Ising model

At the critical point (with H = 0)

At a temperature well below the critical temperature

The Ising model is a lattice model in theoretical physics that Ernst Ising first examined in more detail at the suggestion of his doctoral supervisor Wilhelm Lenz . In particular, it describes ferromagnetism in solids ( crystals ). The Ising model is one of the most studied models in statistical physics .

definition

In the model it is assumed that the spins , which determine the magnetic moment of the atoms or ions, can only assume two discrete states (spin value ). The direction in space remains open; So is vectors (for the classic picture to stay, or quantum mechanics to vector operators ). ${\displaystyle \pm 1}$

The general energy expression (or Hamilton operator ) for such a situation is given by the Heisenberg model :

${\displaystyle {\hat {\mathcal {H}}}=-{\frac {1}{2}}\sum _{i,j}J_{ij}{\vec {s}}_{i}\cdot {\vec {s}}_{j}-{\vec {H}}\cdot \sum _{i=1}^{N}{\vec {s}}_{i}\quad {\text{(Heisenbergmodell)}}}$ . 

Here referred to

• ${\displaystyle {\vec {s}}_{i}}$a (multi-component) spin of the atom at the place of the crystal lattice , ${\displaystyle i}$
• ${\displaystyle J_{ij}}$the coupling constant (strength of the exchange coupling interaction) between the spins at places and , ${\displaystyle i}$ ${\displaystyle j}$
• the point is the scalar product ${\displaystyle \cdot }$
• ${\displaystyle {\vec {H}}}$the strength of the magnetic field .

When Ising model, however, the number is the spinning components at one reduced (i.e. parallel or anti-parallel to an axis excellent - here.. -Axis) : ${\displaystyle z}$ ${\displaystyle s_{i}^{z}=\pm 1}$

${\displaystyle {\hat {\mathcal {H}}}=-{\frac {1}{2}}\sum _{i,j}J_{ij}s_{i}^{z}s_{j}^{z}-H_{z}\sum _{i=1}^{N}s_{i}^{z}\quad {\text{(Isingmodell)}}}$ .

Often it is also assumed that is only non-zero for neighboring spins. If the exchange coupling is positive, one speaks of a ferromagnetic coupling; if it is negative it is called antiferromagnetic . With ferromagnets or antiferromagnets, the respective sign dominates ; Both signs appear equally often in the spin glasses . ${\displaystyle J_{ij}}$

A suitable choice of interactions can u. a. Spin glasses (this is a random variable), thinned magnets with interesting critical properties or spatially modulated magnetic structures (here there are competing couplings , see ANNNI model ) can be modeled. In general, the Ising model describes the magnetic orders at low temperatures , which are broken up at higher temperatures by thermal fluctuations , with a phase transition taking place. A comprehensive theoretical analysis of phase transitions is provided by the theory of renormalization groups , for which Kenneth G. Wilson received the 1982 Nobel Prize in Physics. ${\displaystyle J_{ij}}$ ${\displaystyle J_{ij}}$

In the one-dimensional Ising chain with sufficiently short-range interactions, however, no phase transition is observed . Ernst Ising had already discovered this with regret in his doctoral thesis. He erroneously assumed that this also applies to two and more dimensions, which was initially generally accepted.

Rudolf Peierls showed, however, in 1936 that there was a phase transition in two dimensions. In 1941, Hendrik Anthony Kramers and Gregory Wannier determined the critical temperature using a duality argument . The exact solution of the two-dimensional Ising model with interactions between nearest neighbors and with a vanishing magnetic field was first calculated by Lars Onsager in 1944 . Further improvements came from Bruria Kaufman (partly together with Onsager) and Chen Ning Yang , who exactly calculated the spontaneous magnetization in 1952. A combinatorial treatment comes from Mark Kac and John Clive Ward (1952), and the proof of equivalence to a fermion model from Elliott Lieb , Theodore David Schultz and Daniel Charles Mattis (1964).

There is no analytically exact solution for the three-dimensional Ising model with interactions between neighboring spins . However, its properties can be calculated with the help of the molecular field approximation (or Landau theory ), Monte Carlo simulations , series developments or other numerical solution methods.

Because of its conceptual simplicity and its diverse properties, the Ising model is considered the “ Drosophila ” of statistical physics. It has also found applications in many areas of the natural sciences, including biology and brain research. Michael E. Fisher 's almost programmatic statement 'Ising models still thrive' (for example: 'Ising models are still growing') will probably remain valid for many years to come.

Generalizations of the Ising model are provided by the Blume-Capel model , the Potts model and the Markow network .

Simplified representation

The essential properties of the Ising model can be explained using the two-dimensional Ising model with interaction only between direct neighbors (left, right, above, below) in the absence of an external magnetic field ( ). ${\displaystyle H_{z}=0}$

In this special case the energy of a state can be described by:

${\displaystyle {\begin{aligned}{\mathcal {H}}&=-{\frac {1}{2}}\sum _{i,j}J_{ij}s_{i}^{z}s_{j}^{z}\\&=-JN_{\rm {N}}+2JN_{\rm {A}}\end{aligned}}}$ .

With

• the constant number of possible neighbor pairs ${\displaystyle N_{\rm {N}}}$
• the number of neighboring pairs with different orientations, which depends on the orientation of the individual spins ( ). ${\displaystyle N_{\rm {A}}}$ ${\displaystyle N_{\rm {A}}\leq N_{\rm {N}}}$

The constant energy of the ground state does not contribute to the thermodynamic behavior of the system. Opposite neighboring spins make an energy contribution , parallel spins make no contribution. ${\displaystyle -JN_{\rm {N}}}$ ${\displaystyle 2J}$

Energy, warmth, probability

Very small two-dimensional Ising model

The picture symbolically shows a tiny “magnet” made up of 25 “iron atoms”. Iron atoms behave like small magnets. The magnetic field of the total magnet is the sum of the magnetic fields that emanate from the individual atoms, whereby the fields of oppositely oriented atoms cancel each other out.

Five of the atoms (black) are aligned in one direction, the remaining 20 (white) in the other direction. The net magnetization is therefore units. A particular black and white pattern is called the state of the magnet. ${\displaystyle 5-20=-15}$

The red edges show opposing neighbors. Each red edge corresponds to an amount of energy stored in the magnet , which is named (this does not stand for the energy unit Joule , but for a parameter of the respective material). ${\displaystyle N_{\rm {A}}=14}$ ${\displaystyle 2J}$

Every red edge reduces the likelihood of encountering the condition in nature, the more so the colder it is. This is calculated by multiplying the probability for the state “all atoms in the same direction” for each red edge once . The denominator is the product of the temperature in Kelvin and the Boltzmann constant . ${\displaystyle \exp \left(-{\frac {2J}{Tk_{\mathrm {B} }}}\right)}$

Example: On a warm summer day (27 degrees Celsius, i.e. approx. 300 K) , each red edge in a material with a value of 0.0595 electron volts reduces the  probability by a factor of 10. When it cools down to minus 123 degrees Celsius, i.e. . H. approx. 150 K, the factor is already 100 and at minus 173 degrees, i.e. H. approx. 100 K, even 1000. ${\displaystyle 2J}$

What has been said concerns the probability of an individual condition, which is usually very small. Usually there is also a very large number of states that produce a certain magnetization strength of the magnet (number of black squares minus number of white squares) (think of the numerous possibilities to fill out a lottery ticket).

The large number of states can compensate for the small probability of the individual state. In fact, there is usually a certain strength of magnetization at a given temperature, which is far more likely than any other. This magnetization is found almost exclusively. With increasing temperature, it shifts from “fully magnetized” to “demagnetized”.

Extreme temperatures

To get a feel for the meaning of the above, consider first the borderline cases of very low and very high temperatures. Contrary to intuition, the calculations are not made more difficult by large numbers, but so easy that you can get results with "mental arithmetic".

At extremely low temperatures (temperature approaches absolute zero ) the probability factor becomes so small that no state except “all black” or “all white” can ever be encountered. The magnet thus assumes its full magnetization. ${\displaystyle \exp \left(-{\frac {2J}{Tk_{\mathrm {B} }}}\right)}$

At extremely high temperatures, on the other hand, the probability factor becomes more and more similar to the number 1, so that it does not lead to a reduction in probability and all states are equally likely. Then the number of realizing states applies to every magnetization, and this is highest for “50% white - 50% black”. The magnet is effectively demagnetized.

Moderate temperature

One atom is oriented opposite to the other

The depicted state with a deviating atom has four red edges. At a value of 0.0017 eV, this one state is ten times less likely than full magnetization (at 27 degrees Celsius). However, there are 25 ways to make exactly one atom deviate, and so a magnetization of 24 units (25 - 1 opposite) is 2.5 times as likely as full magnetization. ${\displaystyle 2J}$

Critical temperature

The breakdown of magnetism occurs at a finite temperature, the critical temperature . To justify this requires extensive mathematical analyzes, which cannot be carried out here. ${\displaystyle T_{C}}$

“Interesting” patterns (with regard to the black-and-white distribution) appear near the critical temperature.

Structure formation

Compact structure

Loosened structure

On the way from absolute zero to infinite temperature, one gets from perfect order to perfect noise .

In between you can find “interesting” patterns . With regard to the magnetization value, there is a compromise between a low probability and a large number of a state: A compact structure picked out at will has fewer red edges (and is therefore more likely) than a loosened structure picked out at random; but because there are more loosened structures, the property “loosened” can be more likely overall. So you will find a compromise that is neither completely compact nor completely torn, just an “interesting” structure.

Analogously one can argue with regard to the scattering of black and white squares, given temperature and magnetization.

Applications and interpretations

The original interpretation of the Ising model is the “ magnetic ” one: The spin values ​​point “upwards” or “downwards”. But the Ising model is also suitable for other binary problems.

A prominent example is the "Ising grid gas", which can be used to model liquids : Here, a grid is considered whose places can be either "occupied" or "unoccupied", depending on whether the Isingspin assigned to the grid place has the value +1 or −1.

The Ising model can also be used to describe spin glasses , namely with the energy , where the variables mean the Ising spins and take on fixed but random values. ${\displaystyle {\hat {H}}=-{\tfrac {1}{2}}\sum s_{i}\,J_{ik}\,s_{k}}$ ${\displaystyle s}$ ${\displaystyle J_{ik}}$

Quantum chromodynamics

In addition, there is an interpretation of this Hamilton operator as a greatly simplified model of quantum chromodynamics in elementary particle physics : the variables can be interpreted as quarks and those as gluons if both quantities are allowed to fluctuate. However, in this case one has to add the gluon-gluon couplings called Wilson-Loop variables to the form . ${\displaystyle s}$ ${\displaystyle J_{ik}}$ ${\displaystyle \,J_{ik}J_{kl}J_{lm}J_{mi}}$

One then obtains gauge invariant models that uncorrelated binary values and the coupled gauge transformations  ,  , suffice; d. H. the Hamilton operator remains invariant in these transformations , just as the Lagrangian function of quantum chromodynamics remains invariant with respect to transformations with the elements of group  SU (3) , which are replaced here by the variables. ${\displaystyle \epsilon _{i}=\pm 1}$ ${\displaystyle \epsilon _{k}=\pm 1}$ ${\displaystyle s_{i}\to s_{i}\epsilon _{i}}$ ${\displaystyle s_{k}\to s_{k}\epsilon _{k}}$ ${\displaystyle J_{ik}\to \epsilon _{i}J_{ik}\epsilon _{k}}$ ${\displaystyle \epsilon }$

With this model - a kind of Ising Lattice QCD - the lattice scale theory was introduced. The relevant publication comes from Franz Wegner . 

Nucleation

Homogeneous nucleation (a currently critical nucleation nucleus and an already (far) supercritical nucleation nucleus (oligonucleation))

Another possible application is the simulation of phase transitions through nucleation . Homogeneous nucleation is pretty much the same as ferromagnetism in modeling - some small changes have to be made for heterogeneous nucleation.

${\displaystyle {\hat {\mathcal {H}}}=-J{\frac {1}{2}}\sum _{i,j}{\vec {s}}_{i}\cdot {\vec {s}}_{j}-{\vec {H}}\cdot \sum _{i=1}^{N}{\vec {s}}_{i}-J_{s}{\frac {1}{2}}\sum _{i,j}^{\text{Wand}}{\vec {s}}_{i}\cdot {\vec {s}}_{j}}$

Heterogeneous nucleation in and out of pores (gray = wall, white = spin −1, red = spin +1)

In this case, the first sum is again the interaction between neighbors - but the newly added second summation stands for the interaction with a boundary surface. It turns out that in the area of ​​such boundary surfaces a core of critical size is created many times faster. ${\displaystyle i,j}$

Based on this, simulations for nucleation on porous surfaces were also carried out. Their result was that the pores must have a certain size in order to guarantee the fastest possible nucleation (this is usually most likely with irregular pores): With large pores, the proportion of boundary surfaces is smaller - this means that a nucleation core of critical size is not created for a longer period of time in the pore - if the pore is small, however, the initiation of a phase transition away from the upper edge is less likely.

References and footnotes

1. ^ E. Ising , contribution to the theory of ferromagnetism , Zeitschrift für Physik, Volume 31, 1925, pp. 253-258
2. ↑ There are different conventions regarding taking factor 1/2 with you (it is often left out)
3. ↑ W.Selke : The annni model. In: Physics Reports 170, 1988, pp. 213-264, doi: 10.1016 / 0370-1573 (88) 90140-8
4. ↑ R. Peierls , Ising's model of ferromagnetism, Proc. Cambridge Phil. Soc., Vol. 32, 1936, pp. 477-481
5. ↑ HAKramers , G.Wannier , Statistics of the two dimensional Ferromagnet, 2 parts, Phys. Rev., Vol. 60, 1941, pp. 252-262, 263-276
6. ↑ L. Onsager , Crystal Statistics I, Physical Review, Volume 65, 1944, pp. 117-149
7. ↑ CN Yang , The spontaneous magnetization of the two dimensional Ising model, Phys. Rev., Vol. 85, 1952, pp. 808-816
8. ↑ M. Kac , JC Ward , Physical Review Vol. 88, 1952, p. 1332
9. ↑ TD Schultz, E. Lieb , DC Mattis , Two dimensional Ising model as a soluble model of many fermions, Rev. Mod. Phys., Volume 36, July 1964, pp. 856-871
10. ↑ F. Wegner , Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter , J. Math. Phys. 12 (1971) 2259-2272. Reprinted in Claudio Rebbi (ed.), Lattice Gauge Theories and Monte Carlo Simulations , World Scientific, Singapore (1983), p. 60-73. ( Abstract )
11. ^ AJ Page, RP Sear: Heterogeneous nucleation in and out of pores. In: Physical review letters. Volume 97, number 6, August 2006, p. 065701, doi: 10.1103 / PhysRevLett.97.065701 , PMID 17026175 . (Variable names and signs adapted to ensure consistency on the page)
12. ↑ Calculated with GitHub
13. ↑ Unless the boundary surface directly favors nucleation ( ), the only change that has to be made for the Hamilton function changed in this way is to change the spin of all atoms belonging to the wall to 0. ${\displaystyle J_{s}=0}$
14. ↑ D.Frenkel : Physical chemistry: Seeds of phase change. In: Nature. 443, 2006, p. 641, doi: 10.1038 / 443641a .

literature

• Barry Cipra: An introduction to the Ising model , American Mathematical Monthly, Volume 94, 1987, pp. 937-959, pdf
• Barry McCoy , Tai Tsun Wu : The two dimensional Ising model , Harvard University Press 1973
• John Kogut : An introduction to lattice gauge theory and spin systems , Rev. Mod. Phys., Vol. 51, 1979, pp. 659-713
• Richard Feynman : Statistical mechanics , Benjamin 1972
• Kerson Huang : Statistical mechanics , Wiley 1987
• Stephen G. Brush : History of the Lenz-Ising model , Rev. Mod. Phys., Volume 39, 1967, pp. 883-893