This code implements a Monte-Carlo Markov Chain (MCMC) approach to simulating magnetic moments in 2D lattices. The N=3 Heisenberg model is implemented with anisotropy factors. This includes the magnetic exchange anisotropy (green) and the axis anisotropy (blue).
With an isotropic model (where all gamma factors are equal to zero), no long range order is seen at any finite temperature. Long-range order is explicitly forbidden by the Mermin-Wagner Theorem. This is due to finite thermal fluctuations allowing minute deviations from perfect alignment. Since 3D vectors are used, these deviations can infinitesmally off-axis, having only a tiny energy cost.
However, increasing the any of the gamma factors (adding anisotropy) allows for long range order to occur.
A VASP calculation was run on Mn Octoamine-Phythalocyanine to find the factor gamma for the axis anisotropy.
The lattice symmetry can be arbitrarily defined, where the strength and interaction type (FM/AFM) can be individually defined for each direction. This allows for defining more complex symmetries like a hexagonal lattice.
The LatticeSim_OpenMPI.c requires an OpenMPI installation to operate. OpenMPI is implemented over the temperature range, allowing sections to be calculated independently. This works as the system is a MCMC model and only has large correlation to the lattice step immediately before the current one. There is a tail off in the correlation as the steps between successive sampled lattices increases.
It can be compiled using:
mpicc foo.c -o foo -lm
The math.h library needs to be linked for successful compilation.
- -N : REQUIRED: The dimension of the square lattice NxN. Default is 20.
- -U : REQUIRED: The number of simulation steps taken at each temperature point.
- -W : Set the number of warmup steps for the lattice. Default is 500.
- -a : Set the Axis Anisotropy value. Default is 0.0
- -e : Set the Exchange Anisotropy value. Default is 0.0
- -i : Set the index for the output file. Needed for creating multiple files. Default is 0.
- -T : Centres the temperature range that is simulated over. It is wise to use this as an estimate for the critical temperature. Default is 1.2.
- -I : Set the interval between each temperature point.
- -r : Set the temperature range of the simulation.
Input file for simulation. This should list the nearest neighbours in terms of primitive vector coefficients around the centre point.
The format of the file is given as:
- Simulation name
- Number of basis sets, Number of nearest neighbours
- Basis set number
- X, Y, Interaction Type
- Repeat 4 for all nearest neighbours
- Repeat 3-5 for all basis sets
An exampled for a hexagonal lattice is given below:
Hexagonal Lattice
2,3
1
0,0,1
0,1,1
-1,1,1
2
0,0,1
0,1,1
-1,1,1
The Analysis.py requires Pandas, Numpy & Scipy.stats.moment() to parse the .csv output files. It reads in the raw .csv files, finds the average magnetic moment and standard deviation at each temperature point. The magnetic susceptibility and U2 Binder cumulant for the lattice size is created. The output file is of the following format.
| Temperature | {}x{} Average Magnetic Moment | {}x{} Average Susc | {}x{} Average U2 | {}x{} Magnetic Moment std | Repeated for larger cell sizes |
|---|
{} denotes the lattice size. The lattice sizes to work over are defined using the command line arguments:
- -f : Path to directory containing the input files.
- -n : Number of .csv files to average over for each setting value.
- -o : The output file name.
- -i : The lattice sizes to loop over. Must be passed as a list enclosed in "". e.g: "[20, 40, 60]".
- -a : Number denoting the file number. Can be used to express the used anisotropy value.
Input file format: python '{}_{}_spinDist_{}_{}.csv'.format(lattice_size, lattice_size, a, n)
Output file format: "-o"_"-a".csv
This Jupyter Notebook reads in the output .csv file, unifies the different lattices sizes onto three different graphs.
It automatically formats the graphs to be outputted.
The submit_Combined.sh is an example file for submitting the compiled job to the UCD Sonic cluster. It is also an example of how to write a bash script to create a large number of ensembles and then create the output/averaged script.
The separate submit_PyAnalysis.sh and submit_MPI.sh are separate scripts to run the Python Analysis.py script and the LatticeSim_OpenMPI script respectively.
The headers of these scripts are particular to the UCD Sonic Cluster and its implementation of SLURM.