Phase transitions in the Ising model with random gap defects using the Monte Carlo method

Phase transitions in the Ising model with random gap defects using the Monte Carlo method Kimberly Schultz A senior thesis submitted to the Carthage College Physics & Astronomy Department in partial fulfillment of the requirements for the Bachelor of Arts Degree in Physics Carthage College Kenosha, Wisconsin May 17, 2012 Abstract The Ising model, the simplest model of ferromagnetism, describes a magnetic lattice with nearest-neighbor interactions and is governed by Boltzmann statistics. The Monte Carlo method provides one approach to simulate thermal fluctuations in the Ising Model using a stochastic process to randomly determine dipole-dipole interactions. This allows the Ising model to predict the equilibrium state of a lattice for a given temperature. I estimate the Curie temperature in the two-dimensional zero field Ising model using the Monte Carlo method to be 2.465 ± 0.052. In a real material, however, neighboring dipoles are not always perfect; defects can occur within the substance. I define a defect to be the absence of a dipole at a random point in the lattice. In this thesis, I determine the expected dependence of the Curie temperature on defect density in the two-dimensional Ising model. 1

1 Introduction The Ising model has been studied extensively since it was first proposed in 1920 by Ernst Ising s research director Wilhelm Lenz as a model of magnetic materials [1]. The purpose of the Ising model is to determine the thermal properties of a ferromagnetic lattice. The Ising model is the simplest model of magnetic interactions in a solid and permits analytical solutions for the partition function in one- and two-dimensional lattice, from which properties such as magnetic susceptibility and specific heat can be calculated. In 1925, Ising solved his model in the one-dimensional, zero external magnetic field case. To solve the Ising model for a particular set of parameters, the exact partition function of the lattice must be found using the temperature and all possible energies of the system to predict its thermal behavior. In the one-dimensional Ising model, this is a relatively simple calculation. However, as the dimensionality increases, the calculation of the partition function becomes more difficult. The solution for the two-dimensional case won Lars Onsager a Nobel Prize in physics in 1944 [2]. Lastly, in 2000, Istrail showed that the exact solution of the three-dimensional Ising model cannot cannot be solved in a reasonable amount of time and is therefore computationally intractable [7]. Therefore, in order to study real materials, a different approach must be used; in this case, it is a computational approach. Since the three-dimensional Ising model is impossible to solve analytically, computational simulations are the next step in studying a real ferromagnet. Computational methods use numerical algorithms based on existing theories, either a priori or empirical, to simulate real materials. One particular algorithm, the Monte Carlo method, involves an iterative process that determines the equilibrium state of a ferromagnetic lattice at a given temperature. Many studies have been conducted to investigate the properties of the Ising model [3][4]. In particular, Ferrenberg and Landau performed an extensive study in 1991 on the finite-size behavior of various thermodynamic properties in three-dimensional Ising model using the 2

Monte Carlo method [8]. The Curie temperature, or T c, of a ferromagnet is the temperature at which the magnetic order-disorder phase transition occurs; it depends on the strength of the nearest-neighbor interactions in the material. For a regular two-dimensional lattice, the exact T c is very close to 2.27ɛ/k b, where ɛ is the strength of the interaction energy between dipoles and k b is Boltzmann s constant. However, real materials are not perfect and can have defects. These defects could be random bond defects as studied by Harris and Sknepnek or, as this paper is concerned, with a gap defect [6][5]. The removal of one or more dipoles changes the thermal properties of the system because dipoles next to the defect have fewer neighbors, which affects the partition function of the system. The Monte Carlo method will be used to simulate the Ising model and its nearest-neighbor interactions. In particular, we will study the order-disorder phase transition as a function of defect density, similarly to Sknepnek et al., in two-dimensions [5]. 2 Magnetism 2.1 Origins of Magnetism All matter is magnetic, a phenomenon that is due to the structure of atoms. As is well-known from Maxwell s equations, moving charges create magnetic fields. In an atom, an electron has two types of quantum mechanical angular momentum. One is due to orbital motion about the nucleus and the other is due to electron spin, which is a purely quantum mechanical concept. Electron spin is a fundamental property of electrons, and other elementary particles, that forms the basis of quantum mechanics. Each current loop produced by the moving charge of the electron produces its own magnetic moment. However, the spin moment dominates the magnetic behavior of the atom. On average, the magnetic moment due to the orbiting electron will be zero because 3

of the symmetrical motion of the electron around the nucleus. Spin, on the other hand, is a property that is constant for any electron occupying a given quantum state. Because of this orbital magnetic effects do not contribute to the overall magnetic moment of the atom. In real materials, comprised of many moles of atoms, bulk magnetization effects are observed. Each of the atoms in the material has its own magnetic moment that contributes to the overall magnetic moment of the material. The total magnetic moment of a material is known as magnetization; magnetization, M, is defined as M = N V N m i=1 where m is the magnetic moment of the ith particle, N is the number of particles in the material, V is the volume over which the sum is computed, normalizing the number to find the average magnetization per spin. Each material also has an energy associated with the atomic interactions of neighboring dipoles. These interactions, however, are not the same for all materials. In fact, there are several main types of magnetic materials and they are classified by how their constituent atoms interact with each other and with external magnetic fields. 2.2 Types of Magnetic Materials The most common classes of magnetic materials are: diamagnetism, ferrimagnetism, antiferromagnetism, paramagnetism, and ferromagnetism. 2.2.1 Diamagnetism Of the listed types of magnetism, diamagnetism is the weakest. Diamagnetism is observed in all matter because the magnetic moments of diamagnets are due only to the orbital angular 4

momentum of atomic electrons. Diamagnetism only arises in atoms with an even number of electrons. When two electrons share the same energy level and are in the same orbital, their spins must be opposite in direction since two electrons cannot occupy the same quantum state. This is also known as the Pauli Exclusion Principle. If the spins of two electrons point in opposite directions, their magnetic moments cancel and only the moment of the orbital motion is left. An external electric field affects that orbital motion and induces a change in the magnetic moment of the atoms, which is described by Lenz s law. Since the orbital angular momentum of an electron is less that the spin angular momentum, diamagnetism is the weakest class of magnetism; it is also the only orbital induced magnetization of the list. The other types of magnetism fall into the class of spin-induced magnetization. 2.2.2 Ferrimagnetism Ferrimagnets are generally crystalline in structure. When magnetized, the atoms form alternating sublattices, or domains, of dipoles pointing in similar directions. Neighboring sublattices align antiparallel to one another, but the magnetic moments of those sublattices are unequal as shown in Fig. 1, producing a net magnetization in the material. Ferrimagnetism is one of the weaker types of spin-induced magnetism, a fact which is due to the alternating, unequal sublattices in the material. When it is said that a magnet is weaker, this means that it doesn t react as strongly to external electric fields as other magnets. 5

Figure 1: Schematic of a two-dimensional ferrimagnetic lattice. 2.2.3 Antiferromagnetism Another form of weak magnetization is found in antiferromagnets. In fact, anitferromagnets have no spontaneous, i.e. permanent, magnetization. Like in ferrimagnets, the atoms of antiferromagnets form sublattices within in the material that alternate in direction as shown in Fig. 2. However, neighboring sublattices have equal magnetic moments. So, the overall effect is a zero magnetization within the substance. The reason that antiferromagnetism is a type of magnetism can be seen when the temperature of an antiferromagnet is raised. Above a certain temperature known as the Néel temperature, anitferromagnets enter a paramagnetic state. Figure 2: Schematic of a two-dimensional antiferromagnetic lattice. 6

2.2.4 Paramagnetism Paramagnets, when isolated, have no spontaneous magnetization. Spontaneous magnetization occurs when a material has a net magnetization in the absence of an external magnetic field. As shown in Fig. 3, paramagnetic dipoles are randomly oriented, due to the random fluctuations caused by temperature, and have no tendency to align or misalign with neighboring dipoles to form domains; so, paramagnetic magerials have a net magnetization of zero. However, in the presence of an external magnetic field, the dipoles of a paramagnetic material align themselves with the field and point in the same direction. This produces a magnetization in the material. Figure 3: Schematic of a two-dimensional paramagnetic lattice. 2.2.5 Ferromagnetism Ferromagnetic materials exhibit a spontaneous non-zero magnetization caused by the tendency for neighboring dipoles to align parallel to each other, which can be seen in Fig. 4. This is due to the energy interactions between neighboring dipoles, particularly by nearest neighbors, which will be discussed in depth below. The temperature of the material governs the strength of nearest neighbor interactions. In fact, above a certain temperature called the Curie temperature, the dipoles in a ferromagnet have an increased thermal energy and the direction of their magnetic moments fluctuates; this causes the ferromagnetic material 7

to become demagnetized and enter a paramagnetic state [9]. This phenomenon is known as a phase transition. Figure 4: Schematic of a two-dimensional ferromagnetic lattice. 3 Phase Transitions A discontinuous change in material properties, with only an infinitesimally small change in environment, denotes the presence of a phase change. For example, a phase transition can occur as the result of a temperature or pressure change. The classical way to categorize phase transitions is by using the Ehrenfest classification. Ehrenfest labeled phase transitions based on the discontinuity of the Gibb s free energy with respect to other thermodynamic variables, such as heat capacity and density [16]. A first-order phase transition, like that from water to ice, is characterized by the free energy exhibiting a discontinuity in its first derivative. Similarly, second-order phase transitions are named for the discontinuity of the second derivative of free energy. Higher order transitions are named for the derivative in which the free energy is discontinuous. However, this nomenclature can quickly become uninformative and therefore a different, simpler classification is now used. In the modern classification of phase transitions, there are two types of phase transitions: first order and continuous [16]. A first order phase transition is characterized by the presence of latent heat during the transition. This indicates a discontinuity in the 8

entropy of the system; the change in entropy is heat absorbed or emitted by the system over a change in temperature. Since latent heat is heat absorbed by the system with no change in temperature, there is a discontinuity in entropy. One example of a first-order phase transition is of water changing into steam. The second type of modern phase transitions is the continuous phase transition, which exhibits no latent heat and therefore no discontinuity in the entropy or any other first derivative of the free energy. Also, in continuous phase transitions, such as in the ferromagnetic phase transition, one thermal property, like magnetism, may be continuous, but its derivative is not. In ferromagnets, a continuous phase transition manifests in the change from a non-zero magnetization of the material to a zero value at a particular temperature called the Curie temperature, or T c. The Curie temperature is the temperature at which a ferromagnet enters its paramagnetic state and therefore loses its spontaneous magnetization. Phase transitions which occur at a given temperature, rather than pressure for example, have a critical temperature associated with the transition. The critical temperature is the temperature above which disorder occurs and there are no boundaries between different phases in the material. For a ferromagnet, its critical temperature is the Curie temperature. Below T c, the ferromagnet is fully magnetized, i.e. all of the individual dipoles are aligned with one another. As the temperature is increased and approaches T c, fluctuations in the magnetization appear, another indication that a continuous phase transition is occurring. Then at the critical temperature, the ferromagnet loses its magnetization. This behavior of the magnetization can be described mathematically by a power law relation: M (T c T ) β (1) where β is the critical exponent. This power law describes the magnetization of the lattice 9

near the critical temperature T c which depends on the critical exponent β, a measure of the abruptness of the transition. The important fact about β is that it is independent of lattice size due to universality. In large systems, like a large lattice, the critical behavior of a phenomenon is independent of system size. This successfully models real materials because a large amount of iron has the same critical behavior as a small amount. The power law in Eq. 1 also defines another quantity called the order parameter. The order parameter is the property which, on one side of the transition, is zero and is non-zero on the other. The magnetization is the order parameter of this phase transition; above T c it is zero and below T c it is non-zero. 4 The Ising Model Models of magnetism aim to simulate the behavior of real materials in order to better study and examine how magnetic processes work. They give researchers valuable insights into the inner workings of atomic structure as well. Further, models allow for the prediction of how changes applied to a system will affect its properties, like magnetism or energy. One such system is the Ising model. The Ising model uses Boltzmann statistics to allow for the determination of the most likely state in which a ferromagnet will be found at a given temperature. The state of a magnet is simply a particular arrangement of its dipoles. Consider, for the moment, a one-dimensional array of arrows, like that shown in Fig. 5. Figure 5: Example of a one-dimensional ferromagnetic lattice. These arrows represent the dipole moments of each atom in a ferromagnet and give a schematic view of the one-dimensional Ising model. In the Ising model, it is assumed that 10

the lattice of dipoles has a preferred axis of orientation for magnetization, which is vertical in Fig. 5, a convention that will continue through the document. Let us say that an atom with a magnetic dipole pointing up has a spin s = 1, while a down-pointing dipole has spin s = 1. We further assume that the magnetic dipoles in a ferromagnet align at temperatures below T c, as was previously stated; the alignment produces a non-zero magnetization in the absence of a magnetic field and therefore has a spontaneous magnetization. This spontaneous magnetization is mainly due to nearest neighbor interactions. For example, in the one-dimensional Ising model, any given dipole has only two nearest neighbors, one on each side. This is a valid assumption partly due to comparisons with experimental data, but, given the nature of magnetic fields, we can also see why this assumption can be made. For a loop of current, like that generated by an electron, the magnetic field decreases at a rate proportional to r 3. So, even the second nearest neighbors in the one-dimensional model experience a magnetic field that is 1/8 that of nearest neighbors. Nearest-neighbor interactions between dipoles account for the energy of a given lattice. It is known that this energy is minimized when neighboring dipoles have parallel spins, therefore the energy of a pair of parallel spins can be said to be ɛs 1 s 2 where ɛ is the interaction energy and s 1 and s 2 are neighbors. The value of ɛ determines the strength of the coupling between neighboring dipoles. Given this, the interaction energy between two antiparallel dipoles is ɛs 1 s 2. Determing the total energy of the lattice is simply done by summing over all the nearest neighbor pairs in the system, which can be given by E = ɛ s i s j. (2) neighborpairs:i,j However, thermal effects still need to be taken into effect. The lattice itself is considered to be surrounded by a heat bath. The heat bath provides energy that allows for individual 11

dipoles to flip their orientation and progress into a different state with a different energy. As the lattice is surrounded by a heat bath, its thermal behavior can be predicted by the partition function in the canonical ensemble. The partition function Z is defined as Z = {s} e E/k bt (3) where the sum is over all possible lattice states, k b is Boltzmann s constant (k b T ) 1 and T is in units of ɛ/k b K. If there are N dipoles in the lattice, the number of terms in the sum is 2 N, which is typically a large number. The mathematics become more difficult as the dimensionality of the lattice increases. For the two-dimensional model, each dipole has four nearest-neighbors which drastically inflates the number of terms in the sum. An example of one arrangement of a two-dimensional Ising lattice is shown in Fig. 6. Similarly, dipoles in the three-dimensional cubic model have six neighbors. Determining the partition function for the Ising model in higher dimensions becomes difficult quickly. It is important to note that the Ising model is not the only valid model of ferromagnetism. Figure 6: Example two-dimensional square lattice in the Ising model 4.1 Other Models of Magnetism The Ising model is a classical model of ferromagnetism, considering the microscopic properties of magnetic dipoles in the lattice. Different models of ferromagnetism other than the Ising model have also been developed. One such model, the Hubbard model, uses quantum mechanics to simulate ferromagnetism. In particular, the Hubbard model involves the use of 12

itinerant, or moving, electrons and has two terms in its Hamiltonian: the hopping term and the creation/annihilation term [13]. The hopping term allows for the tunneling of electrons to other lattice sites and exhibits the quantum mechanical nature of this model. The Heisenberg Model is another quantum model of ferromagnetism. In this model, the spins are treated quantum mechanically and have spins of ± 1 2 and have different coupling constants for each direction in the lattice [15]. A limiting case for the Heisenberg model is, in fact, the Ising model when the coupling constant in the Ising takes into account the changed spin values. Another model is known as the Potts model. In the Potts model, the magnetic moments of the dipoles are not constrained to point in just two dimensions; they are able to point in any direction within the plane for the two-dimensional model and are represented by angles from a reference axis [14]. The Ising model is again a limiting case of the Potts model where the spins are constrained to point at one of two allowed angles. Additionally, the partition function of the Potts model is more difficult to solve than for the Ising model, since the angles between neighboring moments must be taken into account as well. 4.2 Solving the Ising Model The Ising model is considered solved when the exact partition function for the lattice is calculated. Evaluating the sum of the Boltzmann factors for each state is relatively simple in the one-dimensional model and is included in most textbooks with the Ising model; Ising himself solved the one-dimensional model in 1925 [9][10]. In fact, the solution will be calculated presently for the zero external magnetic field case. We can write the total energy of a one dimensional lattice with N dipoles as E = ɛ(s 1 s 2 + s 2 s 3 + s 3 s 4 +... + s N 1 s N ). 13

The sum of the partition function must be carried out for all possible dipole arrangements so that Z = s 1 s 2 s 3... s N e ɛs 1s 2 /k b T e ɛs 2s 3 /k b T e ɛs 3s 4 /k b T...e ɛs N 1s N /k b T and each sum is evaluated for s = 1 and s = 1. Consider the sum for the Nth dipole, e ɛs N 1s N /k b T. The product s N 1 s N is either +1 or -1, so the sum is now for e ɛ/k bt + e ɛ/k bt. Through further simplification, we find that s N e ɛs N 1s N /k b T = 2cosh(ɛ/k b T ). By induction, the previous sum over s N 1 is also equal to 2cosh(ɛ/k b T ). This follows for all other preceding terms, the very first modified by a factor of 2. Therefore, we have N 1 factors of the sum simplification. We can then write the partition function for the one-dimensional Ising model as Z = 2 N (cosh(ɛ/k b T )) N 1 which can be approximated to (2ɛ/k b T )) N 1 for N >>> 1. Now that the partition function has been found, the behavior of many thermodynamic properties can be calculated. One such property is the energy of the lattice as a function of temperature. Since Ū = β lnz, where β is (k bt ) 1, Ū = Nɛtanh(βɛ). (4) We can see that this function behaves as expected. As T 0, the energy approaches Nɛ and goes to zero for large values of T, indicating disorder at high temperatures and ordering at low ones. Knowing the energy as a function of temperature is important because the 14

magnetization of the lattice is directly proportional to the energy. Therefore, evidence of a phase transition should be seen in the energy as well as the magnetization. We can determine the expected magnetization of the one-dimensional Ising model using mean field approximation [9]. We select a dipole from the lattice and temporarily freeze its two neighbors. The dipole is then free to point either up or down and have a different energy for each arrangement. If it points up, the energy of the dipole, a version of Eq. 2, is E = ɛ s neighbor = ɛn s (5) neighbors where s is the average arrangement of the neighboring dipoles. For the one-dimensional case, n is two. The energy if the selected dipole pointed down is just the opposite of Eq. 5 and is E = ɛ s neighbor = ɛn s. neighbors For this dipole, the partition function is Z i = e βɛn s + e βɛn s, which simplifies to Z i = 2cosh(βɛn s). To find the average expected value for a thermal property A using the partition function, A = 1 [ ] A(s)e E(s)/k bt. Z s So, the average spin value for the selected dipole is s = 1 ] [(1)e ɛn s + ( 1)e ɛn s. Z This can be simplified to s = tanh(βɛn s, 15

a transcendental equation. From the presence of the tanh function in Eq. mft, it can be inferred that the average magnetization of the one-dimensional lattice is continuous for all intervals of β. Ising determined this as well in 1925 and concluded that there is no phase transition in his model because there is no abrupt transition from the ferromagnetic to the paramagnetic state. There is no phase transition in the one-dimensional Ising model. This lead to Ising believing that his model had no future relavance [1]. However, scientists continued to explore further applications of the Ising model and attempted to solve the two-dimensional case. In 1943, Lars Onsager calculated the exact partition function for the two-dimensional Ising model as N, drawing heavily from methods introduced by Kramers and Wannier [11]. He never published a formal derivation of the solution, but did use quaternian algebra to find the bases of operators on the partition function, rather than the traditional matrix method [2]. These roots have eigenvectors, the largest of which provides the key to finding the partition function. Once the root is found, the limiting case for N simply replaces integrals for sums and Onsager made use of elliptical substitions to complete the integrals. For the three-dimensional lattice, Istrail showed in 2000 that it is computationally intractable to find the partition function [7]. A problem that is intractable is one that, in theory, is solvable, but the solution would take to long to solve in reality. Istrail proved that the solution to the three-dimensional Ising model is part of a group of problems know as NP-complete complexity class. Problems that are NP-complete have solutions that cannot be found in a reasonable amount of time; in fact, the time required to solve NP-complete problems using any known algorithm increases rapidly as the size of the problem increases, reaching millions of years at today s fastest computing speeds. Istrail achieved this proof using the Kuratowski Theorem for forbidden planar graphs. With the challenges of finding exact solutions for the two- and three-dimensional Ising models, the only way to proceed is to rely on thoughtful assumptions which increase the feasibility 16

and preserve the accuracy of the model. To do this for the Ising model, computer simulations involving the Monte Carlo method are a logical choice. 5 Computational Methods As previously described, the Ising model is a simple way to model a ferromagnet. However, in order to simulate a real material, with numbers of atoms measured in moles, analytically determining the partition function of the system is intractable in the three-dimensional case and is difficult even in the two-dimensional case, especially when the system is altered from a pure state. The next logical step is to use a computational approach. The Monte Carlo method is a stochastic algorithm which uses Boltzmann factors to analytically determine the equilibrium state of a lattice for a given temperature. The reason that the Monte Carlo method is necessary for simulating lattices, even on the order of a 10x10 square lattice, the number of possible states for the system is 2 100, or around 10 30. To sample each of these states would take an inordinate amount of time. The Monte Carlo method uses something called the Metropolis algorithm, which creates a random sampling of states that are more likely to have a lower energy [10]. This follows the behavior of real systems since they always settle to a lower energy state. It is important to outline the structure of the code for clarity. The lattice, two-dimensional in this case, starts in an initial state of complete magnetization at T = 0 with all dipoles having a spin of +1 as shown in Fig. 7 with a size of N N where N 2 is the number of dipoles. Over the course of the program, the temperature is increased, causing the final state of the previous temperature to become the initial state of the higher one. This creates no loss of accuracy as the initial state of the lattice has little effect on the final state due to thermal effects; in fact, the lattice reaches an equilibrium position more quickly from state which is 17

Figure 7: Fully magnetized 4x4 lattice. closer in temperature. It is also important to note that the Ising model and the Monte Carlo method are not time dependent. The model provides a virtual snapshot of the lattice as it reaches its equilibrium position, which occurs very quickly, so as to be almost imperceptible, in real materials. Periodic boundary conditions are also implemented to reduce edge effects that are present in any finite lattice. For each temperature, the program successively chooses a randomly generated dipole and analyzes its current state. The order of dipole selection does not matter since each one is selected multiple times over the course of the simulation. With each selected dipole, the code calculates the energy of that dipole if it were to flip its orientation using a modified version of Eq. 2 E dipole = ɛs dipole s i. (6) nearestneighbors,i The sum in Equation 6 is over four dipoles, the number of nearest neighbors for any dipole in a two-dimensional lattice with periodic boundary conditions. To mitigate the edge effects of the lattice, periodic boundary conditions are implemented so that all dipoles have four neighbors. Since the magnetic field of an electron, drops as r 3, we need only include the closest neighbors to the dipole. The program calculates the probability of whether or not to flip the chosen dipole based on the value of E dipole, its energy if it were to flip orientation. If the flipped energy is negative, the program will always flip the magnetic moment to minimize 18

the energy of the dipole. If the energy is positive, Boltzmann statistics determine whether or not the dipole will flip. The Boltzmann factor e Edipole/T is calculated, simulating the lattice being in a heat bath at constant temperature. This factor denotes the probability that the dipole will be found in its current state and is compared to a randomly generated number between zero and one. A Boltzmann factor that is greater than the random number flips the dipole; if the factor is less than the random number, the dipole remains unchanged. Physically, this means that at higher temperatures, the Boltzmann factor of a dipole is smaller than at low temperatures and the dipole has a larger probability of flipping into a state of higher energy. At higher temperatures, above the T c, a ferromagnet s dipoles are randomly aligned due to the higher probability of finding the system in a higher energy state. While at temperatures below T c, the probability of a dipole flipping to higher energy state is very low, making the dipole have a high probability of staying aligned with its neighbors. After each dipole flip, the magnetization of the lattice is calculated and then added to a running total of the magnetization of the lattice for the same temperature. The running total is then divided by the number of Monte Carlo steps to determine the average magnetization of the lattice at a given temperature. When all of the dipoles have been selected, i.e. the calculation of a Boltzmann factor has been implemented N 2 times, one Monte Carlo step has passed. Ultimately, the program produces a data file with average lattice magnetization for each temperature tested, from which the Curie temperature and other quantities can be determined. This will be discussed in the following session. The next step is to modify the program in order to introduce gap defects into the system. This will be done by incrementing the amount of defects and randomly placing them within the lattice while running the program normally. Knowing the exact placement of the defect within the lattice is not necessary because the physics being done on each 19

lattice site is the same, no matter the location, which is partially due to periodic boundary conditions. The gaps themselves will be added into the lattice by changing the spin of the selected site to be zero, which graphically looks like Fig. 8. This simulates the absence of an interacting neighbor and will decrease the energy of its neighbors, making them less likely to stay aligned. Figure 8: 2-dimensional lattice with one gap defect 6 Results The goal of this thesis was to determine the T c of the Ising model in two-dimensions as well as analyze the effects of gap defects on the Curie temperature T c. In order to do this, T c for a perfect lattice, i.e. without defects, must first be determined. However, an idea of what the raw data is will be helpful for discussion purposes. The program outputs temperatures with their corresponding average magnetizations per spin, after normalization by lattice size. Each of the average magnetizations does have a spread in the data. However, as is shown in Fig. 9, we see that the standard deviation of each data point is small; the largest deviation seen is at a temperature of 4.25 with a magnetization of 0.0754 ± 0.0254. However, this point is statistically different from its neighbors. The relatively small size of the vertical error bars shows the precision of the Monte Carlo method. Fig. 9 has the correct shape for the magnetization with respect to temperature because it does go to zero. However, the transitions from a fully magnetized lattice happens over a larger temperature range than in a true ferromagnet. The Curie temperature for the 20

Figure 9: Magnetization vs temperature with vertical error bars denoting the standard deviation in magnetization for each temperature. The lattice size is 50x50 dipoles. perfect two-dimensional Ising model is 2.27ɛ/k b, where the interaction energy ɛ is 1 for the purposes of this experiment. The reason for the non-abrupt transition is that the data for this graph were collected over 100 Monte Carlo steps, to give the reader a sense of the shape of the data. The data presented in the rest of the paper were averaged over 8000 Monte Carlo steps; this would only decrease the size of the standard deviation and lead to more accurate results at the same time. The data for a lattice evaluated over 8000 Monte Carlo steps is shown in Fig. 10. The magnetization goes to zero over a much smaller temperature range than in Fig. 9, even at the same lattice size. Again, this is due to the increase in the number of Monte Carlo steps which allowed the lattice to equilibrate at each temperature, giving us a more accurate set of data. The temperature steps in the code for Fig. 10 is only 0.25. The graph is displayed with lines to more clearly show the behavior of magnetization versus temperature. The next step is to determine the Curie temperature of the square lattice. In order to do this, we need a higher resolution and must decrease the temperature steps in our code. We found 21

Figure 10: Magnetization versus temperature for multiple lattice sizes that the optimum temperature step for our particular program is 0.05. At smaller steps, we saw large fluctuations in the magnetization over a small temperature increase where none should occur. With a temperature step of 0.05, we see a familiar diagram in Fig. 11. With the data from this plot, we can find the Curie temperature of the lattice. Recall that the magnetization is a function of temperature, found in Eq. 1 of Section 3, is M (T c T ) β. When we take the derivative of this equation with respect to temperature, we see that it diverges at the critical temperature. We can also see this would occur by look at Figs. 10 and 11. As the temperature approaches T c, we observe a large change in magnetization for a small change in T. The quantity dm dt will go to infinity for a perfect, infinite lattice since the phase transition happens instantaneously for ideal conditions. 22

Figure 11: Magnetization versus temperature for a 150x150 lattice The derivative of magnetization with respect to temperature of Fig. 11 is shown in Fig. 12 with more runs of the program for the same parameters. Here we can see the temperature derivative of magnetization begin to diverge at around 2.4; however, it does not truly diverge. This is due to the finite size of our lattice and results in the calculation a T c higher than expected. To find the Curie temperature with our data, we ran the Ising code ten times and determined the temperature at which dm dt is the largest because this is the limit of the divergence of the graph. We found that T c is 2.465±0.052, which is higher than the accepted value of 2.27 for an infinite square lattice. The reason for this discrepancy is most likely due to the averaging method used in the program. In the primary literature and in text books, there are nuances in the measuring of observables in the computational Ising Model and my simplified code does not account for these [10]. Differences in the averaging method could lead to the results presented in this paper where the T c found was lower than the accepted value. 23

Figure 12: The derivative of magnetization with respect to temperature for a 150x150 lattice and a temperature step of 0.05 7 Future Work and Effects of Adding Defects We expect to see a decrease in the Curie temperature of the lattice. Consider an individual dipole with four nearest neighbors, all parallel and a dipole with one of those neighbors taken away. The dipole with three neighbors has a higher energy than one with four if they are in the same arrangement. This means a flip in a neighbor dipole to be more likely to cause a flip for the dipole with only three neighbors. This is due to the Boltzmann factor of the dipole, e E/T and the fact that transitioning into a higher energy (less negative) state is less likely. The increased tendency for disorder in a lattice with gap defects will allow for the transition into complete disorder, the paramagnetic state, to happen at a lower temperature. We would then expect to see a decrease in the T c as defects are added. However, data for different numbers of defects in a given lattice would need to be collected. Due to time constraints, no data have been collected for lattices with defects. 24

8 Acknowledgments I would like to thank my thesis advisor Dr. Kevin Crosby for his guidance through the past year as well as Dr. Dahlstrom for her help to all students completing a thesis. Finally, thank you to the Department Physics and Astronomy at Carthage College for my education. References [1] Brush, S. G. (1967). History of the lenz-ising model. Reviews of Modern Physics, 39(4). [2] Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review, 65(3 and 4). [3] Barouch, E., McCoy, B. M., & Wu, T. (1973). Zero-field susceptibility of the twodimensional Ising model near T c. Physical Review Letters, 31(23). [4] Prados, A., & Brey, J. J. (2001). Analytical solution of a one-dimensional Ising model with zero-temperature dynamics. Journal of Physics A, 34, L453-L459. [5] Sknepnek, R., & Vojta, T. (2004). Smeared phase transition in a three-dimensional Ising model with planar defects: Monte Carlo simulations. Physical Review B, 69(17). [6] Harris, A. B. (1974). Effect of random defects on the critical behaviour of Ising models. Journal of Physics C, 7(9). [7] Istrail, S. (2000). Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intractability for the partition function of the Ising model across nonplanar lattices. Proceedings of the 32nd ACM Symposium on the Theory of Computing, 87-96. [8] Ferrenberg, A. M., & Landau, D. P. (1991). Critical behavior of the three-dimensional Ising model: A high-resolution Monte Carlo study. Physical Review B, 44(10). 25

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