Simulation of the 2D Ising Model
Solutions
Problem 1(2 points)
a.) Everyone’
s simulations will be different. Qualitatively, the size of the clusters should decrease as
temperature increases. At higher temperatures, the thermal energy dominates and breaks apart the
clusters. Here is an example of the result.
T=10
T=5
T=1
T=0.1
T=2.5
b.) The same basic behavior should be seen when n = 50, however the differentiation between cluster
sizes should be much more noticable with more sites.
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ising_model_student_solutions.nb
T=10
T=5
T=1
T=0.1
T=2.5
The low temperature structure doesn’
t look as organized at those in part a simply because the number
of iterations is not high enough for it to emerge.
c.) At low temperatures, you should observe symmetry breaking if you give the simulation a high
enough number of steps. With these temperatures, you are below the critical point and the lattice
should exhibit ferromagnetic behavior.
nSteps=1000 ,T=2
nSteps=5000 ,T=2
nSteps=50000 ,T=2
nSteps=100000 ,T=2
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nSteps=10000 ,T=2
ising_model_student_solutions.nb
nSteps=1000 ,T=1
nSteps=5000 ,T=1
nSteps=50000 ,T=1
nSteps=100000 ,T=1
nSteps=1000 ,T=0.5
nSteps=5000 ,T=0.5
nSteps=50000 ,T=0.5
nSteps=100000 ,T=0.5
3
nSteps=10000 ,T=1
nSteps=10000 ,T=0.5
As you can see, when the number of iterations is increased, the symmetry breaking becomes apparent
and occurs more quickly at lower temperatures.
d.) The critical temperature occurs very near T = 2.5. With a small number of sites, you should have
noticed the lattice oscillate between almost all spin-up and all spin-down. With such a small number of
sites, the correlation length reaches across the whole lattice.
e.) As mentioned above, the critical temperature occurs very near T = 2.3. The fractal-like nature of the
system at the crtical temperature in which the average cluster size ranges from a couple of sites to the
whole lattice is more apparent as the number of sites increases.
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ising_model_student_solutions.nb
N=40 , nSteps=100000 , T=2.3
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ising_model_student_solutions.nb
N=100 , nSteps=100000 , T=2.3
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ising_model_student_solutions.nb
N=160 , nSteps=100000 , T=2.3
Out[4]=
Problem 2 (2 points)
To calculate the energy of the lattice, you had to exploit the interaction Hamiltonian
H = -J â Si Sj .
i,j
Since we are dealing with the Ising model, this Hamiltonian only applies to the nearest neighbors. So,
the energy of any given site would be the sum of the product of spins of the site with each of it’
s4
closest neighbors.
In order to get a good result, you should have run the program several times at each temperature and
averaged the values together to find the lattice energy at that temperature.
The result, plotted, should look something like this:
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ising_model_student_solutions.nb
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There is a clear transition at Tc , where the system becomes much less energetic, reflecting the transition from paramagnet to ferromagnet.
Problem 3 (2 points)
To calculate the the average magnetization, one only needs to add up the spins of each point on the
lattice. We are really only concerned with the magnitude of M at this step.
As with energy, there is a clear transition in M at the critical temperature Tc , reflecting the spontaneous
symmetry breaking that occurs when the ferromagnetic domains begin to form. So, M should approach
1 and low temperature and 0 at high temperatures. Plotted, you would see something like this:
Problem 4 (4 points)
Not surprisingly, we excpect the value of the correlation function to increase as the sites in question get
closer and closer. At low values of r, spins are strongly correlated. As r increases, they talk to each
other less and less. When plotted (this graph is near the critical temperature, probably the best choice
of temperature), you shoul see something like this.
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ising_model_student_solutions.nb
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