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Engineering and Technology Journal

The Phase Transition of the 2D-Ising Model By Using Monte Carlo Method

Dhia'a Kudeer, Saad. Abood

Published Jan 1, 2011• 0 citations
Publisher

Abstract

In this project, the location of the phase transition in the two dimensional Ising model will be determined using Monte Carlo simulation with importance sampling. the magnetization per site [m ], energy per site [J ], magnetic susceptibility, specific heat of a Ferromagnetic materials are Calculated as a function of temperature T for 10´10, 20´ 20, 40´ 40 , 50´50 spin lattice interaction by using Monte Carlo Simulation of the 2D Ising Model for some experimental values of ferromagnetic materials such as Gadolinium Chloride ( ) 3 GdCl at Curie temperature c T B = 2.2 J k , and ferromagnetic thin film from Nickel ( ) i N growth on cooper (Cu)at Curie temperature c B T = 2.772 J k , in zero and nonzero magnetic field. It was noticed that above a certain temperature ( ) c T the material will be in a paramagnetic state, this will lead to that the average magnetization will be decrease and the average energy increase, while below that temperature ,it will be in a ferromagnetic state, and the average magnetization will increase and the average energy decrease. Moreover, above a certain temperature spontaneous magnetization will be zero

DOI
10.30684/etj.29.2.7

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In this project, the location of the phase transition in the two dimensional Ising model will be determined using Monte Carlo simulation with importance sampling. the magnetization per site [m ], energy per site [J ], magnetic susceptibility, specific heat of a Ferromagnetic materials are Calculated as a function of temperature T for 10´10, 20´ 20, 40´ 40 , 50´50 spin lattice interaction by using Monte Carlo Simulation of the 2D Ising Model for some experimental values of ferromagnetic materials such as Gadolinium Chloride ( ) 3 GdCl at Curie temperature c T B = 2.2 J k , and ferromagnetic thin film from Nickel ( ) i N growth on cooper (Cu)at Curie temperature c B T = 2.772 J k , in zero and nonzero magnetic field. It was noticed that above a certain temperature ( ) c T the material will be in a paramagnetic state, this will lead to that the average magnetization will be decrease and the average energy increase, while below that temperature ,it will be in a ferromagnetic state, and the average magnetization will increase and the average energy decrease. Moreover, above a certain temperature spontaneous magnetization will be zero
The location of the phase transition in the two dimensional Ising model were determined using Monte Carlo simulation with importance sampling. The magnetization persite ,energy per site , magnetic susceptibility , specific heat of a Ferromagnetic materials were calculated as a function of temperature for spin lattice interaction in zero and nonzero magnetic field . There is thus a phase transition defined by the Curie temperature. The Monte Carlo method was used to check the results and to confirm the phase transition . The internal interaction results were found to be consistent with what was expected. As a magnetic field is applied, the spins tend to align with it for ˃ and its effect is not significant at a very high temperature because of the thermal agitation. For ˂ , the alignment of the spins is possible only if the amplitude of the field is big enough.
The position of the phase transition in the two dimensional Ising model were determined byusing Monte Carlo simulation in a quadratic for area of variable length with external magnetic fieldswitched off (B = 0). The magnetization (M) per site (µ), magnetic susceptibility (x) of aferromagnetic and paramagnetic materials were calculated as a function of temperature T for(20×20,40×40,60×60), (80×80,120×120,200×200) spin lattice interactions. Nearest neighborinteraction is assumed (i.e. each spin has 4 neighbors); uses periodic boundary conditions. The Curietemperature (Tc = 2.27 J/kB ) is determined by measuring the magnetic susceptibility at which theferromagnetic and paramagnetic undergoes a phase change from order to disorder. There is thus aphase transition defined by the Curie temperature. The Monte Carlo method were used to check theseresults and to confirm the phase transition. The data are analyzed using the Curie-Weiss law whichcontains the Curie temperature as a parameter.
In this work, Markov Chain-Monte Carlo technique was used to study the phase transition in two and three dimensional Ising Model (IM) in a square and cubic lattice. The study of temperature dependence of average magnetization and specific heat in different magnetic fields has been carried out in the 3x3 and 3x3x3 lattice with periodic boundary. Critical temperature point kBTc / J for 2D and 3D Ising Model has been observed at around 2.2 and 4.3 respectively at zero field. Our work satisfies Onsager’s critical value in 2D IM. The simulation suggests bifurcation in average magnetization below critical temperature Tc. Temperature plays the role of increasing randomness of spins. We found that Ising Model in small lattice size still retains interesting features like spontaneous magnetization and symmetry breaking below Tc at B = 0. At a non-zero field, the likelihood of spins to prefer certain alignment depends on the direction of the external field and magnitude of magnetization depends on magnitude of field ±B. Specific heat Cv, which gives us fluctuation of energy at particular temperature, has been found to attain maximum value at critical regions.
Many problems in statistical physics resist exact solution once the number of interacting particles becomes large, and the two-dimensional Ising model is a classic case where numerical simulation fills that gap. This paper reports a Monte Carlo study of the two-dimensional Ising model using the Metropolis algorithm, undertaken to examine how magnetization, internal energy, specific heat, and magnetic susceptibility change with temperature. Spin configurations were generated on a square lattice under periodic boundary conditions, with each proposed spin flip accepted or rejected according to the Boltzmann weight of the resulting energy change. The simulation reproduces the expected transition from an ordered ferromagnetic state at low temperature to a disordered paramagnetic state at high temperature, with magnetization falling sharply and both specific heat and susceptibility peaking near a critical temperature of Tc ≈ 2.3–2.4 (in units of J/kB), consistent with the exact value obtained analytically for this system. These results confirm that the Metropolis algorithm, despite its simplicity, remains a dependable tool for locating second-order phase transitions and extracting thermodynamic quantities in lattice spin systems, and the same computational framework can be adapted to more elaborate models used in condensed matter and materials research.
Monte-Carlo simulation can be used to investigate cooperative phenomena, such as phase changes that occur when short-range atomic or molecular interactions give rise to long-range order. For example, some magnetic materials exhibit a paramagnetic phase at high temperatures, where the magnetic structure is disordered, but undergo a phase transition to a more ordered state at a lower temperature. This chapter uses a two-dimensional lattice Ising spin model, where spins on each site can be either up or down, to determine the critical temperature at which the phase transition occurs. In addition, it calculates the variation of energy, specific heat capacity, magnetisation and magnetic susceptibility with temperature. The chapter explains both the Gibbs and Metropolis sampling techniques, and lists a Python program that makes use of the latter.
We simulate the Ising Model with the Monte Carlo method and use the algorithms of Metropolis to update the distribution of spins. We found that, in the specific case of the three-dimensional Ising Model, methods of Metropolis are efficient. Studying the system near the point of phase transition, we observe that the magnetization goes to zero. In our simulations we analyzed the behavior of the magnetization and magnetic susceptibility to verify the phase transition in a paramagnetic to ferromagnetic material. The behavior of the magnetization and of the magnetic susceptibility as a function of the temperature suggest a phase transition around KT/J ≈ 4.5 and was evidenced the problem of finite size of the lattice to work with large lattice.