Week 6 - Adding a magnetic field to the model

Welcome back to my blog! In the previous post I provided some visualizations for what the uniform model looks like physically. In this post, I will be adjusting the Hamiltonian to account for a newly added magnetic field.

Peierls Phase

A magnetic field produces a force on charged particles that are moving around. This force is always perpendicular to the direction of motion, and therefore does not change the speed of the particle. In classical mechanics, we can therefore consider this force to rotate the momentum vector by some angle \theta over some interval. Quantum mechanics adds more nuance to this logic, but we can then consider the magnetic force to twist the phase of the wavefunction by some angle \theta. Mathematically, this is described using a factor of e^{i*\theta}. To apply it to our Hamiltonian, we simply take each of our previous hopping parameters t, and multiply it by this factor. We also assume the magnetic field is uniform everywhere.

Now, we need to find \theta. It is given by this formula [1]:

, where q and hbar are known constants. Here, A is the vector potential, and it can be chosen freely such that the curl of A is equal to the magnetic field vector, though I will not be going into the details on why this works. Basically however, this formula gives a simple relation between the magnetic field and the path that the particle travels along.

I found that \theta was a very clean function of the coordinates of a given start node and a given end node. This was very convenient since my Python program saved the coordinates of each node in the graph. I was able to quickly adjust my code and solve for the eigenvalues of the Hamiltonian once again.

Visualization

Like in the last post, I will show the n=4 iteration of the graph. Here is the most localized eigenstate:

and here is the least localized eigenstate:

Once again, I do not have much formal analysis. However, one can see some clear differences between the probability distribution of this new system compared to the uniform model in the previous post. In the upcoming posts, I will most likely be examining some specific characteristics of both systems, and potentially more into localization effects.

References

[1] https://en.wikipedia.org/wiki/Peierls_substitution