Ising model

You have notes from Prof Hutzler

● Chapter 6: Random numbers and Monte Carlo Integration

● Chapter 7: Statistical Methods

● Chapter 8: Monte-Carlo Simulations

http://www.tcd.ie/Physics/Foams/Lecture_Notes/PY3C01_computer_simulation_1_numerical_and_statistical_methods/py3c01_2016.pdf

Ising model

2D collinear lattice of spins

Assumption● 2D square lattice● Nearest neighbour interactions● h=0 (for now)● collinear

Probability the system is in state v

Ising model

2D collinear lattice of spins

Nearest neighbour sites

{i,j}

{i,j+1}

{i,j-1}

{i+1,j}

{i-1,j}

Contribution of site {i,j} to the total energy =

H i , j=−J S i , j .(Si+1, j+S i−1, j+Si , j+1+S i , j−1)+hSi

Ising model

2D collinear lattice of spins

Nearest neighbour sites

J J

J

J

Ising model

2D collinear lattice of spinsH i , j=−J Si , j .(S i+1, j+S i−1, j+S i , j+1+S i , j−1)+hSi

H i , j=−J 1.(−1−1−1+1)+0

H i , j=+2J

{1,1} {2+1}

{1,2}

{0,1}

{1,0}

E=∑i , jH i , j

= 1

= -1

Ising model

2D collinear lattice of spinsΔ E=H i , j(flip)−H i , j

{1,1} {2+1}

{1,2}

{0,1}

{1,0}

Hi,j

{2+1}

{1,2}

{0,1}

{1,0}

Hi,j(flip)

{1,1}

Δ E=−J (−1).(−1−1−1+1)−+2 J

Δ E=−4 J

Flipping the spin {1,1} lowers the total energy by 4J

Simulating the Ising model

Do we flip the spin

if ΔE < 0 i.e. flipping the spin lowers the energy of the system (accept the change)

Do we flip the spin

if ΔE > 0 i.e. flipping the spin increases the energy of the system:

accept based on the Boltzmann distribution

Boltzmann Distribution

Probability that the electron will have enough energy to flip = Pflip=e−ΔEK B T

Accept if Pflip>randomnumber in range 0-1 , what distribution should be chosen for the random number?

Updating the system

●Now we know how to update one spin you can evolve the system by scanning through the lattice and updating the spins.

Boundary conditions

We cannot have a infinite system so what happens at the edge?

Fixed boundary

Periodic boundary conditions

Think about numerical accuracy

● Has your system reached equilibrium before you collect statistics?

● How do the boundary conditions affect the results?● What about the size of the system?● As the system is evolving in time is one snapshot

enough?● What does time mean anyway for this model?● Does it matter how we initialize the matrix?● Can I reduce the computational resources required?

Physical interpretation

● Can you find a phase change with temperature?● What is the order of the phase change?● What is the critical exponent? ● What if J is negative?● What is the distribution of domain sizes?● What if the system was 1D or 3D?● Can this model be applied to systems other than

magnets?

Phase changes with the Ising model

MarkingPassAttempted implementation of the Ising model demonstrating basic bash and python scripting. The code and report must be submitted through git.

2:2In addition the the pass requirement, a scientific investigation must have been completed looking into a simple aspect e.g. convergence.

2:1 In addition the the 2:2 requirement, a working implementation of the Ising model demonstrating good coding practices: version control history presented on gitlab; use of modules; objects and functions. Sections of the code should be wrapped up in modules or libraries where appropriate. The code should be well commented, documented and clean. A good quality report should be induced looking into a physical problem e.g. phase changes in magnetic systems; affect of different lattice dimensions and connectivity; domain formation.

1st In addition the the 2:1 requirement, the scientific investigation should be a high quality and novel investigation. e.g. annealing a solution to the travelling salesman problem; neural networks; lattice gas; machine learning to find the ground-state in frustrated lattices.

Exercise

Create a flow chart for the 2D Ising model.

There are several flowchart drawing tools, draw.io provides a web based tool to draw flowcharts.