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The '''Ising model''' (or '''Lenz–Ising model'''), named after the physicists [[Ernst Ising]] and [[Wilhelm Lenz]], is a [[mathematical models in physics|mathematical model]] of [[ferromagnetism]] in [[statistical mechanics]]. The model consists of [[discrete variables]] that represent [[Nuclear magnetic moment|magnetic dipole moments of atomic "spins"]] that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a [[lattice (group)|lattice]] (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. 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 model allows the identification of [[phase transition]]s as a simplified model of reality. The two-dimensional [[square-lattice Ising model]] is one of the simplest statistical models to show a [[phase transition]].<ref>See {{harvtxt|Gallavotti|1999}}, Chapters VI-VII.</ref>
The Ising model was invented by the physicist {{harvs|txt|authorlink=Wilhelm Lenz|first=Wilhelm|last=Lenz|year=1920}}, who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by {{harvtxt|Ising|1925}} alone in his 1924 thesis;<ref>[http://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html Ernst Ising, ''Contribution to the Theory of Ferromagnetism'']</ref> it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by {{harvs|txt|authorlink=Lars Onsager|first=Lars |last=Onsager|year=1944}}. It is usually solved by a [[Transfer-matrix method (statistical mechanics)|transfer-matrix method]], although there exist different approaches, more related to [[quantum field theory]].<ref>{{Cite journal |last1=Samuel |first1=Stuart|date=1980 |title=The use of anticommuting variable integrals in statistical mechanics. I. The computation of partition functions|url=https://doi.org/10.1063/1.524404 |journal=Journal of Mathematical Physics |language=en |volume=21|issue=12 |pages= 2806–2814 |doi=10.1063/1.524404}}</ref>
In dimensions greater than four, the phase transition of the Ising model is described by [[mean-field theory]]. The Ising model for greater dimensions was also explored with respect to various tree topologies in the late 1970s, culminating in an exact solution of the zero-field, time-independent {{harvtxt|Barth|1981}} model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible {{pslink|Ising model|applications|nopage=y}}.
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