Origins of the Ising model

TL;DR

This paper reviews the historical development of the Ising model from its inception in 1925 through its key theoretical validation in 1936, highlighting how simple two-state interactions lead to long-range order.

Contribution

It provides a detailed analysis of the origins and early theoretical understanding of the Ising model up to 1936, emphasizing its foundational significance.

Findings

Ising's 1925 analysis of a one-dimensional chain shows no ferromagnetic order.

Peierls' 1936 demonstration confirms the existence of ordered states in two dimensions.

The model's simplicity with two states and short-range interactions explains its long-term success.

Abstract

In 1925, Ernest Ising published a paper analyzing a model proposed in 1920 by Wilhelm Lenz for ferromagnetism. The model is composed of constituent units that take only two states and interact only when they are neighbors. Ising showed that in a linear chain the model does not present an ordered ferromagnetic state, a frustrating but correct result. However, Rudolf Peierls demonstrated in 1936 that the model does in fact present an ordered state in two dimensions, and therefore in three dimensions. This result reveals that short-range interaction and only two states for each constituent unit are sufficient for ordering to occur over long distances. These two elements are the key to understanding the success of the model and its variants even a hundred years after its appearance. Here we analyze the emergence of the model in the period up to 1936.

Equations

$$m=\mu (\coth \alpha -\frac{1}{\alpha }),\alpha =\frac{\mu H}{kT},$$

$$\alpha =\frac{\mu \lambda m}{kT}.$$

$$\frac{kT}{\mu \lambda }\alpha =\frac{1}{3}\alpha (1-c{\alpha }^2),$$

$$c{\alpha }^2=1-\frac{3kT}{\mu \lambda },$$

$$\frac{I}{I_0}=\coth \alpha -\frac{1}{\alpha },\alpha =\frac{3I}{I_0}\frac{\theta }{T},$$

$$\bar{\mu }=\mu \frac{e^a-e^{-a}}{e^a+e^{-a}}.$$

$$m(\sigma )=\mu \sum _i{\sigma }_i.$$

$$E(\sigma )=-J\sum _{i=1}^{N-1}{\sigma }_i{\sigma }_{i+1}-H\mu \sum _{i=1}^N{\sigma }_i.$$

$$P(\sigma )=\frac{1}{Z}{e}^{-\beta E(\sigma )},$$

$$\sum _{{\sigma }_1}\sum _{{\sigma }_2}\dots \sum _{{\sigma }_N}P(\sigma )=1,$$

$$\sum _{\sigma }P(\sigma )=1,$$

$$Z=\sum _{\sigma }{e}^{-\beta E(\sigma )}.$$

$$M=\sum _{\sigma }m(\sigma )P(\sigma ),$$

$$\frac{\partial Z}{\partial H}=\beta \sum _{\sigma }m(\sigma )e^{-\beta E(\sigma )}.$$

$$\frac{\partial }{\partial H}\ln Z=\beta \sum _{\sigma }m(\sigma )P(\sigma ),$$

$$M=\frac{1}{\beta }\frac{\partial }{\partial H}\ln Z.$$

$$E_n(\sigma )=-J\sum _{i=1}^{n-1}{\sigma }_i{\sigma }_{i+1}-H\mu \sum _{i=1}^n{\sigma }_i,$$

$$Z_n({\sigma }_n)=\sum _{{\sigma }_{n-1}}\dots \sum _{{\sigma }_1}{e}^{-\beta E_n(\sigma )},$$

$$Z_{n+1}({\sigma }_{n+1})=\sum _{{\sigma }_n}{e}^{\beta J{\sigma }_{n+1}{\sigma }_n+\beta H{\sigma }_{n+1}}{Z}_n({\sigma }_n).$$

$$A_{n+1}=e^{\beta J+\beta H}{A}_n+e^{-\beta J+\beta H}{B}_n,$$

$$B_{n+1}=e^{-\beta J-\beta H}{A}_n+e^{\beta J-\beta H}{B}_n,$$

$$A_n=a{\lambda }^n,B_n=b{\lambda }^n,$$

$$e^{\beta J+\beta H}a+e^{-\beta J+\beta H}b=\lambda a,$$

$$e^{-\beta J-\beta H}a+e^{\beta J-\beta H}b=\lambda b,$$

$$(e^{\beta J+\beta H}-\lambda )(e^{\beta J-\beta H}-\lambda )-e^{-2\beta J}=0,$$

$${\lambda }^2-2e^{\beta J}(\cosh \beta H)\lambda +e^{2\beta J}-e^{-2\beta J}=0.$$

$${\lambda }_1=e^{\beta J}(\cosh \beta H)+e^{\beta J}\sqrt{(\sinh \beta H{)}^2+e^{-4\beta J}},$$

$${\lambda }_2=e^{\beta J}(\cosh \beta H)-e^{\beta J}\sqrt{(\sinh \beta H{)}^2+e^{-4\beta J}}.$$

$$A_n=a_1{\lambda }_1^n+a_2{\lambda }_2^n,B_n=b_1{\lambda }_1^n+b_2{\lambda }_2^n.$$

$$Z=c_1{\lambda }_1^N+c_2{\lambda }_2^N.$$

Full text

I Introduction

The Ising model domb1960 ; brout1965 ; brush1967 ; ziman1972 ; salinas1997 ; niss2005 was conceived by Wilhelm Lenz in 1920 lenz1920 and proposed to his student Ernst Ising as a model for ferromagnetism. It was analyzed by Ising in his doctoral thesis entitled Contribution to the Theory of Ferromagnetism, defended at the University of Hamburg. The results of the thesis were published the following year in a paper of 1925 ising1925 . Although it is not a model for every type of ferromagnetism, the model is reasonably appropriate for the ferromagnetism of spin-half anisotropic magnetic systems domb1964 .

The Ising model consists of interacting dipoles residing on the sites of a lattice. The basic ingredients of the model, which result in ferromagnetism, are the interaction only between neighboring dipoles and only two states for each dipole. However, these two ingredients are not inherent to magnetic systems and can therefore be used in models to describe the ordering of systems of other natures, such as the ordering in metallic alloys. More generally, the model is capable of explaining and describing the cooperative state of systems composed of interacting elements domb1960 .

Ising solved the one-dimensional model in the presence of a magnetic field. The results showed that the magnetization vanishes when the field vanishes, leading Ising to correctly conclude that the one-dimensional model does not describe the ferromagnetic state. He further argued that the three-dimensional model would not exhibit the ferromagnetic state, a frustrating result given the purpose of the model. However, Herzfeld in 1925 herzfeld1925 referring to Ising paper stated that the result in three dimensions would need to be proven.

Lenz motivation lies in Pierre Weiss theory of ferromagnetism, published in 1907 weiss1907 . In his exposition of the theory, Weiss shows that the interaction between elementary magnetic dipoles cannot be magnetic in nature, a result also commented on by Lenz. The interaction does indeed have a quantum origin and was incorporated into the model for ferromagnetism introduced by Heisenberg in a paper of 1928 heisenberg1928 . In this papaer, Heisenberg comments on Ising result regarding the absence of ferromagnetism in a chain, and, in a letter to Pauli dated 1928, he suggested that if Ising had used a sufficient number of near neighbors, he would have obtained ferromagnetism niss2005 .

In a lecture on magnetism delivered at the 1930 Solvay conference, published in 1932 pauli1932 , Pauli used Heisenberg theory in conjunction with Weiss molecular field to obtain the ferromagnetic state. Pauli also obtains magnetization at low temperatures and comments that the three-dimensionality of the crystal lattice is essential for the emergence of ferromagnetism. He mentions the Ising result on the absence of ferromagnetism in a chain but states that this also occurs with quantum models. He adds that it is very likely that an extension of Ising theory to the case of a three-dimensional lattice would produce ferromagnetism.

In his book of 1932 vanvleck1932 on the theory of electric and magnetic susceptibilities, van Vleck discusses Heisenberg theory of ferromagnetism. He shows how to adapt Weiss molecular field theory to obtain the ferromagnetic state and the critical temperature of Heisenberg model. He claims that Ising results on the absence of ferromagnetism are in apparent contradiction with his own, justifying that this is supposedly due to Ising use of a coupling between elementary magnetic dipoles as being the product of two scalars rather than the scalar product of two vectors.

With the introduction of the Heisenberg model, considered more appropriate for describing ferromagnetism, we should have expected the Ising model to be forgotten brush1967 ; niss2005 . However, the model reappeared as models for the ordering of alloys and for the adsorption of gases on solid surfaces.

Gorsky in 1928 gorsky1928 , Bragg and Williams in 1934 bragg1934 ; bragg1935 , and Bethe in 1935 bethe1935 analyzed models for ordering in alloys that are equivalent to the Ising model, although Ising is not cited. Bethe paper is particularly interesting for two reasons. One is his concern to show that the model he analyzes does not support ordering in one dimension but does have ordering in two dimensions. The other is the use of an approximate method that becomes exact in one dimension.

The model analyzed by Ising was cited by Fowler in his paper of 1935 on the adsorption of gases on solid surfaces fowler1936 . Fowler comments that the problem he analyzes was solved exactly by Ising in a linear chain but not in two or more dimensions. A method equivalent to Bethe was used by Peierls in 1936 peierls1936a to analyze the Fowler model for adsorption.

In another paper of 1936 peierls1936 , Peierls acknowledges that the models analyzed by Bragg and Williams, Fowler, and Bethe are equivalent to the model analyzed by Ising. In this paper, whose title contains the term ’Ising’s model’, Peierls demonstrates that in two dimensions the Ising model does indeed exhibit the ordered state at sufficiently low temperatures. Peierls argument leads us to the conclusion that the ordered state also occurs in three-dimensional models.

With the recognition that the Ising model is equivalent to models of ordering in alloys, it came to be understood as a model for cooperative phenomena niss2005 . In this sense, the model was studied by Kramers and Wannier in 1941 kramers1941 as a model of statistical mechanics. In this paper, they showed that the partition function, and therefore the free energy, is related to the largest eigenvalue of a certain matrix. Furthermore, they managed to determine the exact value of the critical temperature in the square lattice through a symmetry relation between high and low temperatures.

The calculation of the free energy of the two-dimensional model in the absence of a field was performed by Onsager and published in 1944 onsager1944 . The calculation showed that the free energy $f$ as a function of temperature has a singularity at the critical temperature of the type $f-f_{0}\sim\varepsilon\ln|\varepsilon|$ where $\varepsilon$ is the deviation of the temperature from its critical value. From this result, the following singular behavior is obtained for the specific heat $c\sim\ln|\varepsilon|$. As for magnetization, it cannot be determined from the free energy at zero field. However, it can be obtained from correlations, a method used by Yang yang1952 in 1952. Near the critical temperature, he showed that magnetization behaves as $m\sim|\varepsilon|^{1/8}$.

Onsager and Yang results show that the critical behavior represented by critical exponents can be distinct from those obtained using Weiss theory oliveira2012 . For example, from this theory, the critical exponent associated with magnetization is $1/2$, while Yang result gives us the value $1/8$. These results, and others such as spontaneous symmetry breaking, explain the enormous success of the model in the area of phase transitions and critical phenomena, whose developement occurred mainly since the 1960s.

The Ising model, like other statistical mechanics models, is defined not only by the interaction energy between the numerous constituent units but also, and primarily, by the Gibbs probability distribution, usually used implicitly. Since the Gibbs distribution is the one that should occur in thermodynamic equilibrium, the Ising model is understood as a model that describes a system in thermodynamic equilibrium.

Thermodynamic equilibrium can also be understood as a final state of a stochastic dynamic. In 1963, Glauber introduced a dynamic of this nature constructed in such a way that the final distribution is the Gibbs probability distribution associated with the Ising model glauber1963 . This dynamic is defined by transition rates that obey the detailed balancing condition, which ensures that the distribution is equilibrium in the long run.

Stochastic dynamics tome2014 that do not obey detailed balance lead the system in the long run to a steady state out of thermodynamic equilibrium. A model of this type with transition rates containing the two basic ingredients of the Ising model was studied by the author in 1991 oliveira1992 . The study revealed that such a model has an ordered state like the Ising model and that the critical behavior is described by critical exponents identical to those of the Ising model. These results show that thermodynamic equilibrium is not a necessary condition for the occurrence of the ordered state, reaffirming that the ordered state is a consequence of only the two basic ingredients mentioned. This is particularly relevant for the study of systems that are active only when they are outside of thermodynamic equilibrium, as is the case with biological systems.

II Weiss

The theory of ferromagnetic ordering was not established by the introduction of the Ising model. Pierre Weiss proposed in 1907 a theory in which the ferromagnetic state is explained in terms of a molecular field weiss1907 . In a paramagnetic system, magnetization is null but can arise through an external field applied to the system. Weiss argues that in a ferromagnetic system, magnetization arises due to the field generated by the constituent units themselves, the molecules of the system, and therefore he called it the molecular field. More precisely, the field due to the neighbors of a given dipole is responsible for the dipole orientation. Weiss consistently assumed that the molecular field is proportional to the magnetization, so that in the absence of magnetization, there is no molecular field. It is important to note that Weiss emphasized that the molecular field arises from non-magnetic forces.

In presenting the theory, Weiss begins with the Langevin formula for the magnetization $m$ of a paramagnetic system subjected to a magnetic field $H$. We write the Langevin formula as

[TABLE]

where $\mu$ is the magnetic moment of a molecule, $k$ is Boltzmann constant, and $T$ is the absolute temperature. The function of $\alpha$ in parentheses on the right-hand side of equation (1) grows monotonically with $\alpha$ and saturates asymptotically at the value 1. Weiss replaces $H$ with the molecular field $\lambda m$, proportional to $m$, which is equivalent to writing

[TABLE]

One solution of equation (1) with $\alpha$ given by (2) is $m=0$, but there is another non-zero one that occurs at temperatures lower than a critical temperature.

Next we obtain the non-zero solution for the case where $m$ is small and therefore $\alpha$ is small. In this case the function in parentheses in (1) is $\alpha(1-c\alpha^{2})/3$, where $c$ is a positive numerical constant, and therefore equation (1) reduces to

[TABLE]

and we see that one solution is $\alpha=0$. The non-zero solution is given by

[TABLE]

and exists if $T<\mu\lambda/3k$. The non-trivial solution corresponds to spontaneous magnetization and therefore it vanishes when $T=\mu\lambda/3k$, which is the critical temperature.

Using Weiss original notation, the equation for magnetization, which is equation (1) with $\alpha$ given by (2), reads

[TABLE]

where $I$ is the magnetization, $I_{0}$ is the saturation magnetization, and $\theta$ is the critical temperature, called by Weiss the temperature of the loss of spontaneous ferromagnetism. The solution to this equation is shown in figure 1. Weiss also performed experiments to verify formula (5) using a sample of magnetite. Weiss experimental method consisted of shaping the sample into an ellipsoid and suspending it by means of a torsion spring in the presence of an electromagnet field. In equilibrium, the magnetic torque is equal to that exerted by the spring and is proportional to the square of the magnetization intensity. Weiss experimental measurements were in the temperature range between that of dry ice, the solid form of carbon dioxide, ($-79$ ∘C) and that of the disappearance of ferromagnetism in magnetite ($587$ ∘C). The experimental values are presented in figure 1.

III Lenz

The genesis of the Ising model is found in Lenz paper of 1920 lenz1920 on the magnetism of paramagnetic salts. Lenz argues that elementary magnetic dipoles in crystalline solids are free to assume various positions. A quantum treatment, however, shows that certain positions must be distinguished, for example, the extreme positions. Assuming that the potential energy is large between the two extreme positions, then only these should occur equivalently. In the presence of an external magnetic field $H$, the equivalence between these two positions disappears, resulting in a non-zero magnetic moment. The probability of the occurrence of the position such that the elementary dipole moment $\mu$ is in the direction of the field is proportional to $e^{\alpha}$ where $\alpha=\mu H/kT$ and $T$ is the absolute temperature. The probability of the other position, a dipole in oppositition to the field, is smaller and proportional to $e^{-\alpha}$. From these results, Lenz obtains the average dipole moment,

[TABLE]

Lenz then shows how spontaneous magnetization arises in ferromagnetic bodies. He argues that the potential energy between neighboring elementary magnetic dipoles favors one of two extreme positions, thus leading to the emergence of spontaneous magnetization. He further adds that ferromagnetic properties can be explained in terms of non-magnetic forces, in agreement with Weiss view, who established that the force between elementary dipoles is non-magnetic in nature.

IV Ising

Definition of the model

Consider a sequence of $N$ sites along a straight line, equally spaced. The sites are labeled from $1$ to $N$. To each site $i$ we associate a dipole that takes only two values along a given direction. The two values are $+\mu$ and $-\mu$. Defining for each site $i$ a variable $\sigma_{i}$ that takes the values $+1$ or $-1$, then the dipole of site $i$ can be written as $\mu\sigma_{i}$. A state of the system is defined by the vector $\sigma=(\sigma_{1},\sigma_{2},\ldots,\sigma_{N})$. The total magnetization of the system in a given state is therefore

[TABLE]

According to Lenz, only neighboring dipoles interact. Therefore, the contribution to the total energy comes only from neighboring dipoles. When they are parallel, the energy is lower than in the antiparallel configuration. Considering the dipoles located at neighboring sites $i$ and $i+1$, then they will be parallel when $(\sigma_{i},\sigma_{i+1})$ is equal to $(+,+)$ or $(-,-)$. The antiparallel configuration occurs when $(\sigma_{i},\sigma_{i+1})$ is equal to $(+,-)$ or $(-,+)$. In the parallel configuration, the energy is assumed to be $-J<0$ and in the parallel condition as being $J>0$. Thus, the energy between two neighboring sites can be written as $-J\sigma_{i}\sigma_{i+1}$.

The total energy associated with a state $\sigma$ contains two contributions. One is the sum of the interaction energies between neighboring sites. The other is the contribution due to an external field $H>0$. If the dipole located at $i$ is parallel to the field, $\sigma_{i}=+1$, the contribution is $-H\mu$. If it is antiparallel to the field, $\sigma_{i}=-1$, then the contribution is $H\mu>0$. In both cases the contribution is $-H\mu\sigma_{i}$. The total energy $E(\sigma)$ corresponding to a state $\sigma$ is therefore

[TABLE]

Assigning an energy to each state is not sufficient to determine which state should occur. To do this, we assume that the different states occur with a certain probability, which we choose as the one introduced by Gibbs, which describes systems in thermodynamic equilibrium, and is given by

[TABLE]

where $\beta=1/kT$, $k$ is Boltzmann constant and $T$ is the absolute temperature, and $Z$ is a normalization factor such that

[TABLE]

which we write abbreviated in the following form

[TABLE]

and therefore $Z$ is given by

[TABLE]

Therefore, the Ising model is defined not only by the energy function (8), as is usually said, but also by the probability distribution.

To determine the average magnetization, defined by

[TABLE]

we use the following results. Differentiating $Z$ with respect to $H$, we obtain

[TABLE]

Dividing both sides by $Z$, and using (9)

[TABLE]

and therefore

[TABLE]

We therefore see that the calculation of $M$ is reduced to the determination of $Z$, known as the partition function.

Calculation of the partition function

It is convenient to define

[TABLE]

and the quantity

[TABLE]

and note that we are not summing in $\sigma_{n}$. Using these definitions we obtain the following relation

[TABLE]

Using the notation $A_{n}=Z_{n}(+1)$ and $B_{n}=Z_{n}(-1)$, then

[TABLE]

[TABLE]

Assuming solutions of the type

[TABLE]

then

[TABLE]

[TABLE]

which is an eigenvalue equation. Eigenvalues are solutions of

[TABLE]

that is, they are solutions of the equation

[TABLE]

The solutions are

[TABLE]

[TABLE]

The general solution is then

[TABLE]

Since the partiion function is $Z_{N}=A_{N}+B_{N}$ then

[TABLE]

Magnetization

The magnetization per site $m=M/N$ is obtained through

[TABLE]

and we will do this calculation for very large $N$. Using the previous result for $Z$, we can write

[TABLE]

Now we note that $\lambda_{1}>\lambda_{2}$ for $T\neq 0$ and therefore if $N$ is large enough the fraction inside the parentheses can be neglected. Therefore the magnetization per site is given by

[TABLE]

and is therefore related to the larger of the two eigenvalues.

Differentiating $\ln\lambda_{1}$ with respect to $H$ we obtain

[TABLE]

which is the result obtained by Ising ising1925 and contained in his 1925 paper. When $H\to 0$ we see that $m\to 0$ and therefore there is no spontaneous magnetization in the one-dimensional Ising model.

V Gorsky

The ferromagnetic state can be understood as a state with magnetic order. We can imagine that systems of other natures could also exhibit ordered states. This indeed occurs, for example, in metallic alloys. The occurrence of the ordered state in metallic alloys was predicted by Tammann in 1919 and subsequently confirmed by other authors in various alloys. In a paper of 1928 on the CuAu alloy, Vadim Gorsky gorsky1928 introduced a statistical model to describe the transition from the disordered to the ordered state in this crystalline alloy.

A crystalline alloy is a solid composed of two or more types of atoms arranged in a crystalline structure. Typically, each site in the crystal lattice can be occupied by any type of atom, and the alloy is understood as a homogeneous mixture. However, it is possible for atoms of one type to be preferentially located in certain sites of the crystal lattice, as illustrated in figure 2. Consider a binary alloy in a crystal lattice consisting of two intertwined sublattices and assume that half of the atoms are type A and the other half are type B. If the atoms of type A are preferentially located in one sublattice and therefore the atoms of type B preferentially in the other, then we are dealing with an ordered state. The ordered state disappears when the atoms A and B are equally distributed in either of the two sublattices. This state is called disordered and should not be confused with the disordered state that characterizes amorphous solids.

To understand Gorsky model, we consider the crystal lattice just described, composed of two sublattices labeled 1 and 2. The lattice contains $N$ atoms, half of which are type A and the other half are type B. The degree of ordering is characterized by the number of atoms of each type in one of the sublattices, say sublattice 1, since the other will have complementary numbers. Furthermore, it is sufficient to refer to the number of atoms $n$ of one type, say type A, since the number of atoms of another type will be $N-n$. Gorsky chooses the fraction $\alpha=n/N$ to represent the degree of ordering. Ordering occurs for $1\geq\alpha>1/2$, and $\alpha=1/2$ characterizes the disordered state.

In his model, Gorsky refers to “ordered” sites and “disordered” sites which we interpret as the sites of sublattice 1 and sublattice 2, respectively, relative to the atoms of type A. Ordering is reduced to determining the number of atoms in the “ordered” sites, which means determining the number of atoms of type A in sublattice 1, which we denote by $n$.

The Gorsky model is a dynamic model and corresponds to a stochastic process. Following Gorsky, we denote by $a_{1}$ the rate of transition of an atom A from 1 to 2 and by $a_{2}$ that of an atom A from 2 to 1. These rates depend on temperature. The number of A atoms that are transferred per unit time from 1 to 2 is therefore $na_{1}$ and from 2 to 1 is $(N-n)a_{2}$. In equilibrium we must have

[TABLE]

and therefore

[TABLE]

Assuming that in the equilibrium state the probability distribution of the configurations is in agreement with the Gibbs equilibrium distribution then the rates must obey the condition

[TABLE]

where $u_{1}$ and $u_{2}$ are the energies of A at a site on sublattices 1 and 2, respectively. Therefore

[TABLE]

Gorsky introduces the hypothesis that as the system approaches the disordered state, that is, as $\alpha$ approaches $1/2$, the difference $u_{2}-u_{1}$ vanishes. He then admits the following relation

[TABLE]

and therefore

[TABLE]

Substituting this result into (36), Gorsky arrives at the following relation

[TABLE]

which is an equation that determines $\alpha$. This equation has a solution $\alpha=1/2$ that describes the disordered state. But it also has a solution such that $\alpha>1/2$ that describes the ordered state and occurs if $T<c/2k$.

It is worth noting the analogy of this model with Weiss molecular field theory. Gorsky hypothesis (39) is analogous to Weiss hypothesis that the molecular field is proportional to the magnetization. If we use the parameter $m$, defined by $m=2\alpha-1$, then equation (41) can be written in the form

[TABLE]

which is analogous to equation (1) supplemented by equation (2).

VI Bragg and Williams

In a 1934 paper bragg1934 , William L. Bragg and Evan Williams proposed a model for the ordering of alloys that is equivalent to the model proposed by Gorsky. However, they extended the model to more complex lattices than the one considered by Gorsky. In a subsequent paper of 1935 bragg1935 , they acknowledged Gorsky work. Below, we describe the Bragg and Williams model in its simplest form, in which the crystal lattice consists of two sublattices denoted by $\alpha$ and $\beta$ and by two types of atoms, A and B. With respect to the A atoms, the sites of the $\alpha$ sublattice are understood as positions of order and those of the $\beta$ sublattice as positions of disorder. Denoting by $p$ the probability of a site $\alpha$ being occupied by an atom A then the degree of order is defined by $S=2p-1$, and $0<S\leq 1$ and $1/2<p\leq 1$, in the ordered state, and $S=0$ and $p=1/2$ in the disordered state.

Assuming initially that the atoms do not interact, then the potential energy associated with a site depends only on which atom occupies the site. Let $V_{\rm a}$ and $V_{\rm b}$ be the potential energies associated with a site $\alpha$ when it is occupied by an atom A and B, respectively. Using the Boltzmann relation, the probabilities of occupation by an atom A and B are respectively proportional to $e^{-V_{\rm a}/kT}$ and $e^{-V_{\rm b}/kT}$. The ratio $p/(1-p)$ between these probabilities is therefore

[TABLE]

where $V=(V_{\rm a}-V_{\rm b})/2$. From this relation we obtain

[TABLE]

and from it we find $S=2p-1$,

[TABLE]

Bragg and Williams then introduce the hypothesis that $V$ is proportional to $S$,

[TABLE]

This relationship is analogous to that used by Weiss that the molecular field is proportional to the magnetization and also used by Gorsky, as we saw above. From this hypothesis, equation (45) becomes the equation

[TABLE]

and we see that it is equivalent to equation (41) obtained by Gorsky. To do this, simply compare it with (42) which is equivalent to equation (41).

Equation (47) has a solution $S=0$ and also a non-zero solution as long as the temperature is less than the critical temperature. To show the existence of a non-zero solution, Bragg and William use a graph showing the functions $f=\tanh x/4$ and $g=kTx/V_{0}$, as shown in figure 3. The intersection of these two functions gives the solution to equation (47). When $x$ is small, we can use the approximation $f=x/4$. Therefore, if $kT/V_{0}>1/4$, the only solution is $x=0$, or $S=0$. For $kT/V_{0}<1/4$, the non-zero solution appears, and therefore the critical temperature is given by $kT_{c}/V_{0}=1/4$, or

[TABLE]

To explicitly determine the non-zero solution near the critical temperature, simply expand $f$ to cubic order terms and compare and equate to $g$. The result is

[TABLE]

The non-zero solution of this equation is given by

[TABLE]

The general solution of $S$ as a function of temperature is shown in figure 4.

The results presented above correspond to the simplest model in which the crystal lattice divides into two sublattices with the same number of sites in each. Bragg and Williams also consider more complex cases in which the crystal lattice divides into more than two sublattices. Considering $n$ of sublattices, then equation (47) is replaced by

[TABLE]

where $r=1/n$.

VII Bethe

The theoretical approaches of Gorsky and of Bragg and Williams to the ordering of atoms in metal alloys are similar to that employed by Weiss for ferromagnetism. Hans Bethe treatment of this problem on the other hand differs from these approaches and comes closer to that given by Lenz and Ising for ferromagnetism although Bethe does not name either of these two authors bethe1935 .

The model considered by Bethe describes a binary alloy AB consisting of two sublattices. The sublattices are intertwined in such a way that all neighboring sites of a site on one sublattice belong to the other sublattice. The number of A atoms is equal to the number of B atoms, and the completely ordered state is one in which A atoms occupy the sites of one sublattice and B atoms occupy the other sublattice. The atoms interact only when they are in neighboring sites. The interaction energy between an A atom and a B atom is $V_{\rm ab}$, between two A atoms is $V_{\rm aa}$, and between two B atoms is $V_{\rm bb}$. The energy $V_{\rm ab}$ is lower than the other two, so the completely ordered state has the lowest energy.

Bethe solves the model using a statistical method to approximately determine the number of pairs of sites of the various types: AA, BB, and AB. This method has the peculiarity of solving the one-dimensional model exactly, concluding that there is no ordered state. However, Bethe argues before using the method that the interaction between first neighbors is not sufficient to establish the ordered state in the one-dimensional model.

The critical temperature obtained by Bethe for a regular lattice is proportional to the inverse of $\ln[z/(z-2)]$, where $z$ is the number of neighbors of a site. In one dimension $z=2$ and therefore the critical temperature is zero, which means that there is no ordering at finite temperature and the disordered state occurs at any finite temperature in one dimension.

We then derive Bethe results using a different approach from the one originally employed by Bethe. The method we use is a reformulation of that proposed by Ferreira, Salinas, and Oliveira in 1977 ferreira1977 . To do this, we begin by representing the configuration of the atoms by means of a variable $\sigma_{i}$ that assumes the values $+1$ or $-1$ according to the following convention. If site $i$ belongs to one of the sublattices, then $\sigma_{i}=+1$ if the site is occupied by an atom A and $\sigma_{i}=-1$ if it is occupied y B. If site $i$ belongs to the other sublattice, the opposite occurs: $\sigma_{i}=-1$ if the site is occupied by A and $\sigma_{i}=+1$ if the site is occupied by B. Note that using these variables, the ordered state is one in which all variables $\sigma_{i}$ have the same sign.

Using these variables the total energy of the system is

[TABLE]

where $J=(V_{aa}+V_{bb}-2V_{ab})/4>0$ and the sum extends over pairs of neighboring sites. The probability $P(\sigma)$ of the occurrence of a configuration $\sigma$ is given by the Gibbs equilibrium distribution

[TABLE]

where $\beta=1/kT$.

From the Gibbs distribution we can determine several marginal distributions. The one that interests us here is the one associated with a central site labeled zero and its $z$ nearest neighbors, which we denote by $P(\sigma_{0},\sigma^{\rm v})$ where $\sigma^{\rm v}=(\sigma_{1},\ldots,\sigma_{z})$ denotes the collection of variables associated with the neighboring sites of the central site. This distribution is obtained from the Gibbs distribution by summing over all variables other than those associated with the central site and its neighbors.

Next we derive an equation that $P(\sigma_{0},\sigma^{\rm v})$ must obey to ensure that it describes a system in equilibrium, that is, that it is a marginal distribution of $P(\sigma)$ given by (53). To do this we start with the following identity that follows directly from (53)

[TABLE]

which we write in the form

[TABLE]

The term in brackets depends only on site $i$ and its neighbors. Therefore, we can perform the sum on both sides of this equation in the variables of the other sites to obtain for $i=0$ the relation

[TABLE]

where $s$ is an abbreviation for the sum of the variables associated with the neighbors of site 0. We call this equation the thermodynamic equilibrium relation.

The approach we consider consists of using an approximation to $P(\sigma_{0},\sigma^{\rm v})$ obtained as follows. We first write

[TABLE]

and then we approximate the conditional probability distribution by the product of conditional probabilities associated with pairs of neighboring sites, that is,

[TABLE]

Since

[TABLE]

then the approximation to $P(\sigma_{0},\sigma^{\rm v})$ is given by

[TABLE]

We assume that the pair distribution is of the form

[TABLE]

where $K$ and $H_{2}$ are parameters to be determined and

[TABLE]

As for the distribution $P(\sigma_{0})$ it is given by

[TABLE]

where $H_{1}$ is a parameter to be determined and

[TABLE]

Since the one-site distribution must be the marginal distribution of the pairwise distribution, then $m=\langle\sigma_{0}\rangle$ must have the same value whether we use the one-site distribution or the pairwise distribution. Using the former, we obtain

[TABLE]

Using the second one, we get

[TABLE]

Now we use the thermodynamic equilibrium relation (56) to find the relationship between the parameters $K$, $H_{1}$ and $H_{2}$. From (61) and (63), we obtain

[TABLE]

where $s$ represents the sum of the variables associated with the neighbors of the central site. Comparing with (56), we obtain the relations

[TABLE]

Relations (65), (66), and (68) determine $m$ as a function of temperature. One solution of these equations is $m=0$, which corresponds to the disordered state. However, there is a nonzero solution that corresponds to the ordered state that occurs at a temperature lower than a critical temperature. To determine it, we write equations (65) and (66) for small values of $H_{1}$ and $H_{2}$, which are

[TABLE]

Using (68) we obtain the relation

[TABLE]

Remembering that $\beta=1/kT$, we obtain from this relationship the critical temperature

[TABLE]

which is the result obtained by Bethe.

VIII Fowler

The work of Bragg and Williams and Bethe on order-disorder in alloys inspired Ralph Fowler to develop a similar model for gas adsorption on solid surfaces, published in a paper of 1936 fowler1936 . Fowler turns his attention to adsorption by a single monomolecular layer. He acknowledges that the existence of a critical temperature in this system had been previously discussed by Frenkel in 1924 frenkel1924 .

Frenkel analyzes the formation of a layer of atoms adsorbed on a surface. He assumes the following equation for the number of atoms $n$ adsorbed on a surface,

[TABLE]

where $\nu$ is the rate of arrival of atoms at the surface, and the other two terms refer to the departure of atoms from the surface. The term $\alpha n$ refers to isolated atoms and the term $\beta n^{2}$ refers to bonded atoms. In the steady state, $dn/dt=0$, the equation to be solved is $\beta n^{2}-\alpha n+\nu=0$, which has a solution provided $\nu\leq\alpha^{2}/4\beta$. Therefore, there is a critical value

[TABLE]

above which the formation of the adsorbed layer becomes impossible. Since $\alpha$ and $\beta$ depend on temperature, Frenkel concludes hat $\nu_{c}$ must correspond to a critical temperature.

In Fowler model, a solid surface contains $N$ sites where atoms can be deposited. Each site can accommodate only one atom. Fowler assumes that the energy of a layer of $M$ atoms is

[TABLE]

where $-h$ is the energy associated with an atom, $X$ is the number of pairs of nearby neighbors, and $-\varepsilon$ is the interaction energy associated with a pair of neighboring atoms. The partition function is determined by

[TABLE]

where $g$ is the number of arrangements of $M$ atoms in $N$ sites that produce $X$ pairs. Fowler claims that this problem was solved exactly by Ising in a linear chain but not in two-dimensional or higher-dimensional lattices.

The calculation of the function is calculated by an approximate method that consists of assuming that $X=M\theta/2$, $\theta=M/N$ which Fowler claims is the same type of approximation used by Bragg and Williams. With this approximation and considering that

[TABLE]

then

[TABLE]

From this equation Fowler obtains the result

[TABLE]

To simplify Fowler results, we use the abbreviation $\mu=-kT\partial\ln\zeta/\partial\theta$ to write equation (78) as

[TABLE]

The quantity we abbreviate as $\mu$ is related to the pressure and temperature of the gas that is in equilibrium with the adsorbed layer. Therefore, $\mu$ is a function of pressure and temperature, and equation (79) gives $\theta$ as a function of $\mu$.

IX Peierls

Adsorption model

In 1936, Rudolf Peierls published two consecutive papers related to the Ising model. In the first, peierls1936a , he analyzes the model for the adsorption of atoms on solid surfaces that had been introduced by Fowler and published in the same year, 1936. Peierls uses an approximate method that is equivalent to that introduced by Bethe.

The model and approach used by Peierls are as follows.

1. Each site in a regular lattice of $N$ sites can accommodate a single atom. 2) The adsorption energy of each atom is $h$.
2. The interaction energy of each pair of neighboring atoms is $-V$. Using a variable $\eta_{i}$ that takes the values $+1$ if site $i$ is occupied and [math] if site $i$ is empty, then the number $M$ of sites occupied by an atom is

[TABLE]

and the number $X$ of pairs of neighboring sites occupied by atoms is

[TABLE]

in which the sum extends over pairs of neighboring sites. The total energy of a configuration $\eta$ is

[TABLE]

and the probability of a configuration of atoms is given by

[TABLE]

Next, we consider the marginal probability distribution associated with a central site $i=0$ and its $z$ neighboring sites. The occupancy variables for these sites are $\eta_{0},\eta_{1},\ldots,\eta_{z}$. For convenience we define

[TABLE]

and also $E^{\prime}=E-E_{0}$. Therefore, $P(\sigma)$ can be written as

[TABLE]

We note that $E_{0}$ depends only on the variables of the central site and its neighbors. Summing $P(\eta)$ over all variables except these, we obtain

[TABLE]

where $\phi$ is the sum of $e^{-VE^{\prime}/kT}$ in all variables except the variables of the central site and its neighbors. We note that $\phi$ does not depend on $\sigma_{0}$ because $E^{\prime}$ does not depend on that variable. Using (84) we can write

[TABLE]

where we are using the abbreviations $a=e^{-\mu/kT}$ and $b=e^{V/kT}$. Then $\phi$ is approximated by

[TABLE]

where $c$ is a constant to be determined, so that

[TABLE]

Before proceeding, we need to determine $Z$, which is given by

[TABLE]

From the probability distribution (89), we determine the probabilities $P_{0}(\eta_{0})$ and $P_{1}(\eta_{1})$ associated with the central site and one of its neighbors. Summing the probability (89) over all variables except $\eta_{0}$, we obtain

[TABLE]

Summing the probability (89) in all variables except $\eta_{1}$, we obtain

[TABLE]

Since the system is homogeneous, the probability of occupation of a site is independent of the site, which leads us to the condition $P_{0}(1)=P_{1}(1)$. The fraction $\theta=M/N$ of sites occupied by a molecule is therefore equal to $P_{0}(1)$. Using (91), we obtain the equation

[TABLE]

Similarly, $\theta$ is equal to $P_{1}(1)$ and using (92), we obtain the equation

[TABLE]

Equating the equations we obtain

[TABLE]

The solution to equation (95) gives us $c$ in terms of $a$ and $b$, which substituting into (93) gives us $\theta$. To determine the solution to these equations we use the abbreviation

[TABLE]

and therefore $c=(r-1)(b-r)$. Equations (93) and (95) can then be written as

[TABLE]

[TABLE]

Solving this equation we determine $r$, which substituted into (97) gives us $\theta$.

Next we consider the case where $a$ and $b$ are related by $a^{2}b^{z}=1$. Furthermore we consider the case where $ar^{z}$ is close to unity, which means that $\theta$ is close to $1/2$. Defining the deviation $\varepsilon$ by $ar^{z}=1+\varepsilon$, then up to terms of order $\varepsilon^{3}$, equation (98) becomes

[TABLE]

A solution to this equation is $\varepsilon=0$ which gives $\theta=1/2$. There is however a non-zero solution to $\varepsilon$ given by

[TABLE]

that exists if $z>2$ and for $\sqrt{b}>z/(z-2)$. Remembering that $b=e^{V/kT}$ this condition gives us

[TABLE]

Therefore the ordered state occurs above a critical temperature $T_{c}$ given by

[TABLE]

Peierls argument

In the other paper, Peierls presents the proof that the Ising model defined on the square lattice presents the ferromagnetic state at sufficiently low temperatures. Peierls proof is as follows. To each site of a square lattice, we assign a positive or negative sign, representing the two states each site can assume. The most common way to define the Ising model is to designate the values $-J<0$ or $J>0$ for the energy between two neighboring sites when they have the same or different signs, respectively. However, it is more convenient to designate the values 0 or $\varepsilon>0$, respectively, Thus, the total energy is equal to the number of neighboring pairs of opposite signs, multiplied by $\varepsilon$.

It is convenient to construct a line segment of length equal to unity for each pair of neighboring sites with opposite signs, as shown in figure 5. A configuration of the entire lattice is composed of positive and negative signs and lines that determine the boundary between regions of opposite signs. Since each unit segment of the boundary contributes an energy equal to $\varepsilon$, the total energy is proportional to the total length of the boundary. It is important to note that the boundaries completely determine the sign configuration, except for the possibility of replacing all positive signs with negative ones and vice versa, which is unimportant for a zero field.

Peierls observes that there are closed and open boundaries. The latter begin and end at the sides of the lattice. Peierls demonstrates that at sufficiently low temperatures, the area bounded by closed boundaries and intersected by open boundaries is only a small fraction of the total area. Since subtracting the area of regions of distinct signals is proportional do the magnetization, this result shows that this quantity is strictly nonzero, and therefore the model exhibits the ferromagnetic state.

Peierls demonstration, however, turned out to be defective but amenable to correction. This was later done by Griffiths in 1964 griffiths1964 , who stated that Peierls argument is valuable for explaining why the Ising model exhibits phase transitions in two dimensions but not in a linear chain. Furthermore, the argument can easily be extended to the three-dimensional cubic lattice.

In his memoir peierls1985 , published in 1985, almost fifty years after his demonstration, Peierls recalls the motivation that led him to it. He says the inspiration came from his irritation with certain works that seemed incorrect, unnecessarily complicated, and based on unreasonable assumptions. This was the case with a lecture on the Ising model given by a mathematician who argued that the model would not exhibit the ferromagnetic state in two or three dimensions. Peierls realized the inaccuracy, but instead of looking for flaws in the argument, he sought to demonstrate that ferromagnetism can occur in two and three dimensions. He then says he was surprised to learn, a few years before 1985, that his 1936 argument had been extensively used in statistical mechanics as one way to demonstrate the existence of the ordered state.

X Conclusion

We analyzed the emergence of the Ising model from its conception by Lenz to Peierls, who unequivocally demonstrated that the model possesses order in two dimensions. The success of the model is due to the fact that it consists of only the necessary ingredients to bring about the emergence of an ordered state. We saw how equivalent models were introduced within the field of metalic alloys, particularly by Bethe, who was concerned with demonstrating that the model he analyzed does not sustain order in one dimension but does possess order in two dimensions. This was subsequently demonstrated by Peierls for the Ising model in a 1936 paper. Peierls paper is also important for emphasizing that the models used for metalic alloys by Bragg and Williams and by Bethe, as well as the one used by Fowler for adsorption, are equivalent to the Ising model.

The Ising model is usually considered a mathematical model, a model that contains only the essential ingredients to obtain the properties we wish to describe. In general, this term, mathematical model, is used to denote theories that do not include all the factors that can influence the construction of a more realistic theory. The usual justification for introducing the mathematical model is that a realistic theory would be too complicated both from the point of view of construction and of obtaining results. However, we must keep in mind that what is called a realistic theory may be devoid of meaning, or at least needs to be better defined, since a scientific theory is a construction, an invention of our minds. The Ising model is a significant example of this human endeavor.

Acknowledgements

I wish to thank Sílvio Salinas and Jürgen Stilck for their suggestions and critical reading of the manuscript.