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

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Monte Carlo Simulation of Ising Model and Phase Transition Studies By Gelman Evgenii. Introduction to Magnetism. Magnetic susceptibility χ : Types of magnetic materials: 1. D iamagneti c : χ &lt;0 and constant (H elium );

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### Monte Carlo Simulation of Ising Model and Phase Transition StudiesBy Gelman Evgenii

Introduction to Magnetism
• Magnetic susceptibilityχ:
• 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

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.
Ising Model(2D)
• Thermalproperties 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 susceptibilityχ
Phase transitions
• The abrupt sudden change in physical properties of the thermodynamic system around some critical value of thermodynamic variables(such astemperature). 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.
Phase transitions
• C and χaresecond 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 andχ, 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.
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 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],otherwiseP(old->new) = 1.

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

Low temperature

High magnetization

High temperature

Low magnetization

Results: C versus T. Specific heat divergence is shown more clearly at Tc≈2.269 in this figure. Second order phase transition occurs.
Results: Magnetization per spin versus External field H at T= 0.2. It shows a hysteresis loop, characteristic of ferromagnetic materials.
Summary of Results
• Demonstrate that second order phase transition of specific heat C and magnetic susceptibilityχoccur 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.
Plans
• Write parallel algorithm (MPI or Pmatlab)
• Check the performance as a function of:
• Number of processors.
• Lattice size.