Hi Jannis,

thank you for the fast response. You are probably right, that I need to use the k-space Hamiltonian for the infinite case. Would my method work for a finite system then? Potentially with periodic boundary conditions?

For a finite system (with or without PBC) you should directly diagonalize Hamiltonian and count the number of states below the Fermi energy.

The measure, that I am looking for, is the total energy of all occupied particle states. I am not really sure, how to calculate this in an infinite system. First I probably need to isolate the particle Hamiltonian with a projector. But I don't know how to proceed from there in k-space.

If I understand your question correctly then you need to do do a k-space integration; something like:

     ∑_n ∫ E_n(k) f(E_n(k)) dk

Where the sum runs over all the bands, the integral runs over the Brillouin zone and 'f' is the occupation at energy E (e.g. Fermi-Dirac). In this way you count the energy contribution for each state, weighted by its occupation. Note that we have elided the density of states, as we are integrating directly in momentum space, as opposed to in energy, and the density of states in 1D is constant in k.

Does that make sense?

Aside from this I have the suspicion that you might not necessarily get the result you are looking for. You say that you are dealing with an s-wave superconductor, but want to calculate the total energy of the electrons only. However in an s-wave superconductor the eigenstates are not electrons and holes, but a *superposition* of electron and hole (the so-called Bogoliubov quasiparticles). I would therefore recommend taking care to look at what it is that you want to calculate, and asking whether it is meaningful.

Happy Kwanting,

Joe

${\displaystyle \displaystyle \int }$