Editing Ising model

The '''Ising model''' <ref>[http://dx.doi.org/10.1007/BF02980577 Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", 	Zeitschrift für Physik A Hadrons and Nuclei '''31''' pp. 253-258 (1925)]</ref> (also known as the '''Lenz-Ising''' model) is commonly defined over an ordered lattice. 
Each site of the lattice can adopt two states, <math>S \in \{-1, +1 \}</math>. Note that sometimes these states are referred to as ''spins'' and the values are referred to as ''down'' and ''up'' respectively. 
The energy of the system is the sum of pair interactions
between nearest neighbors.

:<math> \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j </math>

where <math>k_B</math> is the [[Boltzmann constant]], <math>T</math> is the [[temperature]],  <math> \langle ij \rangle </math> indicates that the sum is performed over nearest neighbors, and
<math> S_i </math> indicates the state of the i-th site, and <math> K </math> is the coupling constant. 

For a detailed and very readable history of the Lenz-Ising model see the following references:<ref>[http://dx.doi.org/10.1103/RevModPhys.39.883  S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics '''39''' pp. 883-893 (1967)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-004-0088-3 Martin Niss "History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena", Archive for History of Exact Sciences '''59''' pp. 267-318 (2005)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-008-0039-5 Martin Niss "History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance", Archive for History of Exact Sciences '''63''' pp. 243-287 (2009)]</ref>.
==1-dimensional Ising model==
* [[1-dimensional Ising model]] (exact solution)
==2-dimensional Ising model==
The 2-dimensional [[Building up a square lattice |square lattice]] Ising model was solved by [[Lars Onsager]] in 1944
<ref  name="Onsager">[http://dx.doi.org/10.1103/PhysRev.65.117 Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review '''65''' pp. 117-149 (1944)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRev.88.1332 M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review '''88''' pp. 1332-1337 (1952)]</ref>
<ref>Rodney J. Baxter  "Exactly Solved Models in Statistical Mechanics", Academic Press (1982)  ISBN 0120831821 Chapter 7 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>
after [[Rudolf Peierls]] had previously shown  that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition
<ref>[http://dx.doi.org/10.1017/S0305004100019174 Rudolf Peierls "On Ising's model of ferromagnetism", Mathematical Proceedings of the Cambridge Philosophical Society '''32''' pp. 477-481 (1936)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRev.136.A437 Robert B. Griffiths "Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet", Physical Review A '''136''' pp. 437-439 (1964)]</ref>.
====Critical temperature====
The [[Critical points | critical temperature]] of the 2D Ising model is given by <ref  name="Onsager"> </ref>
:<math>\sinh \left( \frac{2S}{k_BT_c} \right) \sinh \left( \frac{2S'}{k_BT_c} \right)  =1</math> 
where <math>S</math> is the interaction energy in the <math>(0,1)</math> direction, and <math>S'</math> is the interaction energy in the <math>(1,0)</math> direction.
If these interaction energies are the same one has 
:<math>k_BT_c = \frac{2S}{ \operatorname{arcsinh}(1)} \approx 2.269 S</math>

====Critical exponents====
The [[critical exponents]] are as follows:
*Heat capacity exponent <math>\alpha = 0</math> (Baxter Eq. 7.12.12)
*Magnetic order parameter exponent <math>\beta = \frac{1}{8}</math> (Baxter Eq. 7.12.14)
*Susceptibility exponent <math>\gamma = \frac{7}{4} </math> (Baxter Eq. 7.12.15)
(see also: [[Universality classes#Ising | Ising universality class]])

==3-dimensional Ising model==
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice
<ref>[http://www.sandia.gov/LabNews/LN04-21-00/sorin_story.html Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown]</ref>
<ref>[http://dx.doi.org/10.1145/335305.335316    	Sorin Istrail 	 "Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces", Proceedings of the thirty-second annual ACM symposium on Theory of computing pp. 87-96   (2000)]</ref>
In three dimensions, the [[critical exponents]] are not known exactly. However, [[Monte Carlo | Monte Carlo simulations]], [[renormalisation group]] analysis and [[conformal bootstrap | conformal bootstrap techniques]] provide accurate estimates <ref name="Campostrini2002">[http://dx.doi.org/10.1103/PhysRevE.65.066127 Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E '''65''' 066127 (2002)]</ref>:

:<math>
\nu=0.63012(16)
</math>

:<math>
\alpha=0.1096(5)
</math>

:<math>
\beta= 0.32653(10)
</math> 

:<math>
\gamma=1.2373(2)
</math>

:<math>
\delta=4.7893(8)
</math>

:<math>
\eta =0.03639(15)
</math>

with a critical temperature of <math>k_BT_c = 4.51152786~S </math><ref>[http://dx.doi.org/10.1088/0305-4470/29/17/042 A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General '''29''' pp. 5727-5733 (1996)]</ref>

==ANNNI model==
The '''axial next-nearest neighbour Ising''' (ANNNI) model <ref>[http://dx.doi.org/10.1016/0370-1573(88)90140-8  Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports  '''170''' pp. 213-264 (1988)]</ref> is used to study spatially modulated structures in alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.
==Cellular automata==
The Ising model can be studied using cellular automata <ref>[http://dx.doi.org/10.1016/0167-2789(84)90253-7 Gérard Y. Vichniac "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena '''10''' pp. 96-116 (1984)]</ref><ref>[http://dx.doi.org/10.1088/0305-4470/17/8/004 Y. Pomeau "Invariant in cellular automata", Journal of Physics A '''17''' pp. L415-L418 (1984)]</ref><ref>[http://dx.doi.org/10.1007/BF01033083 H. J. Herrmann "Fast algorithm for the simulation of Ising models", Journal of Statistical Physics '''45''' pp. 145-151 (1986)]</ref><ref>[http://dx.doi.org/10.1016/S0003-4916(86)80006-9 Michael Creutz "Deterministic ising dynamics", Annals of Physics '''167''' pp. 62-72 (1986)]</ref>.
==See also==
*[[Critical exponents]]
*[[Potts model]]
*[[Mean field models]]

==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1126/science.aab3326 Gemma De las Cuevas, and Toby S. Cubitt "Simple universal models capture all classical spin physics", Science '''351''' pp. 1180-1183 (2016)]

==External links==
*[http://dx.doi.org/10.4249/scholarpedia.10313 Barry McCoy "Ising model: exact results", Scholarpedia, 5(7):10313 (2010)]
[[Category: Models]]