**2. Application of Ising model**

Ising model has been extensively used for solving a variety of problems [3–18]. Some of the problems are discussed, here, with appropriate examples.

#### **2.1 Phase separation and wetting/dewetting**

Ising model was first exploited for investigating spontaneous magnetization in ferromagnetic film (i.e. magnetization in the absence of external magnetic field). An example case of Ising model using metropolis algorithm is shown in **Figure 3**. Transition temperature depends on the strength of the inter-spin exchange coupling; the dominating term governs the kinetics, when long-range interactions are introduced in the calculations. Latter, it was used to study phase separation in binary alloys and liquid-gas phase transitions (i.e., condensation of molecule in one region of space of the box). Binary alloys constitute of two different atoms. At temperature T = 0, Zn-Cu alloy; known as brass, gets completely ordered. This state is said to be β-brass. In β-brass state, each Zn atom is surrounded by eight copper

**Figure 3.**

*Variation in critical temperature vs. next nearest exchange coupling for a bcc lattice (reproduced with permission from Singh [3]).*

atoms, placed at the corners of the unit cell of the body-centered cubic structure and vice versa. The occupation of each site can be represented by:

$$n\_i = \begin{cases} \mathbf{1} & \text{if site i is occupied by atom A} \\ \mathbf{0} & \text{if site i is occupied by atom B} \end{cases} \tag{1}$$

The interaction energy between A-A, B-B, and A-B type of atoms are represented by εAA, εBB, and εAB, respectively. Phase separation has been studied vastly, using Ising model [4–6]. A phase is simply a part of a system, separated from the other part by the formation of an interface; that essentially means that two components aggregate and form rich regions of A and B type of molecules with an interface in between them. The evolution of two distinct phases, when an initial random but homogeneous mixture is annealed below a definite temperature, is known as phase separation. Phase separation leads to discontinuity and inhomogeneity in the systems. This happens because the phase-separated regions are energetically more stable. Phase separation has been an old problem and has been extended to study diverse phenomena ranging from magnetic liquid-liquid phase separation to proteinprotein phase separation in biological systems. This process has also been studied in the presence of external surfaces having affinity to one type of atom or molecule (**Figure 4a**). Both theoretical and experimental methods have been exploited and have been found in close agreement. Formation of long ridges and circular drops has been reported numerous occasions using lattice-based Ising model. For example, one may look into John W. Cahn research paper published in *The Journal of Chemical Physics* in the year 1965. The TEM image taken for Vycor, in which one phase had been leached out and the voids were filled with lead (**Figure 4b**).

#### **2.2 Lattice-based liquid-gas model**

Yang and Lee first coined the term lattice gas in the year 1952. A lattice should have larger volume (V) than the number of lattice molecules (N), so that some of the nodes or lattice vertices are left empty (i.e., N < V). No lattice vertex can be occupied by more than one particle. The interaction potential between two atoms at lattice sites i and j is given by Eq. (2):

$$U(r\ddot{y}) = \begin{cases} -\in & \text{if } \mathbf{S\_i} = \mathbf{S\_j} = \mathbf{1.0} \text{ and } \text{rij} = \mathbf{1.0} \\ \Leftrightarrow & \text{if } \text{rij} = \mathbf{0.0} \\ \mathbf{0} & \text{else} \end{cases} \tag{2}$$

For surface affinity of lower surface to ith liquid molecule, we chose:

$$V(ri) = \begin{cases} -\frac{J\_0}{r(i)} & \text{if } \mathbf{S\_i} = \mathbf{1.0} \\ \mathbf{0} & \text{otherwise} \end{cases} \tag{3}$$

nearest neighbor (i.e., i = n and i + 1 = n + 1). Change in energy is calculated during exchanging of these two sites; the exchange move is accepted, if Exp [ΔE/kBT] is found to be greater than or equal to a random number generated between [0, 1]. For all cases of studies here, ε = 1 and J0 = 12.0, and only the lower surface is functional, while the upper surface has only hard-sphere interaction with the fluid molecules.

*(a) Surface-directed phase separation and dewetting in conserved binary mixture using two-dimensional lattices of size 200 100 nodes. The conserved components are taken in ratio 70:30 at T = 0.70. Majority component is attracted by upper and lower substrates, whereas the minority component has repulsive interaction with the two interfaces. Periodic boundary conditions are applied along X-direction. The micrograph is taken after completion of 30,000 Monte Carlo cycles using Kawasaki exchange method (the figure is reproduced with permission from Singh [5]). (b) Shows Transmission Electron Microscope (TEM) image of unsintered Vycor with one phase replaced by lead (X 200000). Reproduced with permission from W. G. Schmidt and R. J. Charles, Journal of Applied Physics 35, 2552 (1964); doi: 10.1063/1.1702905.*

**Figure 6** shows micrograph of self-aligned liquid columns. The system evolves from an initial homogeneous mixture of liquid- and gas-like molecules obtained by annealing the system at high temperature for few thousand MC cycles. Dynamic Monte Carlo simulation has been used with continuous but random trial movements of the molecules. The lattice-based Ising model using Eqs. (2) and (3) is also

Crystalline solids possess short- and long-range order along its crystal axes and maintain its periodicity in three dimensions. Liquids possess only short-range order, and its molecules have no long-range correlation. Liquid molecules retain only short-range order. Gases possess neither of the two. These are the three phases, in

which any matter may exist. What are the glasses then? Glasses are solids,

Average number density for liquid-like molecules is taken as 0.25 [16].

supposed to give same results, at least qualitatively.

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

**2.3 Spin glasses**

**119**

**Figure 4.**

The occupation number (ni) of a lattice site i is given by:

$$m\_i = \begin{cases} 1 & \text{if site i is occupied} \\ 0 & \text{if site i is un-ocoupled} \end{cases} \tag{4}$$

One example case is shown in **Figure 5**. Here, we chose lattice size of 128 � 128 � 48. The fluid-fluid molecule and wall-liquid molecule interactions are defined, respectively, in Eqs. (2) and (3). In canonical ensemble, the threedimensional lattice is swept one by one; by choosing sites regularly with one of its

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

#### **Figure 4.**

atoms, placed at the corners of the unit cell of the body-centered cubic structure

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

*ni* <sup>¼</sup> 1 if site i is occupied by atom A 0 if site i is occupied by atom B

The interaction energy between A-A, B-B, and A-B type of atoms are represented by εAA, εBB, and εAB, respectively. Phase separation has been studied vastly, using Ising model [4–6]. A phase is simply a part of a system, separated from the other part by the formation of an interface; that essentially means that two components aggregate and form rich regions of A and B type of molecules with an interface in between them. The evolution of two distinct phases, when an initial random but homogeneous mixture is annealed below a definite temperature, is known as phase separation. Phase separation leads to discontinuity and inhomogeneity in the systems. This happens because the phase-separated regions are energetically more stable. Phase separation has been an old problem and has been extended to study diverse phenomena ranging from magnetic liquid-liquid phase separation to proteinprotein phase separation in biological systems. This process has also been studied in the presence of external surfaces having affinity to one type of atom or molecule (**Figure 4a**). Both theoretical and experimental methods have been exploited and have been found in close agreement. Formation of long ridges and circular drops has been reported numerous occasions using lattice-based Ising model. For example, one may look into John W. Cahn research paper published in *The Journal of Chemical Physics* in the year 1965. The TEM image taken for Vycor, in which one phase had

(1)

(2)

(3)

(4)

and vice versa. The occupation of each site can be represented by:

been leached out and the voids were filled with lead (**Figure 4b**).

Yang and Lee first coined the term lattice gas in the year 1952. A lattice should have larger volume (V) than the number of lattice molecules (N), so that some of the nodes or lattice vertices are left empty (i.e., N < V). No lattice vertex can be occupied by more than one particle. The interaction potential between two atoms at

∞ if rij ¼ 0*:*0

For surface affinity of lower surface to ith liquid molecule, we chose:

*ni* <sup>¼</sup> 1 if site i is occupied

One example case is shown in **Figure 5**. Here, we chose lattice size of 128 � 128 � 48. The fluid-fluid molecule and wall-liquid molecule interactions are defined, respectively, in Eqs. (2) and (3). In canonical ensemble, the threedimensional lattice is swept one by one; by choosing sites regularly with one of its

0 else

*V ri* ð Þ¼ � *<sup>J</sup>*<sup>0</sup>

The occupation number (ni) of a lattice site i is given by:

�

8 < :

� ∈ *if* Si ¼ Sj ¼ 1*:*0 and rij ¼ 1*:*0

*r i*ð Þ *if* Si <sup>¼</sup> <sup>1</sup>*:*<sup>0</sup> 0 otherwise

0 if site i is un � occupied

**2.2 Lattice-based liquid-gas model**

lattice sites i and j is given by Eq. (2):

**118**

*U rij* ð Þ¼

8 ><

>:

�

*(a) Surface-directed phase separation and dewetting in conserved binary mixture using two-dimensional lattices of size 200 100 nodes. The conserved components are taken in ratio 70:30 at T = 0.70. Majority component is attracted by upper and lower substrates, whereas the minority component has repulsive interaction with the two interfaces. Periodic boundary conditions are applied along X-direction. The micrograph is taken after completion of 30,000 Monte Carlo cycles using Kawasaki exchange method (the figure is reproduced with permission from Singh [5]). (b) Shows Transmission Electron Microscope (TEM) image of unsintered Vycor with one phase replaced by lead (X 200000). Reproduced with permission from W. G. Schmidt and R. J. Charles, Journal of Applied Physics 35, 2552 (1964); doi: 10.1063/1.1702905.*

nearest neighbor (i.e., i = n and i + 1 = n + 1). Change in energy is calculated during exchanging of these two sites; the exchange move is accepted, if Exp [ΔE/kBT] is found to be greater than or equal to a random number generated between [0, 1]. For all cases of studies here, ε = 1 and J0 = 12.0, and only the lower surface is functional, while the upper surface has only hard-sphere interaction with the fluid molecules. Average number density for liquid-like molecules is taken as 0.25 [16].

**Figure 6** shows micrograph of self-aligned liquid columns. The system evolves from an initial homogeneous mixture of liquid- and gas-like molecules obtained by annealing the system at high temperature for few thousand MC cycles. Dynamic Monte Carlo simulation has been used with continuous but random trial movements of the molecules. The lattice-based Ising model using Eqs. (2) and (3) is also supposed to give same results, at least qualitatively.

#### **2.3 Spin glasses**

Crystalline solids possess short- and long-range order along its crystal axes and maintain its periodicity in three dimensions. Liquids possess only short-range order, and its molecules have no long-range correlation. Liquid molecules retain only short-range order. Gases possess neither of the two. These are the three phases, in which any matter may exist. What are the glasses then? Glasses are solids,

**Figure 5.** *Micrograph for box thickness Hz = 48 after completion of 20 K M C cycles (figure is reproduced with permission from proceedings, Singh [16]).*

possessing no long-range order. Molecules may only locally arrange themselves to minimize its free energy. If the molecular arrangement is completely random, then a term "random media" is assigned to that. Glasses are understood as supercooled liquids. If a liquid is frozen abruptly, so that the molecules do not get sufficient time to organize themselves, some local order can be retained inside the frozen liquid. Glasses have one peculiar property. These retain relatively higher entropy even at quite low temperatures. One example is Mn doped in metals as impurity. Mn atoms interact with other Mn so (i.e. impurity atom) via RKKY interaction *Jij*ð Þ� *r* cos 2ð Þ *kFr* ð Þ *kFr* <sup>3</sup> . Because of the oscillations in it, the interactions remain random. Such spin systems are classified as spin glasses. There is great deal of frustrations in spin orientations; so on many occasions, these are also referred as "frustrated spin glasses."

year 1974, an alloy was found, which first showed that its magnetic behavior exactly

*Self-assembled channels formed in confined geometry; the system starts with a random mixture of square-well fluid (A-type) and hard-sphere (B type) particles. The chemically patterned surface has affinity to (A-type) with interaction range λA-A = 1.5, λA-B = 1.5, λWall-A = 2.0; interaction strengths were taken as εAA = 1.0, εAB = 0.5, and εWall-A = 3.0. Average number density of the system has been taken as ρ = 0.40. Pore width H = 4.0 and composition ratio A:B = 50:50 were taken for all cases of studies. The micrograph and density data were taken after completion of 40* � *105 Monte Carlo cycles (the figure is reproduced with permission from*

Various textbooks are available nowadays, which discuss Ising model and its applications in greater details [19–22]. Here, brief theory of one-dimensional Ising model is presented. H, Q, and A stands for Hamiltonian, partition function, and

> X *n*, *n*

*SiSi*þ<sup>1</sup> � <sup>1</sup> 2 *μB* X *N*

*SiSj* � *<sup>μ</sup> <sup>B</sup>* <sup>X</sup>

*N*

*Si* (5)

ð Þ *Si* þ *Si*þ<sup>1</sup> (6)

*i*¼1

*i*¼1

matched with the Onsager result.

**Figure 6.**

**121**

*Singh et al. [14]).*

free energy of the system, respectively:

**3. Mathematical formulation in one dimension**

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

*HN*f g *Si* ¼ �*J*

*HN*f g *Si* ¼ �*J*

X *N*

*i* ¼ 1 <*i*, *j*>

Lenz-Ising model did not remain limited to above problems only, but it was extensively used to study liquid mixtures, ternary and quaternary alloys, polymer and their mixtures, random walk problem, and many others. The important aspect of Ising model is that a variety of problems (including some problems mentioned above) can be investigated by the similar kind of modeling and approach all together. It is no longer necessary to develop a different kind of theory for each type of cooperative phenomenon. Despite of all the above, it has been ironical that the inventor of the model, Ernst Ising, gave up the idea on working it, any further presuming that his model has no physical significance. He realized after two decades that he had become famous for his model because of the results obtained by other scientist based on his model, rather by his own work. It has been a queer sensation that the results of Ising model matched with any experimental data or the model was bit artificial. As for as the exponents were concerned, they were of universal nature, and a wide variety of systems have the same Ising exponents. The experimental evidence in favor of it remained a challenge, for many decades. In the *The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

#### **Figure 6.**

possessing no long-range order. Molecules may only locally arrange themselves to minimize its free energy. If the molecular arrangement is completely random, then a term "random media" is assigned to that. Glasses are understood as supercooled liquids. If a liquid is frozen abruptly, so that the molecules do not get sufficient time to organize themselves, some local order can be retained inside the frozen liquid. Glasses have one peculiar property. These retain relatively higher entropy even at quite low temperatures. One example is Mn doped in metals as impurity. Mn atoms

*Micrograph for box thickness Hz = 48 after completion of 20 K M C cycles (figure is reproduced with permission*

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

**Figure 5.**

**120**

*from proceedings, Singh [16]).*

interact with other Mn so (i.e. impurity atom) via RKKY interaction *Jij*ð Þ� *r*

so on many occasions, these are also referred as "frustrated spin glasses."

Because of the oscillations in it, the interactions remain random. Such spin systems are classified as spin glasses. There is great deal of frustrations in spin orientations;

Lenz-Ising model did not remain limited to above problems only, but it was extensively used to study liquid mixtures, ternary and quaternary alloys, polymer and their mixtures, random walk problem, and many others. The important aspect of Ising model is that a variety of problems (including some problems mentioned above) can be investigated by the similar kind of modeling and approach all together. It is no longer necessary to develop a different kind of theory for each type of cooperative phenomenon. Despite of all the above, it has been ironical that the inventor of the model, Ernst Ising, gave up the idea on working it, any further presuming that his model has no physical significance. He realized after two decades that he had become famous for his model because of the results obtained by other scientist based on his model, rather by his own work. It has been a queer sensation that the results of Ising model matched with any experimental data or the model was bit artificial. As for as the exponents were concerned, they were of universal nature, and a wide variety of systems have the same Ising exponents. The experimental evidence in favor of it remained a challenge, for many decades. In the

cos 2ð Þ *kFr* ð Þ *kFr* <sup>3</sup> .

*Self-assembled channels formed in confined geometry; the system starts with a random mixture of square-well fluid (A-type) and hard-sphere (B type) particles. The chemically patterned surface has affinity to (A-type) with interaction range λA-A = 1.5, λA-B = 1.5, λWall-A = 2.0; interaction strengths were taken as εAA = 1.0, εAB = 0.5, and εWall-A = 3.0. Average number density of the system has been taken as ρ = 0.40. Pore width H = 4.0 and composition ratio A:B = 50:50 were taken for all cases of studies. The micrograph and density data were taken after completion of 40* � *105 Monte Carlo cycles (the figure is reproduced with permission from Singh et al. [14]).*

year 1974, an alloy was found, which first showed that its magnetic behavior exactly matched with the Onsager result.

### **3. Mathematical formulation in one dimension**

Various textbooks are available nowadays, which discuss Ising model and its applications in greater details [19–22]. Here, brief theory of one-dimensional Ising model is presented. H, Q, and A stands for Hamiltonian, partition function, and free energy of the system, respectively:

$$H\_N\{\mathbf{S}\_i\} = -J\sum\_{n,n} \mathbf{S}\_i \mathbf{S}\_j - \mu \,\,\mathbf{B} \sum\_{i=1}^N \mathbf{S}\_i \tag{5}$$

$$H\_N\{\mathbf{S}\_i\} = -J\sum\_{i=-1 \atop i}^N \mathbf{S}\_i \mathbf{S}\_{i+1} - \frac{1}{2}\mu B \sum\_{i=1}^N (\mathbf{S}\_i + \mathbf{S}\_{i+1}) \tag{6}$$

$$Q\_N(B, T) = -\sum\_{\mathbf{S}\_1 = \pm 1} \dots \dots \dots \sum\_{\mathbf{S}\_{N-\pm 1}} \mathbf{e}^{\rho} \sum\_{i}^{N} \left\{ l \mathbf{S}\_i \mathbf{S}\_{i+1} + \mathbf{\tilde{J}} \boldsymbol{\mu} \mathbf{S} (\mathbf{S}\_i + \mathbf{S}\_{i+1}) \right\} \tag{7}$$

$$
\langle \mathbf{S}\_i | P | \mathbf{S}\_{i+1} \rangle = \sigma^{\beta \left\{ J \mathbf{S}\_i \mathbf{S}\_{i+1} + \frac{1}{2} \mu B(\mathbf{S}\_i + \mathbf{S}\_{i+1}) \right\}} \tag{8}
$$

*<sup>χ</sup>* � *<sup>∂</sup><sup>M</sup>*

*<sup>Q</sup>* ð Þ¼ 0, *<sup>T</sup>* <sup>X</sup>

*S*1

*<sup>σ</sup><sup>i</sup>* <sup>¼</sup> <sup>þ</sup><sup>1</sup> *if Si* <sup>¼</sup> *Si*þ<sup>1</sup>

�

�1 *if Si* ¼ �*Si*þ<sup>1</sup>

In order to consider contributions from all possible configurations {S1, S2, S3 … … … SN}, we need to provide the set of numbers {σ1, σ2, σ3… … ..σN-1}; here each Si can take two values as �1. Configuration in a lattice description means a particular set of values of all spins; if there are N numbers of vertices, there will be 2N different configurations as a result of permutation and combination of spins. The space, thus formed with these configurations, is called configuration space. Here, summing over

… … *::*X *SN*�<sup>1</sup> *e K* P*<sup>N</sup>*�<sup>1</sup>

**3.1 Case A: free boundary with zero field**

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

Partition function is given by:

We now define a new variable:

Then we can assign σ to two values, i.e., �1:

σ<sup>i</sup> will give only half value of Q, henceforth, we can write:

X *σ*1

X *σ*1

… … *::*X *σN*�<sup>1</sup> *e*

… … *::*X *σN*�<sup>1</sup> *e*

*Q* ð Þ¼ 0, *T* 2

*Q* ð Þ¼ 0, *T* 2

**3.2 Case B: periodic boundary with zero field**

Now, the partition function is given by:

*<sup>Q</sup>* ð Þ¼ 0, *<sup>T</sup>* <sup>X</sup>

X *σ*1

Here, SN+1 = S1

Since (Si)

**123**

*Q* ð Þ¼ 0, *T* 2

*<sup>Q</sup>* ð Þ¼ 0, *<sup>T</sup>* <sup>X</sup>

*S*1

… … *::*X *σN*�<sup>1</sup> *e* *S*1

… … *::*X *SN*�<sup>1</sup> *e*

… … *::*X *SN*�<sup>1</sup> *e K* P*<sup>N</sup>*�<sup>1</sup>

<sup>2</sup> = 1, we can write S1SN=S1. S2. S2. S3. S3 ……… ... SN-1. SN-1. SN

X∞ *n*¼0

*K*ð Þ *σ*1þ*σ*2þ ……… *σN*�<sup>1</sup>

Here, K = β J.

*<sup>∂</sup><sup>B</sup>* ! *<sup>χ</sup>*0ð Þ¼ *<sup>T</sup> <sup>N</sup>μ*<sup>2</sup>

*kBT <sup>e</sup>*

*σ<sup>i</sup>* ¼ *SiSi*þ1; *Here i* ¼ 1, 2, … … *::N* � 1 (26)

<sup>2</sup>*J=kBT* (24)

*<sup>i</sup>*¼<sup>1</sup> *SiSi*þ<sup>1</sup> (25)

*<sup>K</sup>*ð Þ *<sup>σ</sup>*1þ*σ*2þ*σ*<sup>3</sup> ………… *::*þ*σN*�<sup>1</sup> (28)

*<sup>K</sup>*ð Þ *<sup>σ</sup>*1þ*σ*2þ*σ*<sup>3</sup> ………… *::*þ*σN*�<sup>1</sup> (29)

*<sup>i</sup>*¼<sup>1</sup> *SiSi*þ1þ*KSNS*<sup>1</sup> (31)

*<sup>K</sup>*ð*σ*1þ*σ*2<sup>þ</sup> ……… *<sup>σ</sup>N*�1Þþ*Kσ*1*σ*<sup>2</sup> ……… *<sup>σ</sup>N*�<sup>1</sup> (32)

*Kσ*1*σ*2*σ*<sup>3</sup> … *:σ<sup>N</sup>*�<sup>1</sup> *n*! � �*<sup>n</sup>*

(33)

*<sup>Q</sup>* ð Þ¼ 0, *<sup>T</sup>* 2 2ð Þ *coshK <sup>N</sup>*�<sup>1</sup> (30)

(27)

$$Q\_N(B, T) = \sum\_{\mathbb{S} = \pm 1} \dots \dots \dots \dots \sum\_{\mathbb{S}\_{N \times \pm 1}} \langle \mathbb{S}\_1 | P | \mathbb{S}\_2 \rangle \langle \mathbb{S}\_2 | P | \mathbb{S}\_3 \rangle \dots \dots \dots \dots \dots \langle \mathbb{S}\_{N-1} | P | \mathbb{S}\_N \rangle \langle \mathbb{S}\_N | P | \mathbb{S}\_1 \rangle \dots \tag{9}$$

$$Q\_N(B, T) = \sum\_{\text{S}\_i = \pm 1} \sum\_{\text{S}\_{i+1} = \pm 1} e^{\rho \| \mathbf{S}\_i \mathbf{S}\_{i+1} + \frac{1}{2} \mu \theta(\mathbf{S}\_i + \mathbf{S}\_{i+1})} \tag{10}$$

$$P = \begin{pmatrix} e^{\beta(J+\mu B)} & & & e^{-\beta f} \\ e^{-\beta f} & & & e^{\beta(J-\mu B)} \end{pmatrix} \tag{11}$$

$$Q\_N(\mathcal{B}, T) = \sum\_{\mathbb{S} = \pm 1} \langle \mathbb{S} | P^N | \mathbb{S} \rangle = \text{Trace } \left( P^N \right) = \boldsymbol{\eta}\_1^N + \boldsymbol{\eta}\_2^N \tag{12}$$

$$
\begin{pmatrix} e^{\beta(J+\mu B)} - \chi & e^{-\beta f} \\ e^{-\beta f} & e^{\beta(J-\mu B)} - \chi \end{pmatrix} = \mathbf{0} \tag{13}
$$

$$
\gamma^2 - 2\eta e^{\beta \mathcal{I}} \cosh\left(\beta \mu B\right) + 2 \sinh\left(2\beta \mathcal{I}\right) = 0 \tag{14}
$$

$$
\begin{pmatrix} \gamma\_1\\ \gamma\_2 \end{pmatrix} = e^{\beta \mathcal{I}} \cosh \left( \beta \mu B \right) + 2 \sinh \left( 2 \beta \mathcal{I} \right) = 0 \tag{15}
$$

$$
\begin{pmatrix} \gamma\_1 \\ \gamma\_2 \end{pmatrix} = e^{\beta \|} \cosh \left( \beta \mu B \right) \pm \left\{ e^{-2\beta \|} + e^{2\beta \|} \sinh^2 \left( \beta \mu B \right) \right\}^{1/2} \tag{16}
$$

$$\gamma\_2 < \gamma\_1; \left(\frac{\gamma\_2}{\gamma\_1}\right)^N \to \mathbf{0} \tag{17}$$

$$
\ln Q\_N(B, T) \cong \text{Nln } \gamma\_1 \tag{18}
$$

$$\frac{1}{N}\ln Q\_N(B,T) \cong \ln \left[ e^{\beta \mathcal{I}} \cosh \left( \beta \mu B \right) + \left\{ e^{-2\beta \mathcal{I}} + e^{2\beta \mathcal{I}} \sinh^2 \left( \beta \mu B \right) \right\}^{1/2} \right] \tag{19}$$

$$A(B,T) = N\!\!\!/ - Nk\_BT \; \ln\left[\cosh\left(\beta\mu B\right) + \left\{e^{-4\beta\dagger} + \sinh^2\left(\beta\mu B\right)\right\}^{1/2}\right] \tag{20}$$

$$U(B,T) \equiv -k\_B T^2 \frac{\partial}{\partial T} \left(\frac{A}{k\_B T}\right) \tag{21}$$

$$\begin{split} U(\boldsymbol{B},T) & \equiv -N\boldsymbol{I} - \frac{N\mu \text{Bsinh}(\boldsymbol{\beta}\mu \boldsymbol{B})}{\left\{\boldsymbol{e}^{-4\boldsymbol{\beta}\boldsymbol{\beta}} + \sinh^{2}(\boldsymbol{\beta}\mu \boldsymbol{B})\right\}^{1/2}} \\ & + \frac{2N\mu \text{e}^{-4\boldsymbol{\beta}\boldsymbol{\beta}}}{\left[\cosh\left(\boldsymbol{\beta}\mu \boldsymbol{B}\right) + \left\{\boldsymbol{e}^{-4\boldsymbol{\beta}\boldsymbol{\beta}} + \sinh^{2}(\boldsymbol{\beta}\mu \boldsymbol{B})\right\}^{1/2}\right] \left\{\boldsymbol{e}^{-4\boldsymbol{\beta}\boldsymbol{\beta}} + \sinh^{2}(\boldsymbol{\beta}\mu \boldsymbol{B})\right\}^{1/2}} \end{split} \tag{22}$$

Some thermodynamic functions are defined as follows:

$$\overline{M}(B,T) \equiv -\left(\frac{\partial A}{\partial B}\right)\_T = \frac{N\mu \, \sinh\left(\beta \mu B\right)}{\left[e^{-4\beta \|} + \sinh^2\left(\beta \mu B\right)\right]^{1/2}}\tag{23}$$

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

$$\chi \equiv \frac{\partial \mathcal{M}}{\partial \mathcal{B}} \rightarrow \chi\_0(T) = \frac{\mathcal{N}\mu^2}{k\_B T} e^{2f/k\_B T} \tag{24}$$

#### **3.1 Case A: free boundary with zero field**

Partition function is given by:

*QN*ð Þ¼� *<sup>B</sup>*, *<sup>T</sup>* <sup>X</sup>

………… X

*QN*ð Þ¼ *<sup>B</sup>*, *<sup>T</sup>* <sup>X</sup>

0 @

*e*

*e*

*<sup>γ</sup>*<sup>2</sup> � <sup>2</sup>*γ<sup>e</sup>*

¼ *e*

*γ*1 *γ*2

¼ *e*

*A B*ð Þ¼ , *T NJ* � *NkBT ln* cosh ð Þþ *βμB e*

*<sup>e</sup>*�4*β<sup>J</sup>* <sup>þ</sup> *sinh* <sup>2</sup> ð Þ *βμ<sup>B</sup>* � �<sup>1</sup>*=*<sup>2</sup>

cosh ð Þþ *βμ<sup>B</sup> <sup>e</sup>*�4*β<sup>J</sup>* <sup>þ</sup> *sinh* <sup>2</sup> ð Þ *βμ<sup>B</sup>* � �<sup>1</sup>*=*<sup>2</sup> h i

Some thermodynamic functions are defined as follows:

*∂B* � �

*T*

*M B*ð Þ�� , *<sup>T</sup> <sup>∂</sup><sup>A</sup>*

*U B*ð Þ�� , *<sup>T</sup> NJ* � *<sup>N</sup>μBsinh*ð Þ *βμ<sup>B</sup>*

*γ*1 *γ*2

*<sup>N</sup> ln QN*ð Þffi *<sup>B</sup>*, *<sup>T</sup>* ln *<sup>e</sup>*

þ

1

**122**

!

!

*QN*ð Þ¼ *<sup>B</sup>*, *<sup>T</sup>* <sup>X</sup>

*Si*¼�1

*S*1¼�1

*SN*¼�<sup>1</sup>

*QN*ð Þ¼ *<sup>B</sup>*, *<sup>T</sup>* <sup>X</sup>

*<sup>P</sup>* <sup>¼</sup> *<sup>e</sup>*

0 @

*S*¼�1

*e*

h i *Si*j j *P Si*þ<sup>1</sup> ¼ *e*

……… *::*

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

*Si*¼�1

*S P<sup>N</sup>* � � �

�*β<sup>J</sup> e*

*<sup>β</sup><sup>J</sup>* cosh ð Þ� *βμ<sup>B</sup> <sup>e</sup>*

*γ*<sup>2</sup> <*γ*1;

*<sup>β</sup><sup>J</sup>* cosh ð Þþ *βμ<sup>B</sup> <sup>e</sup>*

*U B*ð Þ�� , *<sup>T</sup> kBT*<sup>2</sup> *<sup>∂</sup>*

*<sup>β</sup>*ð Þ *<sup>J</sup>*þ*μ<sup>B</sup>* � *<sup>γ</sup> <sup>e</sup>*

X *SN*¼�<sup>1</sup> *e β* P*<sup>N</sup>*

*<sup>β</sup> JSiSi*þ1þ<sup>1</sup>

X *Si*þ1¼�1 *e <sup>β</sup>JSiSi*þ1þ<sup>1</sup> 2

*<sup>β</sup>*ð Þ *<sup>J</sup>*þ*μ<sup>B</sup> e*

�*<sup>S</sup>* � � <sup>¼</sup> *Trace PN* � � <sup>¼</sup> *<sup>γ</sup><sup>N</sup>*

*<sup>β</sup>*ð Þ *<sup>J</sup>*�*μ<sup>B</sup>* � *<sup>γ</sup>*

�2*β<sup>J</sup>* <sup>þ</sup> *<sup>e</sup>*

�2*β<sup>J</sup>* <sup>þ</sup> *<sup>e</sup>*

ð Þ *βμ<sup>B</sup>* � �<sup>1</sup>*=*<sup>2</sup> h i

*∂T*

2*NJe*�4*β<sup>J</sup>*

<sup>¼</sup> *<sup>N</sup><sup>μ</sup>* sinh ð Þ *βμ<sup>B</sup> <sup>e</sup>*�4*β<sup>J</sup>* <sup>þ</sup> *sinh* <sup>2</sup>

*γ*2 *γ*1 � �*<sup>N</sup>*

�*β<sup>J</sup> e*

*<sup>i</sup> JSiSi*þ1þ<sup>1</sup>

<sup>2</sup> f g *<sup>μ</sup>B S*ð Þ *<sup>i</sup>*þ*Si*þ<sup>1</sup> (7)

*μβ*ð Þ *Si*þ*Si*þ<sup>1</sup> (10)

A (11)

A ¼ 0 (13)

<sup>2</sup> (12)

<sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>N</sup>*

(9)

(19)

(20)

(21)

(22)

<sup>2</sup> f g *<sup>μ</sup>B S*ð Þ *<sup>i</sup>*þ*Si*þ<sup>1</sup> (8)

h i *S*1j j *P S*<sup>2</sup> h i *S*2j j *P S*<sup>3</sup> ………… *:*h i *SN*�1j j *P SN* h i *SN*j j *P S*<sup>1</sup>

�*βJ*

1

*β*ð Þ *J*�*μB*

�*βJ*

1

*<sup>β</sup><sup>J</sup>* cosh ð Þþ *βμ<sup>B</sup>* 2 sinh 2ð Þ¼ *<sup>β</sup><sup>J</sup>* <sup>0</sup> (14)

*<sup>β</sup><sup>J</sup>* cosh ð Þþ *βμ<sup>B</sup>* 2 sinh 2ð Þ¼ *<sup>β</sup><sup>J</sup>* <sup>0</sup> (15)

<sup>2</sup>*β<sup>J</sup> sinh* <sup>2</sup>

*ln QN*ð Þffi *B*, *T Nln γ*<sup>1</sup> (18)

<sup>2</sup>*β<sup>J</sup> sinh* <sup>2</sup>

�4*β<sup>J</sup>* <sup>þ</sup> *sinh* <sup>2</sup> ð Þ *βμ<sup>B</sup>* � �<sup>1</sup>*=*<sup>2</sup> h i

> *A kBT* � �

ð Þ *βμ<sup>B</sup>* � �<sup>1</sup>*=*<sup>2</sup> (16)

! 0 (17)

*<sup>e</sup>*�4*β<sup>J</sup>* <sup>þ</sup> *sinh* <sup>2</sup> ð Þ *βμ<sup>B</sup>* � �<sup>1</sup>*=*<sup>2</sup>

ð Þ *βμ<sup>B</sup>* � �<sup>1</sup>*=*<sup>2</sup> (23)

$$Q\left(\mathbf{0}, T\right) = \sum\_{\mathbf{S}\_1} \dots \dots \sum\_{\mathbf{S}\_{N-1}} \mathbf{e}^K \sum\_{i=1}^{N-1} \mathbf{S}\_i \mathbf{S}\_{i+1} \tag{25}$$

Here, K = β J. We now define a new variable:

$$\sigma\_i = \mathbb{S}\_i \mathbb{S}\_{i+1}; \text{Here } i = \mathbf{1}, \mathbf{2}, \dots \dots N - \mathbf{1} \tag{26}$$

Then we can assign σ to two values, i.e., �1:

$$
\sigma\_{\dot{i}} = \begin{cases}
+\mathbf{1} & \text{if } \text{ S}\_{\dot{i}} = \text{S}\_{\dot{i}+\mathbf{1}} \\
\end{cases}
\tag{27}
$$

In order to consider contributions from all possible configurations {S1, S2, S3 … … … SN}, we need to provide the set of numbers {σ1, σ2, σ3… … ..σN-1}; here each Si can take two values as �1. Configuration in a lattice description means a particular set of values of all spins; if there are N numbers of vertices, there will be 2N different configurations as a result of permutation and combination of spins. The space, thus formed with these configurations, is called configuration space. Here, summing over σ<sup>i</sup> will give only half value of Q, henceforth, we can write:

$$Q\_{\bullet}(0, T) = 2 \sum\_{\sigma\_1} \dots \dots \sum\_{\sigma\_{N-1}} e^{K(\sigma\_1 + \sigma\_2 + \sigma\_3 \dots \dots \dots \dots + \sigma\_{N-1})} \tag{28}$$

$$Q\left(0, T\right) = 2\sum\_{\sigma\_1} \dots \dots \sum\_{\sigma\_{N-1}} e^{K(\sigma\_1 + \sigma\_2 + \sigma\_3 \dots \dots \dots \dots + \sigma\_{N-1})} \tag{29}$$

$$Q\left(\mathbf{0},T\right) = \mathbf{2} \left(2cosh \mathbf{K}\right)^{N-1} \tag{30}$$

#### **3.2 Case B: periodic boundary with zero field**

Now, the partition function is given by:

$$Q\left(\mathbf{0}, T\right) = \sum\_{\mathbf{S}\_1} \dots \dots \sum\_{\mathbf{S}\_{N-1}} \mathbf{e}^K \boldsymbol{\sum}^{N-1} \mathbf{S}\_i \mathbf{S}\_{i+1} + K \mathbf{S}\_N \mathbf{S}\_1 \tag{31}$$

Here, SN+1 = S1

$$Q\left(\mathbf{0}, T\right) = \sum\_{\mathbf{S}\_1} \dots \dots \sum\_{\mathbf{S}\_{N-1}} \mathbf{e}^{K(\sigma\_1 + \sigma\_2 + \dots \dots \dots \dots \sigma\_{N-1}) + K\sigma\_1\sigma\_2 \dots \dots \dots \sigma\_{N-1}} \tag{32}$$

Since (Si) <sup>2</sup> = 1, we can write S1SN=S1. S2. S2. S3. S3 ……… ... SN-1. SN-1. SN

$$Q\left(0, T\right) = 2\sum\_{\sigma\_1} \dots \dots \dots \sum\_{\sigma\_{N-1}} \epsilon^{K(\sigma\_1 + \sigma\_2 + \dots \dots \dots \sigma\_{N-1})} \sum\_{n=0}^{\infty} \left(\frac{K\sigma\_1 \sigma\_2 \sigma\_3 \dots \sigma\_{N-1}}{n!}\right)^n \tag{33}$$

Here, second part in exponential has been converted into a summation series:

$$Q\left(0,T\right) = 2\sum\_{n=0}^{\infty} \frac{K^n}{n!} \left[\sum\_{\sigma} \sigma^n e^{K\sigma}\right]^{N-1} \tag{34}$$

**4.1 Scaling hypothesis and renormalization group theory**

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

system, the spatial distances are rescaled by the factor l.

Now, the partition function can be updated as follows:

N<sup>0</sup> spins (**Figure 8**).

Lattice constant:

**Figure 8.**

**125**

*nodal points.*

Kadanoff first suggested that, when a system is near critical temperature, individual spins may be grouped into blocks of spins [23]. It is possible because of the fact that the spin-spin correlation length becomes exceedingly large near Tc and details of individual spins no longer remain important. In transformed system, each block plays the role of a single spin. Now, the spin variable associated with a single block is denoted by symbol σi. σ<sup>i</sup> can take values �1. The new system is composed of

> *N*<sup>0</sup> ¼ *l* �*d*

> > *r* <sup>0</sup> ¼ *l* �1

*<sup>Q</sup>* <sup>¼</sup> <sup>X</sup> f g *σ<sup>i</sup>*

In order to preserve the spatial density of the degrees of freedom of spins in the

This idea was first propounded by Kadanoff, and was later developed by Wilson. This process is also referred as decimation process. A new exchange coupling constant is assigned for interaction between σi. This new construction of lattice does not alter the free energy of the system, and it remains the same as obtained by the original method. The rescaling process helps to find relations between various exponents. More detailed discussion on this topic can be found in standard textbooks of *Statistical Mechanics* by Patharia, Huang, etc. Since this process involves

*Spin decimation process in a two-dimensional square lattice. A small cluster of 36 spins gets transformed into 9*

*N* (37)

*r* (39)

*e* �*βHN*f g *<sup>σ</sup><sup>i</sup>* ½ � (40)

*a*<sup>0</sup> ¼ *la* (38)

$$Q\left(0,T\right) = 2\sum\_{n=0}^{\infty} \frac{K^n}{n!} \left[e^K + (-1)^n e^{-K}\right]^{N-1} \tag{35}$$

$$Q\left(\mathbf{0},T\right) = \left(2\cosh\mathbf{K}\right)^N + \left(2\sinh\mathbf{K}\right)^N\tag{36}$$

It can be shown that in thermodynamic limit, (i.e. N ! ∞), the free energy of the system converge to a finite value. Readers are left with the exercise. So, periodic boundary condition, as shown in **Figure 7** (invented by Ising), really helps one to get rid of constructing infinitely large systems. Using appropriate boundary conditions, one may obtain realistic results using large but finite number of spins.

### **4. Critical phenomena**

A lot of research work has been dedicated to observe system behavior near critical points [23–27]. The relevant thermodynamic variables exhibit power-law dependences on the parameter (T � Tc) specifying the distance away from the critical point. The critical points are marked by the fact that different physical quantities pertaining to the system pose singularities at the critical point. These singularities are expressed in terms of power laws of (T � Tc) characterized by critical exponents. As, for example, magnetization <M > identified as an order parameter in magnetism, shows dependence on critical temperature (Tc), with exponent β as follows other exponents are also listed below.

Reduced temperature *t* � (*T* � *Tc*)/*Tc*. α: specific heat c (*t*) � *t* �α ; B � h = 0. β: spontaneous magnetization M (t) � (�*t*) β ,*T* ≤ *Tc*, B � h = 0. <sup>γ</sup>: magnetic susceptibility <sup>χ</sup> <sup>=</sup> <sup>∂</sup>M/∂ℎ,*<sup>T</sup>* � <sup>|</sup>*t*|�<sup>γ</sup> , B � h = 0. δ: critical Isotherm M (h) � |ℎ| 1/<sup>δ</sup> sgn (ℎ), *t* = 0. <sup>ν</sup>: correlation length, <sup>ξ</sup> � <sup>|</sup>*t*|�<sup>ν</sup> , B � h = 0. η: correlation function G (*r*) � *r* (�*d*+2�*η*) , *t* = 0, B � h = 0.

**Figure 7.** *Representation of periodic boundary conditions in a one-dimensional Ising chain.*

## **4.1 Scaling hypothesis and renormalization group theory**

Kadanoff first suggested that, when a system is near critical temperature, individual spins may be grouped into blocks of spins [23]. It is possible because of the fact that the spin-spin correlation length becomes exceedingly large near Tc and details of individual spins no longer remain important. In transformed system, each block plays the role of a single spin. Now, the spin variable associated with a single block is denoted by symbol σi. σ<sup>i</sup> can take values �1. The new system is composed of N<sup>0</sup> spins (**Figure 8**).

$$N' = l^{-d} N \tag{37}$$

Lattice constant:

Here, second part in exponential has been converted into a summation series:

*Kn n*!

It can be shown that in thermodynamic limit, (i.e. N ! ∞), the free energy of the

X *σ σne Kσ* " #*N*�<sup>1</sup>

*<sup>K</sup>* þ �ð Þ<sup>1</sup> *<sup>n</sup>*

*e*

*<sup>Q</sup>* ð Þ¼ 0, *<sup>T</sup>* ð Þ 2 cosh K *<sup>N</sup>* <sup>þ</sup> ð Þ 2 sinh K *<sup>N</sup>* (36)

�*<sup>K</sup>* � �*N*�<sup>1</sup> (35)

(34)

X∞ *n*¼0

> *Kn <sup>n</sup>*! *<sup>e</sup>*

system converge to a finite value. Readers are left with the exercise. So, periodic boundary condition, as shown in **Figure 7** (invented by Ising), really helps one to get rid of constructing infinitely large systems. Using appropriate boundary conditions,

A lot of research work has been dedicated to observe system behavior near critical points [23–27]. The relevant thermodynamic variables exhibit power-law dependences on the parameter (T � Tc) specifying the distance away from the critical point. The critical points are marked by the fact that different physical quantities pertaining to the system pose singularities at the critical point. These singularities are expressed in terms of power laws of (T � Tc) characterized by critical exponents. As, for example, magnetization <M > identified as an order parameter in magnetism, shows dependence on critical temperature (Tc), with

one may obtain realistic results using large but finite number of spins.

exponent β as follows other exponents are also listed below.

*Representation of periodic boundary conditions in a one-dimensional Ising chain.*

; B � h = 0.

β

1/<sup>δ</sup> sgn (ℎ), *t* = 0.

, B � h = 0.

(�*d*+2�*η*)

,*T* ≤ *Tc*, B � h = 0.

, B � h = 0.

, *t* = 0, B � h = 0.

�α

Reduced temperature *t* � (*T* � *Tc*)/*Tc*.

β: spontaneous magnetization M (t) � (�*t*)

<sup>γ</sup>: magnetic susceptibility <sup>χ</sup> <sup>=</sup> <sup>∂</sup>M/∂ℎ,*<sup>T</sup>* � <sup>|</sup>*t*|�<sup>γ</sup>

α: specific heat c (*t*) � *t*

**Figure 7.**

**124**

δ: critical Isotherm M (h) � |ℎ|

η: correlation function G (*r*) � *r*

<sup>ν</sup>: correlation length, <sup>ξ</sup> � <sup>|</sup>*t*|�<sup>ν</sup>

X∞ *n*¼0

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

*Q* ð Þ¼ 0, *T* 2

*Q* ð Þ¼ 0, *T* 2

**4. Critical phenomena**

$$a' = \text{la} \tag{38}$$

In order to preserve the spatial density of the degrees of freedom of spins in the system, the spatial distances are rescaled by the factor l.

$$r' = l^{-1}r\tag{39}$$

Now, the partition function can be updated as follows:

$$Q = \sum\_{\{\sigma\_i\}} \mathfrak{e}^{[-\beta H\_N \{\sigma\_i\}]} \tag{40}$$

This idea was first propounded by Kadanoff, and was later developed by Wilson. This process is also referred as decimation process. A new exchange coupling constant is assigned for interaction between σi. This new construction of lattice does not alter the free energy of the system, and it remains the same as obtained by the original method. The rescaling process helps to find relations between various exponents. More detailed discussion on this topic can be found in standard textbooks of *Statistical Mechanics* by Patharia, Huang, etc. Since this process involves

#### **Figure 8.**

*Spin decimation process in a two-dimensional square lattice. A small cluster of 36 spins gets transformed into 9 nodal points.*

length transformation or a change of scale, Wilson introduced the concept of renormalization group theory after removing certain deficiencies in Kadanoff's scaling hypothesis. A greater detail of this is omitted here, because that is beyond the scope of the chapter.

correlation function vs. temperature of the system [28]. The critical temperature is marked by the presence of discontinuity in it. Above critical temperature, the magnetization abruptly falls to zero, which is an indication of paramagnetic state. The critical temperature in ferromagnetic thin film is known as Curie temperature. We observe similar kind of behavior with antiferromagnetic films, though below critical point (also known as Neel temperature), the net average magnetization becomes zero, because opposite spins are energetically favored in this case. The schematic diagram is shown in **Figure 11** [28]. Magnetization vs. external magnetic field curves are plotted in **Figure 12(a)**–**(d)** for different sets of parameters [28]. Simulation results obtained for a magnetically striped system as schematically

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

shown in **Figure 13** are reported in **Figures 14–17** [29]. One or two alternate rectangular regions are created, using external field. **Figure 14** shows the gradual transition at the interface, where a definite value of external field suddenly gets zero. The spin polarizations in two regions show sharp boundary. The magnetized film, in presence of magnetic field, induces the magnetic zones in proximity where its close external field is zero. Micrograph also indicates for spin-spin phase separation. The corresponding average magnetization vs. temperature and spin correlation function vs. temperature are also plotted in **Figures 15** and **16**, respectively, but these studies are done using Monte Carlo simulation with semi-infinite free boundary conditions. It has been observed that these systems have relatively high critical transition temperatures. **Figure 17** shows the magnetization process with two alter-

Low-dimensional magnetic heterostructures play vital role in spinotronics. Ferromagnets can induce magnetic ordering through a 40-nm-thick amorphous paramagnetic layer, when placed in its close proximity. One has to reconcile with long-range magnetic interaction to correctly measure the extent of induced magnetization. Readers may go through the *Nature Communications* article of F. Magnus et al. published in the year 2016 [17]. The magnetic properties of ferromagnetic materials with reduced dimensions get altered; when the thickness of a film is

*Schematic representation of ferromagnetic to paramagnetic and antiferromagnetic to paramagnetic transitions.*

nate magnetized zones [29].

**Figure 11.**

**127**
