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The '''Ising model''' <ref>[http://dx.doi.org/10.1007/ | The '''Ising model''' is also known as the '''Lenz-Ising''' model. For a history of the Lenz-Ising model see <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>. | |||
The 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. | 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 | The energy of the system is the sum of pair interactions | ||
between nearest neighbors. | between nearest neighbors. | ||
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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 | 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 | <math> S_i </math> indicates the state of the i-th site, and <math> K </math> is the coupling constant. | ||
==1-dimensional Ising model== | ==1-dimensional Ising model== | ||
* [[1-dimensional Ising model]] (exact solution) | |||
==2-dimensional Ising model== | ==2-dimensional Ising model== | ||
Solved by [[Lars Onsager]] in 1944 | |||
<ref | <ref>[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>[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> | <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>. | ||
[[Rudolf Peierls]] had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition. | |||
==3-dimensional Ising model== | ==3-dimensional Ising model== | ||
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice | 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://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> | <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> | ||
==ANNNI model== | ==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 | 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 alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes. | ||
==See also== | ==See also== | ||
*[[Critical exponents]] | *[[Critical exponents]] | ||
*[[Potts model]] | *[[Potts model]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
[[Category: Models]] | [[Category: Models]] |