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**Monte Carlo Simulation of Ising Model and Phase Transition Studies**

Yu Sun*, Yilin Wu** *Department of Electric Engineering, University of Notre Dame **Department of Physics, University of Notre Dame Instructor: Prof. Mark Alber, Department of Mathematics, University of Notre Dame

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**Outline Describe the Ising model for magnetism;**

Introduce the Monte Carlo simulation method as well as the Metropolis algorithm; Present our Monte Carlo simulation results for Ising model and discuss its properties, especially the phase transition behavior.

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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 Metropolis Algorithm**

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 αfollows 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 δE=E(new)-E(old); 3. If δE>0, the probability to accept the new state P(old->new) = exp[-δE/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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**Results: Energy per spin versus Temperature (Zero external field)**

Results: Energy per spin versus Temperature (Zero external field). The derivative C=dE/dT diverges at around Tc≈2.269.

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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 (Zero external field), T=1. 5, 2**

Results: Magnetization per spin (Zero external field), T=1.5, 2.0. The figures show spontaneous magnetization (most of the spins align in the same direction).

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**Results: Magnetization per spin (Zero external field), T=2. 25, 4**

Results: Magnetization per spin (Zero external field), T=2.25, 4.0. Fluctuations become more significant near Tc≈ For T far above Tc, M oscillates around 0.

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

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**Results: Magnetic susceptibility χ versus T. χ diverges at around Tc≈2**

Results: Magnetic susceptibility χ versus T. χ diverges at around Tc≈ It is second order phase transition. Above Tc, it is paramagnetic.

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**Results: Magnetization per spin versus External field H at T= 0. 2**

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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