Ising Model

The Ising model is a mathematical model of statistical physics designed to describe the magnetization of a material.

Content

Description

Each vertex of the crystal lattice (not only three-dimensional, but also one- and two-dimensional cases are considered) is associated with a number called the spin and equal to +1 or −1 (“field up” / “field down”). To each of $2^{N}$ ${\ displaystyle 2 ^ {N}}$ of possible spins (where N is the number of lattice atoms) is attributed to the energy obtained from the pairwise interaction of the spins of neighboring atoms:

$E(S)=-J\sum _{i\sim j}S_{i}S_{j}\,,$ ${\ displaystyle E (S) = - J \ sum _ {i \ sim j} S_ {i} S_ {j} \ ,,}$

Where $J$ ${\ displaystyle J}$ Is the interaction energy (in the simplest case, the same for all pairs of neighboring atoms). Sometimes an external field is also considered. $h$ ${\ displaystyle h}$ (often considered small):

$E(S)=-J\sum _{i\sim j}S_{i}S_{j}-h\sum _{i}S_{i}.$ ${\ displaystyle E (S) = - J \ sum _ {i \ sim j} S_ {i} S_ {j} -h \ sum _ {i} S_ {i}.}$

Then, for a given return temperature $\beta =1/k_{B}T$ ${\ displaystyle \ beta = 1 / k_ {B} T}$ on the resulting configurations, the Gibbs distribution is considered: the probability of the configuration is assumed to be proportional $e^{-\beta E(S)}$ ${\ displaystyle e ^ {- \ beta E (S)}}$ , and we study the behavior of such a distribution for a very large number of atoms $N$ ${\ displaystyle N}$ .

For example, in models with a dimension greater than 1, there is a second-order phase transition : at sufficiently low temperatures, most of the spins of the ferromagnet (J> 0) will be oriented (with a probability close to 1) in the same way, and at high spins almost certainly “up” and "down" will be almost equally divided. The temperature at which this transition occurs (in other words, at which the magnetic properties of the material disappear) is called the critical, or Curie point . In the vicinity of the phase transition point, a number of thermodynamic characteristics diverge. Experience shows that the divergence is universal, and is determined only by the symmetry of the system. The first critical divergence indices were obtained for the two-dimensional Ising model in the 40s by Onsager . For other dimensions, studies are carried out using computer simulation methods, renormalization groups . The rationale for the use of the renormalization group in this case is the Kadanov block construction and the thermodynamic similarity hypothesis .

Introduced initially to understand the nature of ferromagnetism, the Ising model found itself at the center of various physical theories related to critical phenomena, liquids and solutions, spin glasses, cell membranes, modeling of the immune system, various social phenomena, etc. In addition, this model serves as a testing ground for testing methods of numerical modeling of various physical phenomena.

One-dimensional Ising model

In the case of one measurement, the Ising model can be represented as a chain of interacting spins. An exact solution was found for such a model, but in the general case, the problem does not have an analytical solution.

Monte Carlo Ising Model Implementation Algorithm on a Computer

Create a lattice of spins (two-dimensional array), the spins are oriented randomly.
Randomly select one of the cells of the lattice, erase the value in it.
Calculate the energies of the configurations when this cell is filled with spin up and down (or in all possible states, if there are more than two of them).
Choose one of the options for the “erased” spin randomly, with a probability proportional to $e^{-\beta E(S)}$ ${\ displaystyle e ^ {- \ beta E (S)}}$ where $E(S)$ ${\ displaystyle E (S)}$ - energy in an appropriate state. (Since all terms that do not affect a given spin are the same, in fact, only sums by neighbors need to be calculated).
We return to paragraph 2; after performing a sufficient number of iterations (determining this is a separate and difficult task), the cycle stops.

Applications

In 1982, Hopfield proved the isomorphism of the Ising model and recurrent models of neural networks ^[1] .

Quantum computer company DWave Sys. based on the Ising model. However, the effectiveness of the computer raises questions, which was the reason for new research, the purpose of which is to correctly compare classical algorithms and algorithms for DWave computers. It turned out that there are problems in which an adiabatic quantum computer is obviously not more efficient than the classical one ^[2] .

Notes

Comments

Sources

↑ Khaikin S., 2006 , p. 79.
↑ Katzgraber, Hamze, Andrist, 2014 , p. 6.

Literature

Books

Belavin A.A. , Kulakov A.G. , Usmanov R.A. Lectures on theoretical physics . - M .: ICMMO , 2001 .-- 224 p. - ISBN 5-900916-91-X .
Baxter R. Exactly Solvable Models in Statistical Mechanics. - M .: World , 1985.
Khaikin S. Neural networks: a full course. - M .: Williams Publishing House, 2006. - 1104 p. - ISBN 5-8459-0890-6 .
Zayman J. Principles of solid state theory. - M .: Mir, 1974.- 472 p.

Scientific Articles

Meilikhov E.Z. The tragic and happy life of Ernst Ising // Nature. - 2006. - No. 7 .
Stepanov IA Exact Solutions of the One-Dimensional, Two-Dimensional, and Three-Dimensional Ising Models // Nano Science and Nano Technology: An Indian Journal. - 2012. - Vol. 6, No. 3 . - P. 118-122. Free online ..
Katzgraber HG , Hamze F. , Andrist SR Glassy Chimeras Could Be Blind to Quantum Speedup: Designing Better Benchmarks for Quantum Annealing Machines // Phys. Rev. X. - 2014 .-- Vol. 4. - P. 1-8. - DOI : 10.1103 / PhysRevX.4.021008 .

[_28970afc256059d7-1] Khaikin S., 2006 , p. 79.

[_7bfe0bfac5cd565d-2] Katzgraber, Hamze, Andrist, 2014 , p. 6.