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| 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.
| | Also known as the '''Lenz-Ising''' model. |
| 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. | | |
| | == Ising Model == |
| | |
| | The Ising model is commonly defined over an ordered lattice. |
| | Each site of the lattice can adopt two states: either |
| | UP (S=+1) or DOWN (S=-1). |
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| 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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| :<math> \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j </math>
| | <math> \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j </math> |
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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> \langle ij \rangle </math> indicates that the sum is done 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. |
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| 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>
| | <math> K </math> is called the Coupling constant. |
| <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>
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| <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>.
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| ==1-dimensional Ising model==
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| :''Main article: [[1-dimensional Ising model]]''
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| The 1-dimensional Ising model has an exact solution.
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| ==2-dimensional Ising model==
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| The 2-dimensional [[Building up a square lattice |square lattice]] Ising model was solved by [[Lars Onsager]] in 1944
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| <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>
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| <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>
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| <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>
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| after [[Rudolf Peierls]] had previously shown that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition
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| <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>.
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| ====Critical temperature====
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| The [[Critical points | critical temperature]] of the 2D Ising model is given by <ref name="Onsager"> </ref>
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| :<math>\sinh \left( \frac{2S}{k_BT_c} \right) \sinh \left( \frac{2S'}{k_BT_c} \right) =1</math>
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| 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.
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| If these interaction energies are the same one has
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| :<math>k_BT_c = \frac{2S}{ \operatorname{arcsinh}(1)} \approx 2.269 S</math>
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| ====Critical exponents====
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| The [[critical exponents]] are as follows:
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| *Heat capacity exponent <math>\alpha = 0</math> (Baxter Eq. 7.12.12)
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| *Magnetic order parameter exponent <math>\beta = \frac{1}{8}</math> (Baxter Eq. 7.12.14)
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| *Susceptibility exponent <math>\gamma = \frac{7}{4} </math> (Baxter Eq. 7.12.15)
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| (see also: [[Universality classes#Ising | Ising universality class]])
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| | ==1-dimensional Ising model== |
| | * [[1-dimensional Ising model]] (exact solution) |
| | ==2-dimensional Ising model== |
| | Solved by [[Lars Onsager]] in 1944. |
| | *[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)] |
| ==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>
| | *[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>[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>
| | *[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)] |
| 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>:
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| :<math>
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| \nu=0.63012(16)
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| </math>
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| :<math>
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| \alpha=0.1096(5)
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| </math>
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| :<math>
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| \beta= 0.32653(10)
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| </math>
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| :<math>
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| \gamma=1.2373(2)
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| </math>
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| :<math>
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| \delta=4.7893(8)
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| </math>
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| :<math>
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| \eta =0.03639(15)
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| </math>
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| 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>
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| ==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 spatially modulated structures in alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes. | | The '''axial next-nearest neighbour Ising''' (ANNNI) model is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes. |
| ==Cellular automata== | | *[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)] |
| 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== |
| ==See also==
| | *[[History of the Ising model]] |
| *[[Critical exponents]]
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| *[[Potts model]]
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| *[[Mean field models]]
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| ==References==
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| <references/>
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| ;Related reading
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| *[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)]
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| ==External links==
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| *[http://dx.doi.org/10.4249/scholarpedia.10313 Barry McCoy "Ising model: exact results", Scholarpedia, 5(7):10313 (2010)]
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| [[Category: Models]] | | [[Category: Models]] |