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
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