1. 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

2. 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.

3. 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

4. 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.

5. 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

6. 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.

7. 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.

8. 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.

9. 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

10. Simulation interface

11. 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.

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

13. 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).

14. 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.

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

16. 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.

17. 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.

18. 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.