Ising model using Python : Magnetization vs Time (FIG 8.7)

The exact value of the magnetization of a material at a given point of time is uncertain due to the random nature of the spins. This can be demonstrated by plotting the Magnetization as a function of time for a varying range of temperatures. In this Ising Model simulation a 10 x 10 lattice configuration is used with 1000 Monte Carlo sweeps for each temperature point. The time is not real time but solely a product of the Monte Carlo method, this means that one unit of time is one sweep through the entire lattice or one Monte Carlo sweep. This gives each spin the opportunity to flip during each time step. [1]

The beginning temperature is set to 1.5 and the result of which can be seen in Figure 1. The magnetization of the lattice is in the ferromagnetic state which is to be expected at this low temperature. There are a couple of instances in time where the magnetization drops but these are fairly rare occurrences and just a small fluctuation due to the random nature in the Monte Carlo calculation. For the most part however all the spins are aligned and the lattice is in a ferromagnetic state.

Figure 1. Magnetization vs Time for T = 1.5

When the temperature is slightly increased to 2 as can be seen in figure 2. The fluctuations have increased significantly but the material is still in a ferromagnetic state. The average value of the magnetism has dropped to around to a magnetization value of roughly 0.9. The importance of the increase in fluctuations as they signal that the model is approaching a second order phase transition known as the critical point [1]. This point is sensitive to minor changes in the magnetic field or temperature. This will be shown even clearer when the temperature is raised closer to the Curie temperature.

Figure 2. Magnetization vs Time for T = 2

When this temperature is raised further to 2.25 the fluctuations are much larger and the system fluctuates around ± 0.8. The Curie temperature is 2.27 so the value the simulation is at is very close to that resulting in changes in phase from ferromagnetic to paramagnetic at certain instances of time. This is shown below in figure 3.

Figure 3. Magnetization vs Time for T = 2.25

In figure 4 the temperature is raised to 4 and the fluctuations have decreased and are situated approximately at M = ±0.2. The temperature is now above 2.27 and the system is now in the paragmagnetic state.

Figure 4. Magnetization vs Time for T = 4

The results achieved validated up by the the results found in Giordanos book In figure 5 & 6. The fluctuations due to the random nature of the Ising Model show it is difficult to predict the exact magnetization at a given temperature. However it is possible to get a good idea of what level of magnetization at a given temperature.