1. 2D Ising model
Monte-Carlo (Metropolis) method

2. 2D Ising model
Monte-Carlo (Metropolis) method

3. Some Ferromagnetic Elements Some Paramagnetic Elements
•             Iron •             Uranium
•            Nickel •             Platinum
•      Cobalt •  Aluminum
•              Gadolinium •             Sodium
•              Dysprosium •             Oxygen

5. Magnetic dipole–dipole interaction

6. Energy
Magnetization

15. T<<Tc T ~Tc T>> Tc
Almost at the Close to the phase High temperature
ground state transition (no correlation
between spins)
Single large cluster Broad distribution All clusters of like
of ordered spins of clusters spins are small
with small clusters
of opposite spins

17. Monte-Carlo method

28. Onsanger exact result H=0, kB=1, J=1
for critical temperature
T~Tc=2.269
T<<Tc
T>>Tc

29. The mean average over all configurations=
Yang exact result for H=0
forr magnetization
Ergodic principle
After thermalization
The mean average over all configurations=
The mean time average over 1 configuration

30. The thermalization depends on
the size of the lattice and
the number of steps

32.
A new look on the two-dimensional Ising model: thermal artificial spins
amorphous Co68Fe24Zr8 
NxN lattice , N=10487