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{{Statistical mechanics|cTopic=Models}}
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 (abstract data type)|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 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> Though it is a highly simplified model of a magnetic material, the Ising model can still provide qualitative and sometimes quantitative results applicable to real physical systems.
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 exists a very simple approach relating the model to a non-interacting fermionic [[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|url-access=subscription }}</ref>
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*{{Citation |last=Onsager |first=Lars |author-link=Lars Onsager|title=Discussion|journal=Supplemento al Nuovo Cimento | volume=6|page=261|year=1949}}
* John Palmer (2007), ''Planar Ising Correlations''. Birkhäuser, Boston, {{ISBN|978-0-8176-4248-8}}.
*{{Citation | last1=Istrail | first1=Sorin | title=Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing | chapter-url=
*{{Citation | last1=Yang | first1=C. N. | author-link1=C. N. Yang| title=The spontaneous magnetization of a two-dimensional Ising model | doi=10.1103/PhysRev.85.808 | mr=0051740 | year=1952 | journal=Physical Review | series = Series II | volume=85 | pages=808–816|bibcode = 1952PhRv...85..808Y | issue=5 }}
*{{Citation | last1=Glasser | first1=M. L. | year=1970 | title= Exact Partition Function for the Two-Dimensional Ising Model | journal=American Journal of Physics | volume=38 | issue=8 | pages=1033–1036 | doi=10.1119/1.1976530 | bibcode=1970AmJPh..38.1033G }}
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