Week 5 - Solving the tight-binding model

Welcome back to my blog! In the previous post I introduced an algorithm to create a Hamiltonian matrix in a clean and flexible way. In this post, I will be solving the Hamiltonian for the uniform model and visualizing the results.

np.linalg.eigh

In quantum mechanics, observables are generally assumed to be Hermitian, meaning that the matrix is the transpose of its complex conjugate. This is very clearly true in our case, and therefore we can use numpy’s dedicated linalg.eigh function to compute the eigenvalues and eigenvectors of our Hamiltonian. This function is faster than linalg.eig since it takes advantage of the symmetry of the matrix.

Our Hamiltonian is an NxN matrix, where N = (3+3^(n+1))/2 and n is the iteration of the Sierpinski triangle, starting from n=0. I found that setting n=8 already takes far too long, so I did not analyze any iterations above 7.

Localization visualization

I have not actually given any sort of visualization in my blog yet, so here is where I will explain localization a little bet better.

From solving the Hamiltonian matrix (here I show the n=4 iteration), I got a set of eigenvalues, each of which corresponds to an eigenstate. Now, each eigenstate corresponds to a probability distribution for an electron within the entire graph. Such an eigenstate might look like this (brighter = higher probability of being at that node):

As it turns out, this eigenstate right here is the most localized, and it means what you might think it means (most focused):

And this is the least localized eigenstate (most spread out):

How did I figure out which states were most and least localized? I calculated the participation ratio for each eigenvalue (equation 3 in the referenced paper). A higher participation ratio indicates lower localization.

Finally, how do we determine the current state of the system overall? The general state is what is called a superposition of all the eigenstates. We simply add together a linear combination of all the eigenstates, and they will “overlap” with each other, creating the final probability distribution. How do we know what combination of eigenstates to use? Since I am not in a lab and we are not literally looking at anything, we can just choose arbitrary constant coefficients.

I will provide actual deeper analysis of my results in a later post. For the next two posts though, I will explain how we apply a magnetic field to the system and visualize the localization effects for that as well.

References

[1] Cardoso, A.L., Andrade, R.F.S., Souza, A.M.C. (2008). Localization properties of a tight-binding electronic model on the Apollonian network.