result for the general spin glass model.6.1.4 Onsager's formula for spontaneous magnetization.6.1.3 Spin flip creation and annihilation operators.5.2 One-dimensional solution with transverse field.4.3 Viewing the Ising model as a Markov chain.4 Monte Carlo methods for numerical simulation.3.1 No phase transitions in finite volume.2.2 Phase transition and exact solution in two dimensions.2.1 No phase transition in one dimension.The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization. In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory.
It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by Lars Onsager ( 1944). The one-dimensional Ising model was solved by Ising (1925) alone in his 1924 thesis it has no phase transition. The Ising model was invented by the physicist Wilhelm Lenz ( 1920), who gave it as a problem to his student Ernst Ising.
The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The model allows the identification of phase transitions as a simplified model of reality. Neighboring spins that agree have a lower energy than those that disagree the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The Ising model ( German pronunciation: ) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics.
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