The Ising Model
The Ising Model
PYU33C01 – Computer Simulation
Dr. Steven Power Abstract
The objective of this project was to implement the Metropolis Algorithm and investigate the Ising Model for a 2-Dimensional lattice grid system. Upon the successful implementation of the algorithm, the lattice grids were produced for varying stages and the model was tested above the Curie temperature. The system was shown to be ferromagnetic and it was also shown that above the Curie temperature, the system does not reach equilibrium and becomes paramagnetic. Introduction
The Ising model is a model used in Statistical Mechanics which represents ferromagnetism in a system. Ferromagnetism arises when a collection of atomic spins align in such a way that their associated magnetic moments all point in the same direction. This leads to a net magnetic moment in the macroscopic scale. The Ising spin Hamiltonian is given by:
, where σ is a random variable assuming the values ±1 on sites i=1,2,…….,N of a n-dimensional cubic lattice, represents the nearest neighbour site and h represents the applied magnetic field. Since this project models a simplified 2D Ising model, h is set to zero and so the second term of the Hamiltonian is neglected. This equation represents the interaction energies introduced to bring about an ordered ferromagnetic state (i.e. if J>0, which is assumed in this report).
Ferromagnetism occurs, in some magnetic materials, below the Curie temperature TC. The Curie temperature is the temperature at which certain magnetic materials undergo a sharp change in their magnetic properties, named after Pierre Curie, who in 1895 discovered the laws that relate magnetic properties and temperature. Once above the Curie temperature, the system becomes paramagnetic as the thermal motion of the molecules in enough to disrupt the alignment.
The probability of the system being in any given state can be represented as:
, where β = and Z = , which is the Partition Function
The probability that an electron will have enough energy to flip spins is given by:
The average energy spin is:
, where H is the Hamiltonian of the system as defined earlier in this section and the factor of a half is included to avoid each pair being counted twice in the calculation.
The average magnetisation per unit spin is:
If we calculate M, we can determine the time it takes for the system to reach equilibrium. The magnetisation should be either -1 or +1 for a ferromagnetic material but after the Curie temperature, the magnetisation will be 0.
The specific heat capacity is given by:
The magnetic susceptibility can be represented as:
A simplified model of the 2D Ising model was investigated in this project and was implemented using the metropolis algorithm which is a Modified Monte Carlo Method Method
The Metropolis algorithm was used throughout this project. Firstly, an nxn lattice grid was generated with each site randomly assigned a value of either +1 or -1 to represent the spin in that site, with the initial parameters set to J = 1.0, h = 0.0, size of the grid = 50×50, steps = 1000, temperature = 1K and the Boltzmann constant was defined also from scipy.constants. A lattice grid of this initial state was produced and printed in a figure.
The energy of a given site in the lattice was defined and ΔE for that site was also defined and then calculated. If ΔE was found to be less than or equal to zero, the spin was flipped. If ΔE was found to be greater than zero, the spin is only flipped if the probability of a flip is greater than a random point on the interval [0,1]. This was repeated for every point in the lattice grid and a lattice grid was also produced
Finally, the system is allowed to reach equilibrium by ensuring that enough iterations have occurred and a final lattice grid is produced.
The parameters M, 𝞆 and Cv were also defined.
Results and Analysis
The algorithm was successfully used to iterate through a grid of random spins which were updated until an equilibrium was reached.
Figure 1: Randomly generated lattice grid (size 50×50)
Figure 2: Lattice grid with the Metropolis Algorithm applied before equilibrium has been reached
Figure 3: Lattice Grid at equilibrium.
As can be seen from the figures above, the system is ferromagnetic at a temperature of 1K. If the TC is exceeded, we obtain a lattice grid as below:
Figure 4: Lattice Grid at same conditions as figure 3 but with T = 10000K
As can be seen, when all the iterations and conditions are kept the same but the temperature is raised to 10000K, the system does not reach equilibrium and remains disorganised which implies it is paramagnetic as the Curie Temperature has been exceeded. Conclusion
The 2D lattice grid studied was shown to be ferromagnetic through the use of the metropolis algorithm . The size of the grid was kept small as if a larger grid was used, the computation time required to solve it would be far too large as each site would have to be iterated through and the same logic applies to the number of iteration steps used. However, in this simulation, enough iterations were used in order to ensure that an equilibrium was reached. References
 The Ising Model – Dr. Richard Fitzpatrick, Associate Professor of Physics, University of Texas, http://farside.ph.utexas.edu/teaching/329/lectures/node110.html
 Professor Florent Krzakala, University Pierre et Marie Curie, Laboratoire de Physique Statistique, http://www.lps.ens.fr/~krzakala/ISINGMODEL.pdf
 HyperPhysics, Georgia State University, http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/ferro.html
While no information was directly referenced from the following link, it was a very useful resource and there may be some information used in this report: http://web.mit.edu/ceder/publications/Ising%20Model.pdf
NB: Information and code procedure from the Lecture Notes and extra material provided by Dr. Steven Power throughout the PYUU33C01 module was used multiple times in both this report and the corresponding Jupyter notebook.