Exploring the Ising Model: A Monte Carlo Study on Ferromagnetic Transitions

Abstract

Two dimensional Ising model's partial transition location has been determined with significant samples using Monte Carlo method. The critical inverse temperature (βC=0.435KB/J), the magnetization per site (µ) and the energy per site (j) of a ferromagnetic substance are calculated as a function of inverse temperature (βJ) for different lattice sizes, in magnetic fields (B=0,B≠0). The precise partition functions (i.e. precise solutions) of the Ising model in (L×L= 4, 8, 16, 32) the square lattice sizes with free boarders stipulations are acquired after categorize all 2L×L=216×16 (≈1,157×1077) and 232×32 (≈1,79×10308) shapes of spin with the microcanonical convey matrix.

Moreover, the stage of transitions and critical conditions have been discussed using the precise partition function in (L×L=16, 32) square lattice sizes with free boarders stipulations.

Introduction

Basically the partial a stage transitions and critical conditions are the foremost wide phenomenon in nature. The square-lattice sizes (L2) of the Ising model are the simplest efficient system appears partial transitions (the partial a stage transitions between the paramagnetic and ferromagnetic phase) and critical condition at limited temperatures.

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The square-lattice sizes (L2) Ising model has played a central role in the understanding of stage transitions and critical conditions. The precise solution for square lattice sizes of the ising model with periodic stipulations is well known both in the thermodynamic limit (that is, the unlimited size system) and limited system. However, the precise solution of the square-lattice sizes (L2) Ising model with free boarders stipulations is not known for an arbitrary size system.

Previous study has been well done on two dimensional Ising model of ferromagnetic.

Bhanot evaluated the precise partition functions of the Ising model in (L2) square lattice sizes with free boarders stipulations up to (L=10) using Cray XMP. Bhanot counted all 2L×L= 2100 (≈ 1.27 x 1030) states for (L=10), and start-up get to some beneficial outcomes. Stodolsky and Wosiek [5] obtained the precise partition function for L = 13 (corresponding to 2169 ≈ 7.48 x 1050 states) using IBM RISC 6000, and studied stage transitions based on the entropy as a function of the energy. Seung-Yeon Kim [1] computed the precise partition function of the Ising model on square lattice sizes with free boarders stipulations for (corresponding to states) using the microcanonical convey matrix.

In this study, the precise partition function and emulate the critical conditions of ferromagnetic substances in (L×L) square lattices with free boarders stipulations for (L×L=16) and (L×L=32) have been calculated. The average energy, average magnetization and the average absolute magnetization have been calculated as a function of inverse temperature (βJ) and determining the critical inverse temperature (βC=0.435 KB/J) in a magnetic field (B=0,B≠0) for (L×L= 4, 8, 16, 32) square lattice sizes.

The Ising Model

The Ising model's Hamiltonian, which sums over nearest neighboring spin pairs, and the probability distribution for calculating thermodynamic quantities, are foundational to our analysis. We express the Hamiltonian as:

H=−J∑<i,j>SiSj−μB∑iS

where $J$ represents the interaction energy, Si​ the spin, $B$ the magnetic field, and $\mu$ the magnetic moment. The probability distribution is given by:

P(Ei)=Ze−βEi(2)

with Z as the partition function and β=1/KBT.

Computational Observables

The study calculates average energy, magnetization, heat capacity (Cv), and susceptibility (χ) across various lattice sizes, identifying the critical inverse temperature (βC=0.435 KB/J) through Monte Carlo simulations. The observables are determined by equations (3) to (7), providing insights into the system's equilibrium and phase transitions.

Fluctuations in Monte Carlo Time “Evolution”

In the simulation system, whenever the flip is done, the interaction energy will be reduced. If the energy increases, the flip is only obliged with an eventuality of {exp (-βE)} whereby (β=1/KBT) and (E>0) is the energy variance between upturned and non-upturned case (metropolis algorithm method).The applicable temperature will be in modules of (βJ) named the decreased temperature and is the 'natural' temperature module used for the While of the execution. The emulation time repeatedly calculates what is called (Monte-Carlo Step), generally indicated to as time, each of which involves the potential flipping of all spins within the vicinity.

Simple Sampling

The essential elementary sampling manner comprises of simply randomly selecting points within the configuration space from anywhere. A huge numeral of spins patterns is generated randomly (for the whole lattice) and data are used to calculate the average energy and magnetization. However, this mechanism needs to suffer from precisely the same problems as the quadrature manner, often sampling from unimportant areas of the stage size. The chances of producing a randomly created spin array and up / down spin patterns are remote (~2-L), and high-temperature random spin array is highly likely. The most common way to avoiding this problem is by using the Metropolis importance sampling, which works by applying weights to the microstates .

Boundary conditions for inverse temperature (βJ):

For a given β= (KB T)-1, the initiation lattice is set as the settled lattice of the former (β):

At (β= (KBT)-1=1), i.e. at a fully low temperature, completely aligned spins are obtained. There is maximum magnetization. Then, as the temperature will increase the change of spins gradually.

When (β=(KB T)-1) is such that β ≈ βC there are various clusters of aligned spins, magnetization is utmost in each cluster, however, the magnetization of the group is generally cancelled due to the eventuality of being in the αi configuration is equal to the eventuality of being in the -αi configuration.

The dipoles are routed randomly at a fairly high temperature (β= (KBT)-1=0), while the lattice is initialized at every value of (β= (KBT)-1), the findings differ: At a totally low temperature, there are several clusters of aligned spins. These domains stop evolution: We usually tend to get domain names from Weiss and walls Bloch. As a consequence, magnetization is random. The matrix of size limits the number of possible clusters.

Results and Discussion

Location of the Critical Transition Phenomenon

Critical region shows the maximum temperature of non-zero magnetization. In this situation, the system is subject to transition called partial transition (a stage transition) from order-to-disorder[16]. In order to locate the essential (critical) inverse temperature (βC), the most realistic value is given in the thermodynamic limit, where (L×L=N2), the boundary of infinity is regarded. Thus, we aimed to calculate the essential (critical) inverse temperature with fully different lattice sizes. Tthe critical inverse temperature as N approaches infinity, and has been calculated the critical region for many lattice sizes and it value (βC=0.435 KB/J) without effect of the magnetic field (B=0).

Collected communiques for lattice size (L=4) can be clarified in figure 2. These communiques are done at a temperature minimal than the critical inverse temperature. We can expect a settled state to exist and nonetheless it is plainly displaying a flip-flop which is uncommon, resulting in an entire flip of the magnetization. The flip-flop (fluctuation) happens prior (5000 MCS) and the magnetization summits at (0 from −1). The arrangement is here in the center of the ravine and occurs to return to its prior state. The similar state happens simply after (5000 MCS) however in this case selects to flip to an reverse, but evenly likely, magnetization, from (-1 to 1).

Magnetic field status (B=0)

Impact of the Magnitude at the Distinguishing Amounts

To sight the impacts of the magnitude (size) of the lattice on the transition of the stage, the thermodynamic amounts are plotted in (L×L=4, 8, 16, 32) without effect of the magnetic field (B=0). We began with random spin at the lattice places and used Ising model to calculate magnetization and energy. We executed Metropolis algorithm Monte Carlo simulation of an Ising Model in Fortran 90 Code. The emulations were in (L×L= 4, 8, 16, 32) the square lattice sizes with free boarders stipulations, and the simulation for inverse temperature (β) of a 0.09 KB/J through 1.4KB/J with intervals of 0.02.

As the temperature raise, the system was allowed to equilibrate for 10,000 steps, and then the averages were performed over the entire lattice. The effect of inverse temperature (β) in (L×L= 4, 8, 16, 32) different lattice sizes on average absolute magnetization, average magnetization and susceptibility (x) have been shown in figure (3a, 3b, 3c, 3d).

Our automatic permutation findings, as shown in figure (2), show that this would lead to an averaging out of the mean magnetization. This naturally has a harmful influence on account the disparity of the magnetization and thus the susceptibility. This can be illustrated in the figure (3) , the drawing appears that remain steady or zero at low temperatures (high inverse temperature βJ) for approximately a prolonged duration. This would explain the disparity at βJ greater inverse temperature to summit (peak).

Despite the big size of the lattice the automatic magnetization is fewer probable to happen and the critical point moves gradually to low inverse temperature βJ which means that the summit for the susceptibility will be approaching the Curie temperature (critical inverse temperature βC=0.435 KB/J) from the left. We deduce that the bigger the numeral of MCS we use the more probable we are to insert automatic permutation and then averaging out of the mean magnetization that could change the location of the summit (peak) of the susceptibility further to the left. Workaround for this issue, that is, there is egalitarian eventuality for the magnetization to change to an adverse arrangement or revert to its prior arrangement.

To calculate the average absolute magnetization only one likelihood summit (positive one) will be actively considered and the multipliers will be overcome by averaging out the mean magnetization. This is evident from the reality that formerly we have recurrent flip-flops, in figure (2), between peak positive and negative magnetization at low inverse temperatures resulting in a zero mean magnetization. However positive worths to regard for the averaging of the mean magnetization producing a non zero average.

This influene is fewered somewhat because the ravine is raised at low inverse temperatures resulting in the magnetization having a rise likelihood of being close to zero. Happily, this nonzero average for the magnetization at higher temperatures is illogical since it does not impact the Curie temperature and shows solely in the zone which is above it. Figure (4a,4b), the average energyand heat capacity (CV) against the inverse temperature in (L×L= 4, 8, 16, 32) lattice sizes are shown. As the lattice size smaller, the more complicated to see an abrupt increasing as in (L×L=4, 8) lattice sizes, the sharp gradient of smaller lattices indicates at a probable but not obviously explained phase change.

The curve of the diagram gets clearer with the increase in lattice size thus, (L×L=16, 32) lattice sizes their results were nearly identical. The average energy is minimal and it increases slowly as the temperature rises. At a given temperature, the energy increases significantly finally approaches 0J. The heat capacity (CV) has a summit that symbolizes the phase change. At low inverse temperature βJ (high temperatures), it decreases until it reaches 0. While at a high inverse temperature βJ (low temperatures), the more of the peaks marking the phase change clearly.

Magnetic field status (B≠0)

Thermodynamic amounts against the magnetic field (B≠0)

The effects of the presence of the magnetic field (B≠0) on the thermodynamic quantities have been also investigated. The effect of the magnetic field on the average absolute magnetization, average magnetization, and average energyas a function of inverse temperature (βJ) for (L×L=16, 32) lattice sizes have been shown in figures (5 and 6). The average energy per spin as a function for inverse temperature βJ. At (ββC) settle to a Figure 5. The impact of the external magnetic field in the average energy & heat capacity in two lattice sizes.

(a) L=16, (b) L=32.

This indicates that all spins are line up in the parallel. but regarding to the heat capacity (CV) gets with clearer for different lattice sizes, the peaks a decrease for the heat capacity can be observed in (L×L=16) and (L×L=32) lattice sizes which leads to a little dislodging in the locations of the peaks at the low values for the inverse temperature βJ. Figure (6) shows the average magnetizationand the average absolute magnetizationas a function for inverse temperature βJ. At (β>βC ), the average magnetization and average absolute magnetization are high values and is consistent with the results because the sequence particles is long in lattice and the system takes greater time reach equilibrium, which means it evolves over bigger numeral of steps, while at (ββC), wherever the magnetization (M) is maximum, the spins are aligned.

Table 1: Critical Inverse Temperature and Thermodynamic Quantities for Different Lattice Sizes

Note: TBD - To be determined.

Conclusion

The Monte Carlo method, applied to the Ising model for ferromagnetic substances, effectively identifies critical transitions and conditions for various lattice sizes. This study highlights the importance of considering both zero and non-zero magnetic fields in analyzing ferromagnetic transitions, providing valuable insights into the critical phenomena of ferromagnetic materials.