Ising model: Difference between revisions

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==2-dimensional Ising model==
==2-dimensional Ising model==
The 2-dimensional Ising model was solved by [[Lars Onsager]] in 1944
The 2-dimensional [[Building up a square lattice |square lattice]] Ising model was solved by [[Lars Onsager]] in 1944
<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 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>[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>
after [[Rudolf Peierls]] had previously shown  that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition
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>.
<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==
==3-dimensional Ising model==
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<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>
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==
==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 alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.
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==
==See also==
*[[Critical exponents]]
*[[Critical exponents]]
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==References==
==References==
<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]]
[[Category: Models]]

Latest revision as of 13:10, 11 March 2016

The Ising model [1] (also known as the Lenz-Ising model) is commonly defined over an ordered lattice. Each site of the lattice can adopt two states, . 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.

where is the Boltzmann constant, is the temperature, indicates that the sum is performed over nearest neighbors, and indicates the state of the i-th site, and is the coupling constant.

For a detailed and very readable history of the Lenz-Ising model see the following references:[2] [3] [4].

1-dimensional Ising model[edit]

Main article: 1-dimensional Ising model

The 1-dimensional Ising model has an exact solution.

2-dimensional Ising model[edit]

The 2-dimensional square lattice Ising model was solved by Lars Onsager in 1944 [5] [6] [7] after Rudolf Peierls had previously shown that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition [8] [9].

Critical temperature[edit]

The critical temperature of the 2D Ising model is given by [5]

where is the interaction energy in the direction, and is the interaction energy in the direction. If these interaction energies are the same one has

Critical exponents[edit]

The critical exponents are as follows:

  • Heat capacity exponent (Baxter Eq. 7.12.12)
  • Magnetic order parameter exponent (Baxter Eq. 7.12.14)
  • Susceptibility exponent (Baxter Eq. 7.12.15)

(see also: Ising universality class)

3-dimensional Ising model[edit]

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice [10] [11] In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations, renormalisation group analysis and conformal bootstrap techniques provide accurate estimates [12]:

with a critical temperature of [13]

ANNNI model[edit]

The axial next-nearest neighbour Ising (ANNNI) model [14] is used to study spatially modulated structures in alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.

Cellular automata[edit]

The Ising model can be studied using cellular automata [15][16][17][18].

See also[edit]

References[edit]

  1. Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei 31 pp. 253-258 (1925)
  2. S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics 39 pp. 883-893 (1967)
  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)
  4. 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)
  5. 5.0 5.1 Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review 65 pp. 117-149 (1944)
  6. M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review 88 pp. 1332-1337 (1952)
  7. Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available pdf)
  8. Rudolf Peierls "On Ising's model of ferromagnetism", Mathematical Proceedings of the Cambridge Philosophical Society 32 pp. 477-481 (1936)
  9. Robert B. Griffiths "Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet", Physical Review A 136 pp. 437-439 (1964)
  10. Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown
  11. 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)
  12. 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)
  13. 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)
  14. Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports 170 pp. 213-264 (1988)
  15. Gérard Y. Vichniac "Simulating physics with cellular automata", Physica D: Nonlinear Phenomena 10 pp. 96-116 (1984)
  16. Y. Pomeau "Invariant in cellular automata", Journal of Physics A 17 pp. L415-L418 (1984)
  17. H. J. Herrmann "Fast algorithm for the simulation of Ising models", Journal of Statistical Physics 45 pp. 145-151 (1986)
  18. Michael Creutz "Deterministic ising dynamics", Annals of Physics 167 pp. 62-72 (1986)
Related reading

External links[edit]