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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]].
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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where ''N'' is the number of time slices. The sum over all paths is given by a product of matrices, each matrix element is the transition probability from one slice to the next.
Similarly, one can divide the sum over all partition function configurations into slices, where each slice is the one-dimensional configuration at time 1. This defines the [[Transfer-matrix method (statistical mechanics)|transfer matrix]]:
<math display="block">T_{C_1 C_2}.</math>
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