Content deleted Content added
m Reference added |
m Reference correction. Onsager did not know or use the connection to fermionic field theory. |
||
Line 4:
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
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}}.
Line 111:
This was first proven by [[Rudolf Peierls]] in 1936,<ref>{{Cite journal |doi=10.1017/S0305004100019174 |title=On Ising's model of ferromagnetism |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=32 |issue=3 |pages=477 |year=1936 |last1=Peierls |first1=R. |last2=Born |first2=M. |bibcode=1936PCPS...32..477P|s2cid=122630492 }}</ref> using what is now called a '''Peierls argument'''.
The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by {{harvs|txt|authorlink=Lars Onsager|first=Lars |last=Onsager|year=1944}}. Onsager
=== Correlation inequalities ===
Line 451:
* In the ferromagnetic case there is a phase transition. At low temperature, the [[Peierls argument]] proves positive magnetization for the nearest neighbor case and then, by the [[Griffiths inequality]], also when longer range interactions are added. Meanwhile, at high temperature, the [[cluster expansion]] gives analyticity of the thermodynamic functions.
* In the nearest-neighbor case, the free energy was exactly computed by Onsager
==== Onsager's exact solution ====
| |||