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. 7.12.145)


==3-dimensional Ising model==
==3-dimensional Ising model==

Revision as of 16:09, 29 April 2010

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

Main article: 1-dimensional Ising model

The 1-dimensional Ising model has an exact solution.

2-dimensional Ising model

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

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

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. 7.12.145)

3-dimensional Ising model

Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice [10] [11]

ANNNI model

The axial next-nearest neighbour Ising (ANNNI) model [12] is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.

See also

References

  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. Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports 170 pp. 213-264 (1988)