ising model - trinity college, dublin...ising model 2d collinear lattice of spins assumption 2d...
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![Page 1: Ising model - Trinity College, Dublin...Ising model 2D collinear lattice of spins Assumption 2D square lattice Nearest neighbour interactions h=0 (for now) collinear Probability the](https://reader033.vdocuments.net/reader033/viewer/2022041816/5e5ab8ebd0f0bc3b3956d703/html5/thumbnails/1.jpg)
Ising model
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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
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Ising model
2D collinear lattice of spins
Assumption● 2D square lattice● Nearest neighbour interactions● h=0 (for now)● collinear
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Probability the system is in state v
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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
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Ising model
2D collinear lattice of spins
Nearest neighbour sites
J J
J
J
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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
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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
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Simulating the Ising model
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Do we flip the spin
if ΔE < 0 i.e. flipping the spin lowers the energy of the system (accept the change)
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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
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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?
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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.
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Boundary conditions
We cannot have a infinite system so what happens at the edge?
Fixed boundary
Periodic boundary conditions
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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?
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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?
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Phase changes with the Ising model
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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.
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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.