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Monte Carlo Simulation of Ising Model and Phase Transition Studies By Gelman Evgenii

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Introduction to Magnetism Magnetic susceptibility χ : Types of magnetic materials: 1. Diamagnetic: χ <0 and constant (Helium); 2. Paramagnetic: magnetic susceptibility χ >0 and χ ∝ 1/T (Rare earth); 3. Ferromagnetic: Iron. Below a critical temperature (Curie temperature), χ depends on magnetic field, and the M-H diagram shows a hysteresis loop; above this temperature, the material becomes paramagnetic; 4. Anti-Ferromagnetic: Below a critical temperature, χ ∝ T; above this temperature, the material becomes paramagnetic. (MnO) Hysteresis loop

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Ising Model(2D) A lattice model proposed to interpret ferromagnetism in materials(1925). Basic idea: Elementary particles have an intrinsic property called “spin”. Spins carry magnetic moments. The magnetism of a bulk material is made up of the magnetic dipole moments of the atomic spins inside the material. Ising model postulates a lattice with a spin σ (or magnetic dipole moment) on each site, defining the following Hamiltonian: E is total energy of the system, J is the nearest spin-spin interaction energy, H is external magnetic field. σ =+1 or -1.

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Ising Model(2D) Thermal properties are defined, and computed, by the partition function, which is the normalization factor of the probability of a thermodynamic state: Using Z(T), we can calculate the specific heat C, and magnetic susceptibility χ

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Phase transitions The abrupt sudden change in physical properties of the thermodynamic system around some critical value of thermodynamic variables (such as temperature). A particular quantity is the specific heat. Ehrenfest classification of Phase Transition: First-order phase transitions exhibit a discontinuity in the first derivative of the chemical potential with a thermodynamic variable. Such as solid/liquid/gas transitions. Second-order phase transitions (also called continuous phase transition) have a discontinuity or divergence in a second derivative of the chemical potential with thermodynamic variables.

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Phase transitions C and χ are second derivative of chemical potential with T and H separately. Onsager (1944) obtained the exact solution for 2D Ising model without external field. The solution shows that there exists second order phase transition in C and χ, because they diverge at some critical value of temperature (Tc ≈ 2.269 in unit of (1/Boltzmann constant)). The studies can explain the ferromagnetic to paramagnetic transition of materials. Monte Carlo simulations also reveal the phase transition properties of Ising model.

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Monte Carlo method and Monte Carlo: A method using pseudorandom number to simulate the random thermal fluctuation from state to state of a system; The probability of a particular state α follows Boltzmann distribution: In theory, sum over all possible states to calculate the statistical mean values of a physical quantity, weighing each state based on its Boltzmann factor; Metropolis algorithm (importance sampling technique): 1.Flip one randomly picked spin; 2.Calculate the total energy difference between new and old spin state δ E=E(new)-E(old); 3. If δ E>0, the probability to accept the new state P(old->new) = exp[- δ E/kT], otherwise P(old->new) = 1.

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Simulation settings Set the spin-spin interaction energy J=1, Boltzmann constant k=1, Bohr magneton The unit of Energy is J; the unit of temperature T is

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Simulation interface

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Low temperature High magnetization High temperature Low magnetization

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Results: C versus T. Specific heat divergence is shown more clearly at Tc≈2.269 in this figure. Second order phase transition occurs.

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Results: Magnetization per spin versus Temperature (Zero external field).

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Results: Magnetization per spin versus External field H at T= 0.2. It shows a hysteresis loop, characteristic of ferromagnetic materials.

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Summary of Results Demonstrate that second order phase transition of specific heat C and magnetic susceptibility χ occur at Tc≈2.269, as predicted by Onsager’s exact solution. Demonstrate the existence of spontaneous magnetization and hysteresis loop below Tc≈2.269 (J>0). These show that the system is ferromagnetic below Tc. Combing these results, the ferromagnetic to paramagnetic phase transition of 2D Ising model is demonstrated.

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Plans Write parallel algorithm (MPI or Pmatlab) Check the performance as a function of: 1.Number of processors. 2.Lattice size. 3.Load equalization.

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