Superconductivity and BdG, Part 1

There are moments in research where you suddenly have an euphoric rush once you truly understand the beauty of something. First such moment happened to me when I got the beauty of spinor group. Couple of days ago, I had the same cathartic moment when I finally got the brilliance of BdG Hamiltonians. This post is just to clarify to my future self if I get stuck understanding this again.

Let's try to understand a rudimentary superconductor that is made up of 2 sites with no hopping between them. The Hamiltonian can be written as

First part with and terms are the cost for the electron to stay in each site while is the superconducting pairing term that pairs the electron creating or destroying them in both sites. This term forces us to forgo our faithful single particle basis, as one can clearly see that involves two particle terms. So, as a many body theorist would do, let's solve it in the two particle basis.

The basis order chosen is . Because of the block diagonal nature of H, it is clear to see that two of the eigen values are .

Parity Symmetry

A block diagonal form means we have a symmetry in our system, and that symmetry is Parity. This symmetry lets our eigenvectors be labeled by a number specifying if it belongs to the single particle block or the two particle block. The parity operator is:

This operator when acted on our basis spits either 1 or -1 based on if the eigenvalue is in single particle or multi particle subspace.

The Phase Transition

Plotting the eigenvalues projected on the parity subspace as a function of for , we find that the ground state of this system after around switches in parity and thus starts having one unpaired electron that has a finite energy gap to be paired. This gives the amazing physical result of superconductivity.

We will look at how this entire discussion can be mapped in BdG formalism in the next post.