Ising Model Phase Transition Calculation for Ferro-Paramagnetic Lattice

The position of the phase transition in the two dimensional Ising model were determined by using Monte Carlo simulation in a quadratic for area of variable length with external magnetic field switched off ( B = 0). The magnetization   M per site    , magnetic susceptibility    of a ferromagnetic and paramagnetic materials were calculated as a function of temperature T for   60 60, 40 40, 20 20    ,   200 200 , 120 120 , 80 80    spin lattice interactions. Nearest neighbor interaction is assumed (i.e. each spin has 4 neighbors); uses periodic boundary conditions. The Curie temperature   B C k J T 27 . 2  is determined by measuring the magnetic susceptibility at which the ferromagnetic and paramagnetic undergoes a phase change from order to disorder. There is thus a phase transition defined by the Curie temperature. The Monte Carlo method were used to check these results and to confirm the phase transition. The data are analyzed using the Curie-Weiss law which contains the Curie temperature as a parameter.


INTRODUCTION
Ferromagnetic materials show ferromagnetic behavior only below a critical temperature called the Curie temperature, above which the material has normal paramagnetic behavior.
The approach to ferromagnetism as a function of temperature from above is described by the Curie-Weiss Law which gives the magnetic susceptibility as a function of temperature.
where  and  are the magnetic susceptibility and relative magnetic permeability of the material respectively. C is a constant characteristic for a given substance and C T is the Curie temperature. Equetion 1 is only valid above the Curie temperature [1,2]. Imagine a quadratic 2d area with 2 L spins on a grid. Each spin can either point up (+1) or down (-1). The average magnetization of the area is the average spin value and hence between 1 (completely ordered state) and 0. Neighboring spins S and S' interact with an interaction energy of Since each spin has 4 nearest neighbors (periodic boundary conditions), the interaction energy per spin can be between   J 4  (all neighbors parallel to the center spin) and   J 4  (all 4 neighbors antiparallel to center spin) where   J is the coupling strength. (There is no external magnetic field present) [3,4] .
Generally, states with less energy are preferred, so the system stays in completely ordered state for zero temperature. However, as we increase temperature, each spin has a thermal energy of k B T (where k B is Boltzmann's constant and T is the absolute temperature). Due to this thermal energy, the system does not stay in completely ordered state but spins start to fluctuate ("flip") randomly.
There is a phase transition at the critical temperature of   : Starting with a completely ordered state, the system stays mostly ordered below the critical temperature while it goes completely unordered above it.
Hence, above the critical temperature, the average magnetization is (about) zero while it is non-zero below it. On the two-dimensional lattice each spin interacts with its four neighbors as shown in Figure 1

THE ISING MODEL
The Hamiltonian for a system that is dependent on the arrangement of spins on a lattice and from that we can deduce properties such as magnetization and susceptibility [5,6,7]. Suppose that the Hamiltonian is The nature of the interaction in the model is all contained in the sign of the interaction coupling constant J . If J is positive it would mean that the material has a ferromagnetic nature (parallel alignment) while a negative sign would imply that the material is antiferromagnetic (favors anti-parallel alignment). J will be taken to be 1  in our discussion and the values for spins will be 1  for spin up and 1  for spin down. A further simplification is made in that B k J is taken to be unity. The relative positioning of nearest neighbors of spins is shown in Figure 2 with the darker dot being interacted on by its surrounding neighbors [9,10].
International Letters of Chemistry, Physics and Astronomy Vol. 15 In the simulation, whenever flipping a spin lowers the interaction energy, the flip is done. If it increases the energy, the flip is only committed with a probability of exp (-ßE) where ß=1/k B T and (E >0) is the energy difference between flipped and unflipped state (Metropolis algorithm). As one can see, the relevant temperature can be expressed in units of ßJ which is called the reduced temperature and is the "natural" temperature unit used throughout the implementation [4].
The simulation repeatedly computes so-called MCS (Monte-Carlo step), commonly also referred to as time, each of whom involves the potential flipping (as explained above) of all spins in the area [4].

1. Effect of the size
In order to see the effect of the size of the lattice on the transition of the phase, the square of the magnetization against T, for each size and for five temperatures in the range (2.25-2.29 J/k B ) is plotted in Figure 3 for three lattice of sizes   60 , 40 , 20 for (200000 steps each), in the absence of the external magnetic field.
In Figure 3, at low temperature below ( T c = 2.27 J/k B ) the square of the magnetization were all most stable for all lattice sizes.
While at higher temperatures above ( T c = 2.27 J/k B ) the square of the magnetization disorder and the fluctuation are larger for all lattice sizes .

2. Relaxation
Starting from a completely ordered state, this displays the average magnetization over time (i.e. MCS) at a reduced temperature for lattice   has been shown in Figure 4. In the first case (green graph) where the temperature is below the critical temperature the magnetization and the system stays mostly ordered, while in the second case (red graph) where the temperature is above the critical temperature the magnetization fluctuating thermically around zero and the system rapidly goes in to completely unordered state .

Magnetization
The phase transition can most easily be seen when starting with a completely ordered state and computing the average magnetization after lots of (MCS) for different temperatures.

4. Susceptibility
At ( T > T C ) the susceptibility is in phase paramagnetic where the Curie-Weiss law At high temperature ( T > T C )the paramagnetic susceptibility decrease and the effect of the thermal agitation appears to cause in neglecting the effect of the intrinsic molecular field.   In Figure 8, where ( T < T C ) the Curie-Weisslaw as the susceptibility has negative values.

ILCPA Volume 15
This can be clear in Figure 9 where at 

CONCLUSION
The Monte Carlo method applied to the Ising model which describes the magnetic properties of materials (lattices) allows to obtain the thermodynamic quantities variations with . Above a certain temperature (T > T C ) and in the absence of the magnetic field   B , the spins are randomly oriented, a phase transition will be paramagnetic state, leading to decreasing the susceptibility and magnetization and it has no effect because of the thermal agitation. In a certain temperature at (T =T C ) where there will be a phase transition, the magnetization decreases causing the magnetic susceptibility to goes to infinity. At (T < T C ) the spins are aligned, hence the average magnetization will increased and the phase transition will be in a ferromagnetic state.