Simulating a two-dimensional Ising model using the Metropolis-Hastings algorithm.

Abstract

The two-dimensional Ising model is used to study the phase transition of ferromagnets. To simulate the model, we use a parallelized version of the Metropolis-Hastings algorithm. First, we calculate the analytical solution of a model with a square $2\times2$ lattice and use the analytical results to verify our implementation of the model. We found that for a $20\times20$ lattice, the model needs approximately 105 Monte Carlo cycles to reach equilibrium. We estimate the probability distribution of the expected energy per spin and found that the variance for the temperature $T = 1$ $J/kB$ was of order 103 smaller than for the temperature $T = 2.4$ $J/kB$. When using OpenMP to parallelize our code we made it run approximately 4 times faster using 8 threads. Lastly, we simulate the model for four different lattice sizes to study the phase transition. We use these simulated values to find an estimate of the critical temperature for an infinite Ising model. We find this to be $\approx 2.269282$ $J/kB$, which is close to the analytical result of $1 / \ln(1+\sqrt{2}) \approx 2.269185$ $J/kB$.

Authors

• Sophus B Gullbekk (sophusbg@math.uio.no)
• Erlend Kristensen (erlek@math.uio.no)
• Tov Uberg Tyvold (tovut@math.uio.no)
• Jonathan Larsen (jonathel@math.uio.no)

Compile and run

For Linux users, in the file src write $make OMP For Mac users, in the file src write $make omp

To recreate the results we found in our report the laticeSizeComp function in main.cpp has to be run with 8 threads.

File structure

All the code for the project is located in the src folder.

The code for implementing the Ising model is located in the IsingModel.cpp and IsingModel.hpp files.

Other helpful functions for the Ising model is located in the utilities.cpp and utilities.hpp files.

The random number generator is located in the omp_rng.cpp and omp_rng.hpp files.

The code for generating results is located in the main.cpp file.

The code for plotting the figures used in the report is located in plot.py.

The data from our simulations is stored in the textfiles folder.