Part 3: External Magnetic Field
The Monte Carlo simulation of an Ising model can also be performed in the presence of an external magnetic field, which adds an additional term to the Hamiltonian of the Ising model:
efield = -Hm
From Part 2, a 20x20 crystal is described by ⟨e⟩=-1.75 and ⟨m⟩=0.91 for βJ = 0.5 (below the critical temperature) without an external magnetic field. Consider the strength of a magnetic field corresponding to efield of ± 5% of ⟨e⟩ (which is .0875):H = ± ⟨e⟩/(20⟨m⟩) = ± 0.0962 J/μ
The model was simulated in the presence of this magnetic field and ⟨e⟩ and ⟨m⟩ were recorded:| ⟨H⟩ | ⟨e⟩ | ⟨m⟩ |
|---|---|---|
| 0.0 | -1.88 | +0.91 |
| -0.0962 | -1.75 | -0.91 |
| +0.0962 | -1.88 | +0.93 |
This result shows that the previous assumption of a constant ⟨m⟩ is only valid in a single direction.
When the external field is aligned with the spins, ⟨m⟩ only increases by 0.02. The change in energy is roughly 7%, slightly above the desired 5% change. The actual change is larger because the increase in ⟨m⟩ lowers the energy of the system by increasing the conformity of the spins.
When the external field is aligned against the spins, the sign of ⟨m⟩ completely flips. Rather than slightly shift the all-up system in the down direction (which would make down spins less uncommon but still the minority), the field is strong enough that the system becomes all-down, which is more energetically favourable than all-up in this situation. This explains how energy manages to decrease by 7% instead of increase by 5%
Note that the presence of an external field breaks the symmetry of the spins, biasing the system in favour of either up or down. See how the average magnetization and energy change with H depending on whether the state in the all-up or all-down state:
This graph shows exactly how the strength and direction of the magnetic field affects the minimum in which the system ends up at. With no field, the system will stay all-up or all-down. When the field aligned with the spin only slightly increases magnitude of ⟨m⟩. When the field is not aligned with the initial spin direction and weak, it decreases ⟨m⟩ slightly. A stronger field allows the system to hop to the other minima (aligning it with the field).
This graph shows how the symmetry-breaking magnetic field removes changes the degeneracy of the two minima. As before, an aligned field makes the initial minimum more favourable, while an anti-aligned spin makes the initial minimum less favourable, possibly to a point that the system reaches the other (now very favourable) minimum.